支持向量机通俗导论（理解SVM的三层境界）

支持向量机通俗导论（理解SVM的三层境界）

作者：July、pluskid ；致谢：白石、JerryLead
出处：结构之法算法之道blog。

前言
第一层、了解SVM
1.0、什么是支持向量机SVM
1.1.、线性分类
1.1.1、分类标准
1.1.2、1或-1分类标准的起源：logistic回归
1.1.3、形式化标示类
1.2、线性分类的一个例子
1.3、函数间隔Functional margin与几何间隔Geometrical margin
1.3.1、函数间隔Functional margin
1.3.2、点到超平面的距离定义：几何间隔Geometrical margin
1.4、最大间隔分类器Maximum Margin Classifier的定义
1.5、到底什么是Support Vector
第二层、深入SVM
2.1、从线性可分到线性不可分
2.1.1、从原始问题到对偶问题的求解
2.1.2、线性不可分的情况
2.2、核函数Kernel
2.2.1、如何处理非线性数据
2.2.2、特征空间的隐式映射：核函数
2.2.3、几个核函数
2.2.4、核函数的本质
2.3、使用松弛变量处理 outliers 方法
2.4、小结
第三层、证明SVM
3.1、线性学习器
3.1.1、感知机算法
3.1.2、松弛变量
3.2、最小二乘法
3.3、核函数特征空间
3.4、SMO算法
3.5、SVM的应用
3.5.1、文本分类
参考文献及推荐阅读

前言

动笔写这个支持向量机(support vector machine)是费了不少劲和困难的，从5月22日凌晨两点在微博上说我要写了，到此刻真正动笔要写此文，中间竟然隔了近半个月(而后你会发现，我写完此文得花一个半月，修改完善又得再花一个月，故前后加起来至8月底，写这个SVM便要花足足近3个月)。原因很简单，一者这个东西本身就并不好懂，要深入学习和研究下去需花费不少时间和精力，二者这个东西也不好讲清楚，尽管网上已经有朋友已经写得不错了(见文末参考链接)，但在描述数学公式的时候还是显得不够。得益于同学白石的数学证明，我还是想尝试写一下，希望本文在兼顾通俗易懂的基础上，真真正正能足以成为一篇完整概括和介绍支持向量机的导论性的文章。

本文作为Top 10 Algorithms in Data Mining系列第二篇文章，将主要结合支持向量机导论、数据挖掘导论及网友Free Mind的支持向量机系列而写(于此，还是一篇学习笔记，只是加入了自己的理解，有任何不妥之处，还望海涵)，宏观上整体认识支持向量机的概念和用处，微观上深究部分定理的来龙去脉，证明及原理细节，力求深入浅出 & 通俗易懂。

在本文中，你将看到，理解SVM分三层境界，

第一层、了解SVM(你只需要对SVM有个大致的了解，知道它是个什么东西便已足够)；
第二层、深入SVM(你将跟我一起深入SVM的内部原理，通宵其各处脉络，以为将来运用它时游刃有余)；
第三层、证明SVM(当你了解了所有的原理之后，你会有大笔一挥，尝试证明它的冲动)；

以此逐层深入，从而照顾到水平深浅度不同的读者，在保证浅显直白的基础上尽可能深入，还读者一个较为透彻清晰的SVM。

同时，阅读本文之前，请读者注意以下两点：

若读者用~~IE6~~浏览器阅读本文，将有大部分公式无法正常显示(显示一半或者完全无法显示)，故若想正常的阅读本文请尽量使用chrome等浏览器，谢谢大家。
本文中出现了诸多公式，若想真正理解本文之内容，我希望读者，能拿张纸和笔出来，把本文所有定理.公式都亲自推导一遍或者直接打印下来，在文稿上演算（读本blog的最好办法便是直接把某一篇文章打印下来，随时随地思考.演算.讨论）。

Ok，还是那句原话，有任何问题，欢迎任何人随时不吝指正 & 赐教，感谢。

第一层、了解SVM

1.0、什么是支持向量机SVM

要明白什么是SVM，便得从分类说起。

分类作为数据挖掘领域中一项非常重要的任务，目前在商业上应用最多(比如分析型CRM里面的客户分类模型，客户流失模型，客户盈利等等，其本质上都属于分类问题)。而分类的目的则是学会一个分类函数或分类模型(或者叫做分类器)，该模型能吧数据库中的数据项映射到给定类别中的某一个，从而可以用于预测未知类别。

其实，若叫分类，可能会有人产生误解，以为凡是分类就是把一些东西或样例按照类别给区分开来，实际上，分类方法是一个机器学习的方法，分类也成为模式识别，或者在概率统计中称为判别分析问题。

你甚至可以想当然的认为，分类就是恰如一个商场进了一批新的货物，你现在要根据这些货物的特征分门别类的摆放在相关的架子上，这一过程便可以理解为分类，只是它由训练有素的计算机程序来完成。
来举个例子，比如心脏病的确诊中，如果我要完全确诊某人得了心脏病，那么我必须要进行一些高级的手段，或者借助一些昂贵的机器，那么若我们没有那些高科技医疗机器，怎么判断某人是否得了心脏病呢？

当然了，古代中医是通过望、闻、问、切“四诊”，但除了这些，我们在现代医学里还是可以利用一些比较容易获得的临床指标进行推断某人是否得了心脏病。如作为一个医生，他可以根据他以往诊断的病例对很多个病人(假设是500个)进行彻底的临床检测之后，已经能够完全确定了哪些病人具有心脏病，哪些没有。因为，在这个诊断的过程中，医生理所当然的记录了他们的年龄，胆固醇等10多项病人的相关指标。那么，以后，医生可以根据这些临床资料，对后来新来的病人通过检测那10多项年龄、胆固醇等指标，以此就能推断或者判定病人是否有心脏病（虽说这个做法不能达到100%的标准，但也能达到80、90%的正确率），而这一根据以往临场病例指标分析来推断新来的病例的技术，即成为分类classification技术。

OK，既然讲到了病例诊断这个例子，接下来咱们就以这个例子来简单分析下SVM。
假定是否患有心脏病与病人的年龄和胆固醇水平密切相关，下表对应10个病人的临床数据(年龄用[x1]表示，胆固醇水平用[x2]表示)：

这样，问题就变成了一个在二维空间上的分类问题，可以在平面直角坐标系中描述如下：根据病人的两项指标和有无心脏病，把每个病人用一个样本点来表示，有心脏病者用“+”形点表示，无心脏病者用圆形点，如下图所示：

如此我们很明显的看到，是可以在平面上用一条直线把圆点和“+”分开来的。当然，事实上，还有很多线性不可分的情况，下文将会具体描述。

So，本文将要介绍的支持向量机SVM算法便是一种分类方法。

所谓支持向量机，顾名思义，分为两个部分了解，一什么是支持向量(简单来说，就是支持 or 支撑平面上把两类类别划分开来的超平面的向量点，下文将具体解释)，二这里的“机”是什么意思。我先来回答第二点：这里的“机（machine,机器）”便是一个算法。在机器学习领域，常把一些算法看做是一个机器，如分类机(当然，也叫做分类器)，而支持向量机本身便是一种监督式学习的方法(至于具体什么是监督学习与非监督学习，请参见此系列Machine L&Data Mining第一篇)，它广泛的应用于统计分类以及回归分析中。

支持向量机(SVM)是90年代中期发展起来的基于统计学习理论的一种机器学习方法，通过寻求结构化风险最小来提高学习机泛化能力，实现经验风险和置信范围的最小化，从而达到在统计样本量较少的情况下，亦能获得良好统计规律的目的。

对于不想深究SVM原理的同学(比如就只想看看SVM是干嘛的)，那么，了解到这里便足够了，不需上层。而对于那些喜欢深入研究一个东西的同学，甚至究其本质的，咱们则还有很长的一段路要走，万里长征，咱们开始迈第一步吧(相信你能走完)。

1.1、线性分类

OK，在讲SVM之前，咱们必须先弄清楚一个概念：线性分类器(也可以叫做感知机，这里的机表示的还是一种算法，本文第三部分、证明SVM中会详细阐述)。

1.1.1、分类标准

这里我们考虑的是一个两类的分类问题，数据点用<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-4" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0.59em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.95em 1000em 2.73em -0.45em); 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BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-14" class="mrow"><span id="MathJax-Span-15" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">n<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">维的数据空间中找到一个超平面，其方程可以表示为：

上面给出了线性分类的定义描述，但或许读者没有想过：为何用y取1 或者 -1来表示两个不同的类别呢？其实，这个1或-1的分类标准起源于logistic回归，为了完整和过渡的自然性，咱们就再来看看这个logistic回归。

1.1.2、1或-1分类标准的起源：logistic回归

Logistic回归目的是从特征学习出一个0/1分类模型，而这个模型是将特性的线性组合作为自变量，由于自变量的取值范围是负无穷到正无穷。因此，使用logistic函数（或称作sigmoid函数）将自变量映射到(0,1)上，映射后的值被认为是属于y=1的概率。

形式化表示就是

假设函数

其中x是n维特征向量，函数g就是logistic函数。

而

的图像是

可以看到，将无穷映射到了(0,1)。

而假设函数就是特征属于y=1的概率。

当我们要判别一个新来的特征属于哪个类时，只需求，若大于0.5就是y=1的类，反之属于y=0类。

再审视一下

，发现

只和

有关，

>0，那么

，g(z)只不过是用来映射，真实的类别决定权还在支持向量机通俗导论（理解SVM的三层境界）

。还有当时

，

=1，反之

=0。如果我们只从

出发，希望模型达到的目标无非就是让训练数据中y=1的特征支持向量机通俗导论（理解SVM的三层境界）

，而是y=0的特征

。Logistic回归就是要学习得到支持向量机通俗导论（理解SVM的三层境界）

，使得正例的特征远大于0，负例的特征远小于0，强调在全部训练实例上达到这个目标。

1.1.3、形式化标示

我们这次使用的结果标签是y=-1,y=1，替换在logistic回归中使用的y=0和y=1。同时将支持向量机通俗导论（理解SVM的三层境界）

替换成w和b。以前的支持向量机通俗导论（理解SVM的三层境界）

，其中认为

。现在我们替换为b，后面替换支持向量机通俗导论（理解SVM的三层境界）

为

（即

）。这样，我们让

，进一步

。也就是说除了y由y=0变为y=-1，只是标记不同外，与logistic回归的形式化表示没区别。

再明确下假设函数

上面提到过我们只需考虑支持向量机通俗导论（理解SVM的三层境界）

的正负问题，而不用关心g(z)，因此我们这里将g(z)做一个简化，将其简单映射到y=-1和y=1上。映射关系如下：

于此，想必已经解释明白了为何线性分类的标准一般用1 或者-1 来标示。

注：上小节来自jerrylead所作的斯坦福机器学习课程的笔记。

1.2、线性分类的一个例子

下面举个简单的例子，一个二维平面(一个超平面，在二维空间中的例子就是一条直线)，如下图所示，平面上有两种不同的点，分别用两种不同的颜色表示，一种为红颜色的点，另一种则为蓝颜色的点，红颜色的线表示一个可行的超平面。

从上图中我们可以看出，这条红颜色的线把红颜色的点和蓝颜色的点分开来了。而这条红颜色的线就是我们上面所说的超平面，也就是说，这个所谓的超平面的的确确便把这两种不同颜色的数据点分隔开来，在超平面一边的数据点所对应的<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-33" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0.53em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.95em 1000em 2.92em -0.46em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-34" class="mrow"><span id="MathJax-Span-35" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">y<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">全是 -1 ，而在另一边全是 1 。

接着，我们可以令分类函数（提醒：下文很大篇幅都在讨论着这个分类函数）：

f(x)=<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-36" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 6.78em; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.77em 1000em 3.19em -0.43em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.77em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-37" class="mrow"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 1.33em; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.95em 1000em 2.73em -0.46em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial">w<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.38em; PADDING-TOP: 0px; LEFT: 0.82em; border-image: initial">T<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.02em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">x+b<span style="font-family:MathJax_Main, MathJax_Size1, MathJax_AMS;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.77em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">

显然，如果<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-49" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 3.77em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.86em 1000em 3.19em -0.43em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.77em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial">f(x)=0，那么<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-57" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0.59em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.95em 1000em 2.73em -0.45em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-58" class="mrow">x<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">是位于超平面上的点。我们不妨要求对于所有满足<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-60" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 3.77em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.86em 1000em 3.19em -0.43em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.77em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-61" class="mrow">f<span id="MathJax-Span-63" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">(x<span id="MathJax-Span-65" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">)<span id="MathJax-Span-66" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.27em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><<span id="MathJax-Span-67" class="mn" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.27em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">0<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.77em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">的点，其对应的<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-68" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0.53em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.95em 1000em 2.92em -0.46em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-69" class="mrow">y<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">等于 -1 ，而<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-71" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 3.77em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.86em 1000em 3.19em -0.43em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.77em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-72" class="mrow">f<span id="MathJax-Span-74" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">(x<span id="MathJax-Span-76" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">)<span id="MathJax-Span-77" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.27em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">><span id="MathJax-Span-78" class="mn" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.27em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">0<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.77em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">则对应<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-79" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 2.37em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.95em 1000em 3.14em -0.46em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.77em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-80" class="mrow">y<span id="MathJax-Span-82" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.27em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">=<span id="MathJax-Span-83" class="mn" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.27em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">1<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.77em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">的数据点。

（有一朋友飞狗来自Mare_Desiderii，看了上面的定义之后，问道：请教一下SVM functional margin 为 γˆ=y(wTx+b)=yf(x)中的Y是只取1和-1 吗？y的唯一作用就是确保functional margin的非负性？真是这样的么？当然不是，详情请见本文评论下第43楼）

当然，有些时候（或者说大部分时候）数据并不是线性可分的，这个时候满足这样条件的超平面就根本不存在(不过关于如何处理这样的问题我们后面会讲)，这里先从最简单的情形开始推导，就假设数据都是线性可分的，亦即这样的超平面是存在的。

更进一步，我们在进行分类的时候，将数据点<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-84" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0.59em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.95em 1000em 2.73em -0.45em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-85" class="mrow">x<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="font-family:MathJax_Main, MathJax_Size1, MathJax_AMS;BORDER-LEFT: 0em solid; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; OVERFLOW-X: hidden; OVERFLOW-Y: hidden; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0.58em; VERTICAL-ALIGN: -0.01em; PADDING-TOP: 0px; border-image: initial">代入<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-87" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 1.88em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.86em 1000em 3.19em -0.43em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.77em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-88" class="mrow">f<span id="MathJax-Span-90" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">(x<span id="MathJax-Span-92" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">)<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.77em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="font-family:MathJax_Main, MathJax_Size1, MathJax_AMS;BORDER-LEFT: 0em solid; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; OVERFLOW-X: hidden; OVERFLOW-Y: hidden; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 1.29em; VERTICAL-ALIGN: -0.32em; PADDING-TOP: 0px; border-image: initial">中，如果得到的结果小于 0 ，则赋予其类别 -1 ，如果大于 0 则赋予类别 1 。如果<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-93" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 3.77em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.86em 1000em 3.19em -0.43em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.77em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-94" class="mrow">f<span id="MathJax-Span-96" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">(x<span id="MathJax-Span-98" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">)<span id="MathJax-Span-99" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.27em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">=<span id="MathJax-Span-100" class="mn" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.27em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">0<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.77em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="font-family:MathJax_Main, MathJax_Size1, MathJax_AMS;BORDER-LEFT: 0em solid; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; OVERFLOW-X: hidden; OVERFLOW-Y: hidden; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 1.29em; VERTICAL-ALIGN: -0.32em; PADDING-TOP: 0px; border-image: initial">，则很难办了，分到哪一类都不是。

请读者注意，下面的篇幅将按下述3点走：

咱们就要确定上述分类函数f(x) = w.x + b（w.x表示w与x的内积）中的两个参数w和b，通俗理解的话w是法向量，b是截距；
那如何确定w和b呢？答案是寻找两条边界端或极端划分直线中间的最大间隔（之所以要寻最大间隔是为了能更好的划分不同类的点，下文你将看到：为寻最大间隔，导出1/2||w||^2，继而引入拉格朗日函数和对偶变量a，化为对单一因数对偶变量a的求解，当然，这是后话），从而确定最终的最大间隔分类超平面hyper plane和分类函数；
进而把寻求分类函数f(x) = w.x + b的问题转化为对w，b的最优化问题。

总结成一句话即是：从最大间隔出发（目的本就是为了确定法向量w），转化为求对变量w和b的凸二次规划问题。亦或如下图所示（有点需要注意，如读者@酱爆小八爪所说：从最大分类间隔开始，就一直是凸优化问题）：

1.3、函数间隔Functional margin与几何间隔Geometrical margin

一般而言，一个点距离超平面的远近可以表示为分类预测的确信或准确程度。在超平面w*x+b=0确定的情况下，|w*x+b|能够相对的表示点x到距离超平面的远近，而w*x+b的符号与类标记y的符号是否一致表示分类是否正确，所以，可以用量y*(w*x+b)的正负性来判定或表示分类的正确性和确信度，于此，我们便引出了函数间隔functional margin的概念。

1.3.1、函数间隔Functional margin

我们定义函数间隔functional margin为：

接着，我们定义超平面(w，b)关于训练数据集T的函数间隔为超平面(w，b)关于T中所有样本点(xi，yi)的函数间隔最小值，其中，x是特征，y是结果标签，i表示第i个样本，有：

γˆ=minγˆi (i=1，...n)

然与此同时，问题就出来了。上述定义的函数间隔虽然可以表示分类预测的正确性和确信度，但在选择分类超平面时，只有函数间隔还远远不够，因为如果成比例的改变w和b，如将他们改变为2w和2b，虽然此时超平面没有改变，但函数间隔的值f(x)却变成了原来的4倍。其实，我们可以对法向量w加些约束条件，使其表面上看起来规范化，如此，我们很快又将引出真正定义点到超平面的距离--几何间隔geometrical margin的概念。

1.3.2、点到超平面的距离定义：几何间隔Geometrical margin

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POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-152" class="mrow"><span id="MathJax-Span-153" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">x<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="font-family:MathJax_Main, MathJax_Size1, MathJax_AMS;BORDER-LEFT: 0em solid; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; OVERFLOW-X: hidden; OVERFLOW-Y: hidden; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0.58em; VERTICAL-ALIGN: -0.01em; PADDING-TOP: 0px; border-image: initial">，令其垂直投影到超平面上的对应的为<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-154" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 1.07em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(0.39em 1000em 1.32em -0.45em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -1em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-155" class="mrow"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-156" class="msubsup"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 1.11em; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.95em 1000em 2.73em -0.45em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span id="MathJax-Span-157" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">x<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.03em; PADDING-TOP: 0px; LEFT: 0.66em; border-image: initial"><span id="MathJax-Span-158" class="mn" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; FONT-SIZE: 13px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">0<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.18em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 1em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="font-family:MathJax_Main, MathJax_Size1, MathJax_AMS;BORDER-LEFT: 0em solid; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; OVERFLOW-X: hidden; OVERFLOW-Y: hidden; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0.78em; VERTICAL-ALIGN: -0.21em; PADDING-TOP: 0px; border-image: initial">，由于<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-159" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0.75em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.95em 1000em 2.73em -0.46em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-160" class="mrow"><span id="MathJax-Span-161" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">w<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="font-family:MathJax_Main, MathJax_Size1, MathJax_AMS;BORDER-LEFT: 0em solid; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; OVERFLOW-X: hidden; OVERFLOW-Y: hidden; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0.58em; VERTICAL-ALIGN: -0.01em; PADDING-TOP: 0px; border-image: initial">是垂直于超平面的一个向量，我们有

<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-162" class="math"><span style="position: relative; padding: 0px; border-width: 0px; margin: 0px; width: 7.05em; display: inline-block; font-family: MathJax_Main, MathJax_Size1, MathJax_AMS; height: 0px; font-size: 19px; vertical-align: 0px;"><span style="clip: rect(1.49em 1000em 3.91em -0.45em); position: absolute; padding: 0px; border-width: 0px; margin: 0px; vertical-align: 0px; top: -2.77em; left: 0em;"><span style="padding: 0px; border-width: 0px; margin: 0px; display: inline; vertical-align: 0px;" id="MathJax-Span-163" class="mrow"><span id="MathJax-Span-164" class="mi" style="font-family: MathJax_Math; padding: 0px; border-width: 0px; margin: 0px; display: inline; vertical-align: 0px;">x</span><span id="MathJax-Span-165" class="mo" style="font-family: MathJax_Main; padding: 0px 0px 0px 0.27em; border-width: 0px; margin: 0px; display: inline; vertical-align: 0px;">=</span><span style="padding: 0px 0px 0px 0.27em; border-width: 0px; margin: 0px; display: inline; vertical-align: 0px;" id="MathJax-Span-166" class="msubsup"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 1.11em; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.95em 1000em 2.73em -0.45em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span id="MathJax-Span-167" class="mi" style="font-family: MathJax_Math; padding: 0px; border-width: 0px; margin: 0px; display: inline; vertical-align: 0px;">x</span><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"></span></span><span style="POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.03em; PADDING-TOP: 0px; LEFT: 0.66em; border-image: initial"><span id="MathJax-Span-168" class="mn" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; FONT-SIZE: 13px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">0</span><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.18em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"></span></span></span></span><span id="MathJax-Span-169" class="mo" style="font-family: MathJax_Main; padding: 0px 0px 0px 0.22em; border-width: 0px; margin: 0px; display: inline; vertical-align: 0px;">+</span><span id="MathJax-Span-170" class="mi" style="font-family: MathJax_Math; padding: 0px 0px 0px 0.22em; border-width: 0px; margin: 0px; display: inline; vertical-align: 0px;">γ</span><span style="padding: 0px 0.12em 0px 0.28em; border-width: 0px; margin: 0px; display: inline; vertical-align: 0px;" id="MathJax-Span-171" class="mfrac"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 1.89em; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.95em 1000em 2.73em -0.46em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px 0px 0px -0.37em; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -3.23em; PADDING-TOP: 0px; LEFT: 50%; border-image: initial"><span id="MathJax-Span-172" class="mi" style="font-family: MathJax_Math; padding: 0px; border-width: 0px; margin: 0px; display: inline; vertical-align: 0px;">w</span><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"></span></span><span style="CLIP: rect(1.86em 1000em 3.18em -0.35em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px 0px 0px -0.88em; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.05em; PADDING-TOP: 0px; LEFT: 50%; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-173" class="mrow"><span id="MathJax-Span-174" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; 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margin: 0px; width: 0px; display: inline-block; height: 2.77em; vertical-align: 0px;"></span></span></span><span style="font-family: MathJax_Main, MathJax_Size1, MathJax_AMS; border-width: 0px 0px 0px 0em; border-left-style: solid; padding: 0px; overflow: hidden; margin: 0px; width: 0px; display: inline-block; height: 2.7em; vertical-align: -1.25em;"></span></span></nobr>

（||w||表示的是范数，关于范数的概念参见：http://baike.baidu.com/view/637132.htm）

又由于<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-177" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 1.07em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(0.39em 1000em 1.32em -0.45em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -1em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-178" class="mrow"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-179" class="msubsup"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 1.11em; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.95em 1000em 2.73em -0.45em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span id="MathJax-Span-180" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">x<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.03em; PADDING-TOP: 0px; LEFT: 0.66em; border-image: initial"><span id="MathJax-Span-181" class="mn" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; FONT-SIZE: 13px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">0<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.18em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 1em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="font-family:MathJax_Main, MathJax_Size1, MathJax_AMS;BORDER-LEFT: 0em solid; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; OVERFLOW-X: hidden; OVERFLOW-Y: hidden; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0.78em; VERTICAL-ALIGN: -0.21em; PADDING-TOP: 0px; border-image: initial">是超平面上的点，满足<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-182" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 4.25em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.86em 1000em 3.19em -0.43em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.77em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-183" class="mrow"><span id="MathJax-Span-184" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">f<span id="MathJax-Span-185" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">(<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-186" class="msubsup"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 1.11em; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.95em 1000em 2.73em -0.45em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span id="MathJax-Span-187" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">x<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.03em; PADDING-TOP: 0px; LEFT: 0.66em; border-image: initial"><span id="MathJax-Span-188" class="mn" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; FONT-SIZE: 13px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">0<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.18em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span id="MathJax-Span-189" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">)<span id="MathJax-Span-190" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.27em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">=<span id="MathJax-Span-191" class="mn" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.27em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">0<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.77em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="font-family:MathJax_Main, MathJax_Size1, MathJax_AMS;BORDER-LEFT: 0em solid; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; OVERFLOW-X: hidden; OVERFLOW-Y: hidden; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 1.29em; VERTICAL-ALIGN: -0.32em; PADDING-TOP: 0px; border-image: initial">，代入超平面的方程即可算出：<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-113" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 10.44em; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.77em 1000em 3.19em -0.47em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.77em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-114" class="mrow"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-115" class="munderover"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0.53em; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="font-family:MathJax_Main, MathJax_Size1, MathJax_AMS;CLIP: rect(1.95em 1000em 2.94em -0.47em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span id="MathJax-Span-116" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">γ

（有的书上会写成把||w|| 分开相除的形式，如本文参考文献及推荐阅读条目9，其中，||w||为w的二阶泛数）

不过，这里的<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-219" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0.53em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.95em 1000em 2.94em -0.47em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-220" class="mrow"><span id="MathJax-Span-221" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">γ<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="font-family:MathJax_Main, MathJax_Size1, MathJax_AMS;BORDER-LEFT: 0em solid; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; OVERFLOW-X: hidden; OVERFLOW-Y: hidden; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0.84em; VERTICAL-ALIGN: -0.27em; PADDING-TOP: 0px; border-image: initial">是带符号的，我们需要的只是它的绝对值，因此类似地，也乘上对应的类别<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-222" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0.53em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.95em 1000em 2.92em -0.46em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-223" class="mrow"><span id="MathJax-Span-224" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">y<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="font-family:MathJax_Main, MathJax_Size1, MathJax_AMS;BORDER-LEFT: 0em solid; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; OVERFLOW-X: hidden; OVERFLOW-Y: hidden; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0.83em; VERTICAL-ALIGN: -0.26em; PADDING-TOP: 0px; border-image: initial">即可，因此实际上我们定义几何间隔geometrical margin 为(注：别忘了，上面ˆγ的定义，ˆγ=y(wTx+b)=yf(x))：

（代人相关式子可以得出：yi*(w/||w|| + b/||w||)）

正如本文评论下读者popol1991留言：函数间隔y*(wx+b)=y*f(x)实际上就是|f(x)|，只是人为定义的一个间隔度量；而几何间隔|f(x)|/||w||才是直观上的点到超平面距离。
想想二维空间里的点到直线公式：假设一条直线的方程为ax+by+c=0,点P的坐标是(x0,y0)，则点到直线距离为|ax0+by0+c|/sqrt(a^2+b^2)。如下图所示：
支持向量机通俗导论（理解SVM的三层境界）
那么如果用向量表示，设w=(a,b),f(x)=wx+c,那么这个距离正是|f(p)|/||w||。

1.4、最大间隔分类器Maximum Margin Classifier的定义

于此，我们已经很明显的看出，函数间隔functional margin 和几何间隔geometrical margin 相差一个<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-242" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 1.77em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.86em 1000em 3.18em -0.35em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.77em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-243" class="mrow"><span id="MathJax-Span-244" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">∥<span id="MathJax-Span-245" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">w<span id="MathJax-Span-246" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">∥<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.77em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="font-family:MathJax_Main, MathJax_Size1, MathJax_AMS;BORDER-LEFT: 0em solid; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; OVERFLOW-X: hidden; OVERFLOW-Y: hidden; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 1.28em; VERTICAL-ALIGN: -0.32em; PADDING-TOP: 0px; border-image: initial">的缩放因子。按照我们前面的分析，对一个数据点进行分类，当它的 margin 越大的时候，分类的 confidence 越大。对于一个包含<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-247" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0.64em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.95em 1000em 2.73em -0.46em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-248" class="mrow"><span id="MathJax-Span-249" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">n<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="font-family:MathJax_Main, MathJax_Size1, MathJax_AMS;BORDER-LEFT: 0em solid; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; OVERFLOW-X: hidden; OVERFLOW-Y: hidden; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0.58em; VERTICAL-ALIGN: -0.01em; PADDING-TOP: 0px; border-image: initial">个点的数据集，我们可以很自然地定义它的 margin 为所有这<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-250" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0.64em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.95em 1000em 2.73em -0.46em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-251" class="mrow"><span id="MathJax-Span-252" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">n<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="font-family:MathJax_Main, MathJax_Size1, MathJax_AMS;BORDER-LEFT: 0em solid; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; OVERFLOW-X: hidden; OVERFLOW-Y: hidden; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0.58em; VERTICAL-ALIGN: -0.01em; PADDING-TOP: 0px; border-image: initial">个点的 margin 值中最小的那个。于是，为了使得分类的 confidence 高，我们希望所选择的超平面hyper plane 能够最大化这个 margin 值。

通过上节，我们已经知道：

1、functional margin 明显是不太适合用来最大化的一个量，因为在 hyper plane 固定以后，我们可以等比例地缩放<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-253" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0.75em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.95em 1000em 2.73em -0.46em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-254" class="mrow"><span id="MathJax-Span-255" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">w<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="font-family:MathJax_Main, MathJax_Size1, MathJax_AMS;BORDER-LEFT: 0em solid; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; OVERFLOW-X: hidden; OVERFLOW-Y: hidden; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0.58em; VERTICAL-ALIGN: -0.01em; PADDING-TOP: 0px; border-image: initial">的长度和<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-256" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0.48em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.7em 1000em 2.73em -0.44em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-257" class="mrow"><span id="MathJax-Span-258" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">b<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="font-family:MathJax_Main, MathJax_Size1, MathJax_AMS;BORDER-LEFT: 0em solid; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; OVERFLOW-X: hidden; OVERFLOW-Y: hidden; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0.9em; VERTICAL-ALIGN: -0.01em; PADDING-TOP: 0px; border-image: initial">的值，这样可以使得<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-259" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 6.78em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.77em 1000em 3.19em -0.43em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.77em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-260" class="mrow"><span id="MathJax-Span-261" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">f<span id="MathJax-Span-262" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">(<span id="MathJax-Span-263" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">x<span id="MathJax-Span-264" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">)<span id="MathJax-Span-265" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.27em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">=<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.27em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-266" class="msubsup"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 1.33em; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.95em 1000em 2.73em -0.46em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span id="MathJax-Span-267" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">w<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.38em; PADDING-TOP: 0px; LEFT: 0.82em; border-image: initial"><span id="MathJax-Span-268" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; FONT-SIZE: 13px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">T<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.02em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span id="MathJax-Span-269" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">x<span id="MathJax-Span-270" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.22em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">+<span id="MathJax-Span-271" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0.22em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">b<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.77em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="font-family:MathJax_Main, MathJax_Size1, MathJax_AMS;BORDER-LEFT: 0em solid; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; OVERFLOW-X: hidden; OVERFLOW-Y: hidden; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 1.41em; VERTICAL-ALIGN: -0.32em; PADDING-TOP: 0px; border-image: initial">的值任意大，亦即 functional margin<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-272" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0.53em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(0.1em 1000em 1.37em -0.47em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -1em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-273" class="mrow"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-274" class="munderover"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0.53em; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.95em 1000em 2.94em -0.47em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span id="MathJax-Span-275" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">γ<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.92em 1000em 2.4em -0.37em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.82em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span id="MathJax-Span-276" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">ˆ<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.77em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 1em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="font-family:MathJax_Main, MathJax_Size1, MathJax_AMS;BORDER-LEFT: 0em solid; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; OVERFLOW-X: hidden; OVERFLOW-Y: hidden; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 1.23em; VERTICAL-ALIGN: -0.27em; PADDING-TOP: 0px; border-image: initial">可以在 hyper plane 保持不变的情况下被取得任意大，

2、而 geometrical margin 则没有这个问题，因为除上了<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-277" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 1.77em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.86em 1000em 3.18em -0.35em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.77em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-278" class="mrow"><span id="MathJax-Span-279" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">∥<span id="MathJax-Span-280" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">w<span id="MathJax-Span-281" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">∥<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.77em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="font-family:MathJax_Main, MathJax_Size1, MathJax_AMS;BORDER-LEFT: 0em solid; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; OVERFLOW-X: hidden; OVERFLOW-Y: hidden; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 1.28em; VERTICAL-ALIGN: -0.32em; PADDING-TOP: 0px; border-image: initial">这个分母，所以缩放<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-282" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0.75em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.95em 1000em 2.73em -0.46em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-283" class="mrow"><span id="MathJax-Span-284" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">w<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="font-family:MathJax_Main, MathJax_Size1, MathJax_AMS;BORDER-LEFT: 0em solid; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; OVERFLOW-X: hidden; OVERFLOW-Y: hidden; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0.58em; VERTICAL-ALIGN: -0.01em; PADDING-TOP: 0px; border-image: initial">和<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-285" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0.48em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.7em 1000em 2.73em -0.44em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-286" class="mrow"><span id="MathJax-Span-287" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">b<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="font-family:MathJax_Main, MathJax_Size1, MathJax_AMS;BORDER-LEFT: 0em solid; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; OVERFLOW-X: hidden; OVERFLOW-Y: hidden; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0.9em; VERTICAL-ALIGN: -0.01em; PADDING-TOP: 0px; border-image: initial">的时候<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-288" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0.53em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(0.16em 1000em 1.37em -0.47em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -1em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-289" class="mrow"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-290" class="munderover"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0.53em; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.95em 1000em 2.94em -0.47em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span id="MathJax-Span-291" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">γ<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.94em 1000em 2.37em -0.4em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.78em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span id="MathJax-Span-292" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">˜<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.77em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 1em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="font-family:MathJax_Main, MathJax_Size1, MathJax_AMS;BORDER-LEFT: 0em solid; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; OVERFLOW-X: hidden; OVERFLOW-Y: hidden; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 1.15em; VERTICAL-ALIGN: -0.27em; PADDING-TOP: 0px; border-image: initial">的值是不会改变的，它只随着 hyper plane 的变动而变动，因此，这是更加合适的一个 margin 。

这样一来，我们的 maximum margin classifier 的目标函数可以定义为：

当然，还需要满足一些条件，根据 margin 的定义，我们有

其中<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-333" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 4.2em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.86em 1000em 3.18em -0.47em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.77em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-334" class="mrow"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-335" class="munderover"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0.53em; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.95em 1000em 2.94em -0.47em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span id="MathJax-Span-336" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">γ<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.92em 1000em 2.4em -0.37em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.82em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span id="MathJax-Span-337" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">ˆ<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.77em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span id="MathJax-Span-338" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.27em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">=<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.27em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-339" class="munderover"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0.53em; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.95em 1000em 2.94em -0.47em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span id="MathJax-Span-340" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">γ<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.94em 1000em 2.37em -0.4em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.78em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span id="MathJax-Span-341" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">˜<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.77em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span id="MathJax-Span-342" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">∥<span id="MathJax-Span-343" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">w<span id="MathJax-Span-344" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">∥<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.77em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="font-family:MathJax_Main, MathJax_Size1, MathJax_AMS;BORDER-LEFT: 0em solid; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; OVERFLOW-X: hidden; OVERFLOW-Y: hidden; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 1.28em; VERTICAL-ALIGN: -0.32em; PADDING-TOP: 0px; border-image: initial">(等价于 ˜γ = ˆγ / ∥w∥，故有稍后的 γˆ =1 时， ˜γ = 1 / ||w||)，处于方便推导和优化的目的，我们可以令<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" class="math" id="MathJax-Span-385"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 2.42em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.87em 1000em 3.15em -0.47em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.77em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-386" class="mrow"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-387" class="munderover"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0.53em; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.95em 1000em 2.94em -0.47em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span id="MathJax-Span-388" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">γ<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.92em 1000em 2.4em -0.37em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.82em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span id="MathJax-Span-389" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">ˆ<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.77em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span id="MathJax-Span-390" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.27em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">=<span id="MathJax-Span-391" class="mn" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.27em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">1<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.77em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="font-family:MathJax_Main, MathJax_Size1, MathJax_AMS;BORDER-LEFT: 0em solid; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; OVERFLOW-X: hidden; OVERFLOW-Y: hidden; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 1.23em; VERTICAL-ALIGN: -0.27em; PADDING-TOP: 0px; border-image: initial">(对目标函数的优化没有影响，至于为什么，请见本文评论下第42楼回复，貌似博客系统故障，原来评论在第42楼的回复现在第42+81楼，即第123楼显示)，此时，上述的目标函数˜γ转化为(其中，s.t.，即subject to的意思，它导出的是约束条件)：

通过求解这个问题，我们就可以找到一个 margin 最大的 classifier ，如下图所示，中间的红色线条是 Optimal Hyper Plane ，另外两条线到红线的距离都是等于

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POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -1em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-432" class="mrow"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-433" class="munderover"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0.53em; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.95em 1000em 2.94em -0.47em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span id="MathJax-Span-434" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">γ<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; 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HEIGHT: 2.77em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 1em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="font-family:MathJax_Main, MathJax_Size1, MathJax_AMS;BORDER-LEFT: 0em solid; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; OVERFLOW-X: hidden; OVERFLOW-Y: hidden; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 1.15em; VERTICAL-ALIGN: -0.27em; PADDING-TOP: 0px; border-image: initial">的(<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; 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POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span id="MathJax-Span-434" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">γ<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.94em 1000em 2.37em -0.4em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.78em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span id="MathJax-Span-435" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">˜<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.77em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 1em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">便是上文所定义的geometrical margin，当令<nobr style="padding-bottom: 0px; border-right-width: 0px; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-left: 0px; padding-right: 0px; max-width: none; border-top-width: 0px; border-bottom-width: 0px; max-height: none; vertical-align: 0px; border-left-width: 0px; padding-top: 0px; border-image: initial; "><span id="MathJax-Span-385" class="math" style="padding-bottom: 0px; border-right-width: 0px; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-left: 0px; padding-right: 0px; display: inline; border-top-width: 0px; border-bottom-width: 0px; vertical-align: 0px; border-left-width: 0px; padding-top: 0px; border-image: initial; "><span style="position: relative; padding-bottom: 0px; border-right-width: 0px; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-left: 0px; width: 2.42em; padding-right: 0px; display: inline-block; font-family: MathJax_Main, MathJax_Size1, MathJax_AMS; border-top-width: 0px; border-bottom-width: 0px; height: 0px; font-size: 19px; vertical-align: 0px; border-left-width: 0px; padding-top: 0px; border-image: initial; "><span style="clip: rect(1.87em 1000em 3.15em -0.47em); position: absolute; padding-bottom: 0px; border-right-width: 0px; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-left: 0px; padding-right: 0px; border-top-width: 0px; border-bottom-width: 0px; vertical-align: 0px; border-left-width: 0px; top: -2.77em; padding-top: 0px; left: 0em; border-image: initial; "><span id="MathJax-Span-386" class="mrow" style="padding-bottom: 0px; border-right-width: 0px; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-left: 0px; padding-right: 0px; display: inline; border-top-width: 0px; border-bottom-width: 0px; vertical-align: 0px; border-left-width: 0px; padding-top: 0px; border-image: initial; "><span id="MathJax-Span-387" class="munderover" style="padding-bottom: 0px; border-right-width: 0px; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-left: 0px; padding-right: 0px; display: inline; border-top-width: 0px; border-bottom-width: 0px; vertical-align: 0px; border-left-width: 0px; padding-top: 0px; border-image: initial; "><span style="position: relative; padding-bottom: 0px; border-right-width: 0px; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-left: 0px; width: 0.53em; padding-right: 0px; display: inline-block; border-top-width: 0px; border-bottom-width: 0px; height: 0px; vertical-align: 0px; border-left-width: 0px; padding-top: 0px; border-image: initial; "><span style="clip: rect(1.95em 1000em 2.94em -0.47em); position: absolute; padding-bottom: 0px; border-right-width: 0px; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-left: 0px; padding-right: 0px; border-top-width: 0px; border-bottom-width: 0px; vertical-align: 0px; border-left-width: 0px; top: -2.56em; padding-top: 0px; left: 0em; border-image: initial; "><span id="MathJax-Span-388" class="mi" style="font-family:MathJax_Math;padding-bottom: 0px; border-right-width: 0px; font-style: italic; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-left: 0px; padding-right: 0px; display: inline; border-top-width: 0px; border-bottom-width: 0px; vertical-align: 0px; border-left-width: 0px; padding-top: 0px; border-image: initial; ">γ<span style="padding-bottom: 0px; border-right-width: 0px; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-left: 0px; width: 0px; padding-right: 0px; display: inline-block; border-top-width: 0px; border-bottom-width: 0px; height: 2.56em; vertical-align: 0px; border-left-width: 0px; padding-top: 0px; border-image: initial; "><span style="clip: rect(1.92em 1000em 2.4em -0.37em); position: absolute; padding-bottom: 0px; border-right-width: 0px; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-left: 0px; padding-right: 0px; border-top-width: 0px; border-bottom-width: 0px; vertical-align: 0px; border-left-width: 0px; top: -2.82em; padding-top: 0px; left: 0em; border-image: initial; "><span id="MathJax-Span-389" class="mo" style="font-family:MathJax_Main;padding-bottom: 0px; border-right-width: 0px; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-left: 0px; padding-right: 0px; display: inline; border-top-width: 0px; border-bottom-width: 0px; vertical-align: 0px; border-left-width: 0px; padding-top: 0px; border-image: initial; ">ˆ<span style="padding-bottom: 0px; border-right-width: 0px; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-left: 0px; width: 0px; padding-right: 0px; display: inline-block; border-top-width: 0px; border-bottom-width: 0px; height: 2.77em; vertical-align: 0px; border-left-width: 0px; padding-top: 0px; border-image: initial; "><span id="MathJax-Span-390" class="mo" style="font-family:MathJax_Main;padding-bottom: 0px; border-right-width: 0px; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-left: 0.27em; padding-right: 0px; display: inline; border-top-width: 0px; border-bottom-width: 0px; vertical-align: 0px; border-left-width: 0px; padding-top: 0px; border-image: initial; ">=<span id="MathJax-Span-391" class="mn" style="font-family:MathJax_Main;padding-bottom: 0px; border-right-width: 0px; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-left: 0.27em; padding-right: 0px; display: inline; border-top-width: 0px; border-bottom-width: 0px; vertical-align: 0px; border-left-width: 0px; padding-top: 0px; border-image: initial; ">1<span style="padding-bottom: 0px; border-right-width: 0px; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-left: 0px; width: 0px; padding-right: 0px; display: inline-block; border-top-width: 0px; border-bottom-width: 0px; height: 2.77em; vertical-align: 0px; border-left-width: 0px; padding-top: 0px; border-image: initial; ">时，<nobr style="padding-bottom: 0px; border-right-width: 0px; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-left: 0px; padding-right: 0px; max-width: none; border-top-width: 0px; border-bottom-width: 0px; max-height: none; vertical-align: 0px; border-left-width: 0px; padding-top: 0px; border-image: initial; "><span id="MathJax-Span-431" class="math" style="padding-bottom: 0px; border-right-width: 0px; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-left: 0px; padding-right: 0px; display: inline; border-top-width: 0px; border-bottom-width: 0px; vertical-align: 0px; border-left-width: 0px; padding-top: 0px; border-image: initial; "><span style="position: relative; padding-bottom: 0px; border-right-width: 0px; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-left: 0px; width: 0.53em; padding-right: 0px; display: inline-block; font-family: MathJax_Main, MathJax_Size1, MathJax_AMS; border-top-width: 0px; border-bottom-width: 0px; height: 0px; vertical-align: 0px; border-left-width: 0px; padding-top: 0px; border-image: initial; "><span style="clip: rect(0.16em 1000em 1.37em -0.47em); position: absolute; padding-bottom: 0px; border-right-width: 0px; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-left: 0px; padding-right: 0px; border-top-width: 0px; border-bottom-width: 0px; vertical-align: 0px; border-left-width: 0px; top: -1em; padding-top: 0px; left: 0em; border-image: initial; "><span id="MathJax-Span-432" class="mrow" style="padding-bottom: 0px; border-right-width: 0px; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-left: 0px; padding-right: 0px; display: inline; border-top-width: 0px; border-bottom-width: 0px; vertical-align: 0px; border-left-width: 0px; padding-top: 0px; border-image: initial; "><span id="MathJax-Span-433" class="munderover" style="padding-bottom: 0px; border-right-width: 0px; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-left: 0px; padding-right: 0px; display: inline; border-top-width: 0px; border-bottom-width: 0px; vertical-align: 0px; border-left-width: 0px; padding-top: 0px; border-image: initial; "><span style="position: relative; padding-bottom: 0px; border-right-width: 0px; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-left: 0px; width: 0.53em; padding-right: 0px; display: inline-block; border-top-width: 0px; border-bottom-width: 0px; height: 0px; vertical-align: 0px; border-left-width: 0px; padding-top: 0px; border-image: initial; "><span style="clip: rect(1.95em 1000em 2.94em -0.47em); position: absolute; padding-bottom: 0px; border-right-width: 0px; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-left: 0px; padding-right: 0px; border-top-width: 0px; border-bottom-width: 0px; vertical-align: 0px; border-left-width: 0px; top: -2.56em; padding-top: 0px; left: 0em; border-image: initial; "><span id="MathJax-Span-434" class="mi" style="font-family:MathJax_Math;padding-bottom: 0px; border-right-width: 0px; font-style: italic; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-left: 0px; padding-right: 0px; display: inline; border-top-width: 0px; border-bottom-width: 0px; vertical-align: 0px; border-left-width: 0px; padding-top: 0px; border-image: initial; ">γ<span style="padding-bottom: 0px; border-right-width: 0px; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-left: 0px; width: 0px; padding-right: 0px; display: inline-block; border-top-width: 0px; border-bottom-width: 0px; height: 2.56em; vertical-align: 0px; border-left-width: 0px; padding-top: 0px; border-image: initial; "><span style="clip: rect(1.94em 1000em 2.37em -0.4em); position: absolute; padding-bottom: 0px; border-right-width: 0px; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-left: 0px; padding-right: 0px; border-top-width: 0px; border-bottom-width: 0px; vertical-align: 0px; border-left-width: 0px; top: -2.78em; padding-top: 0px; left: 0em; border-image: initial; "><span id="MathJax-Span-435" class="mo" style="font-family:MathJax_Main;padding-bottom: 0px; border-right-width: 0px; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-left: 0px; padding-right: 0px; display: inline; border-top-width: 0px; border-bottom-width: 0px; vertical-align: 0px; border-left-width: 0px; padding-top: 0px; border-image: initial; ">˜<span style="padding-bottom: 0px; border-right-width: 0px; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-left: 0px; width: 0px; padding-right: 0px; display: inline-block; border-top-width: 0px; border-bottom-width: 0px; height: 2.77em; vertical-align: 0px; border-left-width: 0px; padding-top: 0px; border-image: initial; "><span style="padding-bottom: 0px; border-right-width: 0px; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-left: 0px; width: 0px; padding-right: 0px; display: inline-block; border-top-width: 0px; border-bottom-width: 0px; height: 1em; vertical-align: 0px; border-left-width: 0px; padding-top: 0px; border-image: initial; ">便为1/||w||，而我们上面得到的目标函数便是在相应的约束条件下，要最大化这个1/||w||值)：

通过最大化 margin ，我们使得该分类器对数据进行分类时具有了最大的 confidence 。但，这个最大分类间隔器到底是用来干嘛的呢？很简单，SVM 通过使用最大分类间隙Maximum Margin Classifier来设计决策最优分类超平面，而为何是最大间隔，却不是最小间隔呢？因为最大间隔能获得最大稳定性与区分的确信度，从而得到良好的推广能力(超平面之间的距离越大，分离器的推广能力越好，也就是预测精度越高，不过对于训练数据的误差不一定是最小的.2012.08.21updated)。

So，对于什么是Support Vector Machine ，我们可以先这样理解，如上图所示，我们可以看到 hyper plane 两边的那个 gap 分别对应的两条平行的线（在高维空间中也应该是两个 hyper plane）上有一些点，显然两个超平面hyper plane 上都会有点存在，否则我们就可以进一步扩大 gap ，也就是增大<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-436" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0.53em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(0.16em 1000em 1.37em -0.47em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -1em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-437" class="mrow"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-438" class="munderover"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0.53em; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.95em 1000em 2.94em -0.47em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span id="MathJax-Span-439" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">γ<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.94em 1000em 2.37em -0.4em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.78em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span id="MathJax-Span-440" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">˜<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.77em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 1em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">的值了。这些点，就叫做 support vector。下文1.5节将更为具体描述。

1.5、到底什么是Support Vector

上节，我们介绍了Maximum Margin Classifier，但并没有具体阐述到底什么是Support Vector，本节，咱们来重点阐述这个概念。咱们不妨先来回忆一下上节1.4节最后一张图：

可以看到两个支撑着中间的 gap 的超平面，它们到中间的纯红线separating hyper plane 的距离相等，即我们所能得到的最大的 geometrical margin<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-10" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0.53em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(0.16em 1000em 1.37em -0.47em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -1em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-11" class="mrow"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-12" class="munderover"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0.53em; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.95em 1000em 2.94em -0.47em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span id="MathJax-Span-13" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">γ<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.94em 1000em 2.37em -0.4em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.78em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span id="MathJax-Span-14" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">˜<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.77em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 1em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">。而“支撑”这两个超平面的必定会有一些点，而这些“支撑”的点便叫做支持向量Support Vector。

很显然，由于这些 supporting vector 刚好在边界上，所以它们是满足<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-18" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 5.49em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.77em 1000em 3.19em -0.46em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.77em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-19" class="mrow"><span id="MathJax-Span-20" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">y<span id="MathJax-Span-21" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">(<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-22" class="msubsup"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0.85em; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.95em 1000em 2.73em -0.46em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span id="MathJax-Span-23" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">w<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.38em; PADDING-TOP: 0px; LEFT: 0.34em; border-image: initial"><span id="MathJax-Span-24" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; FONT-SIZE: 13px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">T<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.02em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span id="MathJax-Span-25" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">x<span id="MathJax-Span-26" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.22em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">+<span id="MathJax-Span-27" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0.22em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">b<span id="MathJax-Span-28" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">)<span id="MathJax-Span-29" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.27em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">=<span id="MathJax-Span-30" class="mn" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.27em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">1<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.77em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">（还记得我们把 functional margin 定为 1 了吗？上节中：“处于方便推导和优化的目的，我们可以令<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-385" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 2.42em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.87em 1000em 3.15em -0.47em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.77em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-386" class="mrow"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-387" class="munderover"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0.53em; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.95em 1000em 2.94em -0.47em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span id="MathJax-Span-388" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">γ<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.92em 1000em 2.4em -0.37em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.82em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span id="MathJax-Span-389" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">ˆ<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.77em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span id="MathJax-Span-390" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.27em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">=<span id="MathJax-Span-391" class="mn" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.27em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">1<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.77em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">”），而对于所有不是支持向量的点，也就是在“阵地后方”的点，则显然有<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-31" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 5.49em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.77em 1000em 3.19em -0.46em); POSITION: absolute; 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border-image: initial">)<span id="MathJax-Span-42" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.27em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">><span id="MathJax-Span-43" class="mn" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.27em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">1<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.77em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">。当然，通常除了 K-Nearest Neighbor 之类的Memory-based Learning算法，通常算法也都不会直接把所有的点记忆下来，并全部用来做后续 inference 中的计算。不过，如果算法使用了Kernel 方法进行非线性化推广的话，就会遇到这个问题了。Kernel 方法在下文第二部分2.2节中介绍）。

OK，到此为止，算是了解到了SVM的第一层，对于那些只关心怎么用SVM的朋友便已足够，不必再更进一层深究其更深的原理。

第二层、深入SVM

2.1、从线性可分到线性不可分

2.1.1、从原始问题到对偶问题的求解

当然，除了在上文中所介绍的从几何直观上之外，支持向量的概念也可以从其优化过程的推导中得到。虽然上文1.4节给出了目标函数，却没有讲怎么来求解。现在就让我们来处理这个问题。回忆一下之前得到的目标函数（subject to导出的则是约束条件）：

由于求

的最大值相当于求

的最小值，所以上述问题等价于（w由分母变成分子，从而也有原来的max问题变为min问题，很明显，两者问题等价）：

到这个形式以后，就可以很明显地看出来，它是一个凸优化问题，或者更具体地说，它是一个二次优化问题——目标函数是二次的，约束条件是线性的。这个问题可以用任何现成的QP (Quadratic Programming)的优化包进行求解；
虽然这个问题确实是一个标准的 QP 问题，但是它也有它的特殊结构，通过Lagrange Duality变换到对偶变量 (dual variable) 的优化问题之后，可以找到一种更加有效的方法来进行求解，而且通常情况下这种方法比直接使用通用的 QP 优化包进行优化要高效得多。

也就说，除了用解决QP问题的常规方法之外，还可以应用拉格朗日对偶性，通过求解对偶问题得到最优解，这就是线性可分条件下支持向量机的对偶算法，这样做的优点在于：一者对偶问题往往更容易求解；二者可以自然的引入核函数，进而推广到非线性分类问题。
ok，接下来，你将看到“对偶变量dual variable的优化问题”等类似的关键词频繁出现，便是解决此凸优化问题的第二种更为高效的解--对偶变量的优化求解。

至于上述提到，关于什么是Lagrange duality？简单地来说，通过给每一个约束条件加上一个 Lagrange multiplier(拉格朗日乘值)，即引入拉格朗日对偶变量α，如此我们便可以通过拉格朗日函数将约束条件融和到目标函数里去(也就是说把条件融合到一个函数里头，现在只用一个函数表达式便能清楚的表达出我们的问题)：

然后我们令

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PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.38em; PADDING-TOP: 0px; LEFT: 0.34em; border-image: initial"><span id="MathJax-Span-249" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; FONT-SIZE: 13px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">T<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.02em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-250" class="msubsup"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0.68em; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.95em 1000em 2.73em -0.45em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span id="MathJax-Span-251" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">x<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -1.87em; PADDING-TOP: 0px; LEFT: 0.34em; border-image: initial"><span id="MathJax-Span-252" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; FONT-SIZE: 13px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">i<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.02em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span id="MathJax-Span-253" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.22em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">+<span id="MathJax-Span-254" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0.22em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">b<span id="MathJax-Span-255" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">)<span id="MathJax-Span-256" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.27em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><<span id="MathJax-Span-257" class="mn" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.27em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">1<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.77em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">，那么我们显然有<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-258" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 3.93em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.86em 1000em 3.19em -0.45em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.77em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-259" class="mrow"><span id="MathJax-Span-260" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">θ<span id="MathJax-Span-261" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">(<span id="MathJax-Span-262" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">w<span id="MathJax-Span-263" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">)<span id="MathJax-Span-264" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.27em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">=<span id="MathJax-Span-265" class="mi" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.27em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">∞<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.77em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">（只要令<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-266" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 3.44em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(2.17em 1000em 3.09em -0.45em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.77em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-267" class="mrow"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-268" class="msubsup"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 1.11em; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.95em 1000em 2.73em -0.45em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span id="MathJax-Span-269" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">α<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -1.87em; PADDING-TOP: 0px; LEFT: 0.77em; border-image: initial"><span id="MathJax-Span-270" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; FONT-SIZE: 13px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">i<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.02em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span id="MathJax-Span-271" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.27em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">=<span id="MathJax-Span-272" class="mi" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.27em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">∞<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.77em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">即可）。而当所有约束条件都满足时，则有<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-273" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 5.54em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.66em 1000em 3.32em -0.45em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.77em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-274" class="mrow"><span id="MathJax-Span-275" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">θ<span id="MathJax-Span-276" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">(<span id="MathJax-Span-277" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">w<span id="MathJax-Span-278" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">)<span id="MathJax-Span-279" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.27em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">=<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.39em; PADDING-RIGHT: 0.12em; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-280" class="mfrac"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0.49em; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.55em 1000em 2.34em -0.42em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px 0px 0px -0.18em; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.59em; PADDING-TOP: 0px; LEFT: 50%; border-image: initial"><span id="MathJax-Span-281" class="mn" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; FONT-SIZE: 13px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">1<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.18em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.55em 1000em 2.34em -0.45em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px 0px 0px -0.18em; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -1.8em; PADDING-TOP: 0px; LEFT: 50%; border-image: initial"><span id="MathJax-Span-282" class="mn" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; FONT-SIZE: 13px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">2<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.18em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(0.82em 1000em 1.21em -0.48em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -1.27em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="BORDER-LEFT: 0.49em solid; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; OVERFLOW-X: hidden; OVERFLOW-Y: hidden; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0.06em; VERTICAL-ALIGN: 0em; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 1.05em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span id="MathJax-Span-283" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.16em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">∥<span id="MathJax-Span-284" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">w<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-285" class="texatom"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-286" class="mrow"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-287" class="msubsup"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 1.06em; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.86em 1000em 3.18em -0.35em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.77em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span id="MathJax-Span-288" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">∥<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.77em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.66em; PADDING-TOP: 0px; LEFT: 0.61em; border-image: initial"><span id="MathJax-Span-289" class="mn" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; FONT-SIZE: 13px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">2<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.18em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.77em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">，亦即我们最初要最小化的量。因此，在要求约束条件得到满足的情况下最小化<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-290" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 2.69em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.66em 1000em 3.32em -0.36em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.77em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-291" class="mrow"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.12em; PADDING-RIGHT: 0.12em; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-292" class="mfrac"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0.49em; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.55em 1000em 2.34em -0.42em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px 0px 0px -0.18em; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.59em; PADDING-TOP: 0px; LEFT: 50%; border-image: initial"><span id="MathJax-Span-293" class="mn" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; FONT-SIZE: 13px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">1<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.18em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.55em 1000em 2.34em -0.45em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px 0px 0px -0.18em; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -1.8em; PADDING-TOP: 0px; LEFT: 50%; border-image: initial"><span id="MathJax-Span-294" class="mn" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; FONT-SIZE: 13px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">2<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.18em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(0.82em 1000em 1.21em -0.48em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -1.27em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="BORDER-LEFT: 0.49em solid; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; OVERFLOW-X: hidden; OVERFLOW-Y: hidden; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0.06em; VERTICAL-ALIGN: 0em; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 1.05em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span id="MathJax-Span-295" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.16em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">∥<span id="MathJax-Span-296" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">w<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-297" class="texatom"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-298" class="mrow"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-299" class="msubsup"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 1.06em; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.86em 1000em 3.18em -0.35em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.77em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span id="MathJax-Span-300" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">∥<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.77em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.66em; PADDING-TOP: 0px; LEFT: 0.61em; border-image: initial"><span id="MathJax-Span-301" class="mn" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; FONT-SIZE: 13px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">2<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.18em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.77em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"> 实际上等价于直接最小化<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-302" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 1.56em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.86em 1000em 3.19em -0.45em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.77em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-303" class="mrow"><span id="MathJax-Span-304" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">θ<span id="MathJax-Span-305" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">(<span id="MathJax-Span-306" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">w<span id="MathJax-Span-307" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">)<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.77em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"> （当然，这里也有约束条件，就是<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-308" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 8.02em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.95em 1000em 3.13em -0.45em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.77em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-309" class="mrow"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-310" class="msubsup"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 1.11em; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.95em 1000em 2.73em -0.45em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span id="MathJax-Span-311" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">α<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -1.87em; PADDING-TOP: 0px; LEFT: 0.77em; border-image: initial">i<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.02em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span id="MathJax-Span-313" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.27em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">≥<span id="MathJax-Span-314" class="mn" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.27em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">0<span id="MathJax-Span-315" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">,<span id="MathJax-Span-316" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0.16em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">i<span id="MathJax-Span-317" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.27em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">=<span id="MathJax-Span-318" class="mn" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.27em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">1<span id="MathJax-Span-319" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">,<span id="MathJax-Span-320" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.16em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">…<span id="MathJax-Span-321" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.16em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">,<span id="MathJax-Span-322" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0.16em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">n<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.77em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">），因为如果约束条件没有得到满足，<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-323" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 1.56em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.86em 1000em 3.19em -0.45em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.77em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-324" class="mrow"><span id="MathJax-Span-325" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">θ<span id="MathJax-Span-326" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">(<span id="MathJax-Span-327" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">w<span id="MathJax-Span-328" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">)<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.77em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"> 会等于无穷大，自然不会是我们所要求的最小值。具体写出来，我们现在的目标函数变成了：

这里用<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-382" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0.75em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(0.14em 1000em 1.35em -0.52em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -1em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-383" class="mrow"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-384" class="msubsup"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0.79em; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.95em 1000em 2.91em -0.52em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span id="MathJax-Span-385" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">p<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.54em; PADDING-TOP: 0px; LEFT: 0.34em; border-image: initial"><span id="MathJax-Span-386" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; FONT-SIZE: 13px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">∗<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.18em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 1em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">表示这个问题的最优值，这个问题和我们最初的问题是等价的。不过，现在我们来把最小和最大的位置交换一下（稍后，你将看到，当下面式子满足了一定的条件之后，这个式子d便是上式P 的对偶形式表示）：

当然，交换以后的问题不再等价于原问题，这个新问题的最优值用<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-425" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0.75em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(0.08em 1000em 1.17em -0.45em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -1em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-426" class="mrow"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-427" class="msubsup"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0.79em; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.7em 1000em 2.73em -0.45em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span id="MathJax-Span-428" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">d<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.6em; PADDING-TOP: 0px; LEFT: 0.34em; border-image: initial"><span id="MathJax-Span-429" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; FONT-SIZE: 13px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">∗<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.18em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 1em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">来表示。并，我们有<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-430" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 2.85em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.86em 1000em 3.13em -0.45em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.77em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-431" class="mrow"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-432" class="msubsup"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0.79em; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.7em 1000em 2.73em -0.45em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span id="MathJax-Span-433" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">d<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.6em; PADDING-TOP: 0px; LEFT: 0.34em; border-image: initial"><span id="MathJax-Span-434" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; FONT-SIZE: 13px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">∗<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.18em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span id="MathJax-Span-435" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.27em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">≤<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.27em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-436" class="msubsup"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0.79em; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.95em 1000em 2.91em -0.52em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span id="MathJax-Span-437" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">p<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.54em; PADDING-TOP: 0px; LEFT: 0.34em; border-image: initial"><span id="MathJax-Span-438" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; FONT-SIZE: 13px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">∗<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.18em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.77em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">，这在直观上也不难理解，最大值中最小的一个总也比最小值中最大的一个要大吧！总之，第二个问题的最优值<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-439" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0.75em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(0.08em 1000em 1.17em -0.45em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -1em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-440" class="mrow"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-441" class="msubsup"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0.79em; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.7em 1000em 2.73em -0.45em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span id="MathJax-Span-442" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">d<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.6em; PADDING-TOP: 0px; LEFT: 0.34em; border-image: initial"><span id="MathJax-Span-443" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; FONT-SIZE: 13px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">∗<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.18em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 1em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">在这里提供了一个第一个问题的最优值<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-444" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0.75em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(0.14em 1000em 1.35em -0.52em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -1em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-445" class="mrow"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-446" class="msubsup"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0.79em; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.95em 1000em 2.91em -0.52em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span id="MathJax-Span-447" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">p<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.54em; PADDING-TOP: 0px; LEFT: 0.34em; border-image: initial"><span id="MathJax-Span-448" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; FONT-SIZE: 13px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">∗<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.18em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 1em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">的一个下界，在满足某些条件的情况下，这两者相等，这个时候我们就可以通过求解第二个问题来间接地求解第一个问题。

注：上段说“在满足某些条件的情况下”，这所谓的“满足某些条件”就是要先满足Slater's Condition，进而就满足KKT条件。理由如下3点所述（观点来自frestyle）：

在convex problem中，d*和p*相同的条件是Slater's Condition，Slater's condition保证临界点saddle point存在。
至于KKT条件，首先原问题的最优值可以通过求Lagrangian的临界点saddle point（如果有的话）来得到，再者，KKT theorem里面进一步引入了更强的前提，也就是在满足Slater condition的同时（前面说了，Slater's condition保证临界点saddle point存在），f和gi都是可微的，这样saddle point不仅存在，而且能通过对Lagrangian求导得到，
所以KKT条件是一个点是最优解的条件，而不是d*=p*的条件，当然这个KKT条件对后边简化dual problem很关键。

那KKT条件的表现形式是什么呢？据*：KKT 条件的介绍，一般地，一个最优化数学模型能够表示成下列标准形式：

所谓 Karush-Kuhn-Tucker 最优化条件，就是指上式的最小点x*必须满足下面的条件：

经过论证，我们这里的问题是满足 KKT 条件的（首先已经满足Slater condition，再者f和gi也都是可微的，即L对w和b都可导），因此现在我们便转化为求解第二个问题。也就是说，现在，咱们的原问题通过满足一定的条件，已经转化成了对偶问题。而求解这个对偶学习问题，分为3个步骤，首先要让L(w，b，a)关于<nobr style="padding-bottom: 0px; border-right-width: 0px; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-left: 0px; padding-right: 0px; max-width: none; border-top-width: 0px; border-bottom-width: 0px; max-height: none; vertical-align: 0px; border-left-width: 0px; padding-top: 0px; border-image: initial; "><span id="MathJax-Span-454" class="math" style="padding-bottom: 0px; border-right-width: 0px; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-left: 0px; padding-right: 0px; display: inline; border-top-width: 0px; border-bottom-width: 0px; vertical-align: 0px; border-left-width: 0px; padding-top: 0px; border-image: initial; "><span style="position: relative; padding-bottom: 0px; border-right-width: 0px; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-left: 0px; width: 0.26em; padding-right: 0px; display: inline-block; font-family: MathJax_Main, MathJax_Size1, MathJax_AMS; border-top-width: 0px; border-bottom-width: 0px; height: 0px; font-size: 19px; vertical-align: 0px; border-left-width: 0px; padding-top: 0px; border-image: initial; "><span style="clip: rect(1.95em 1000em 2.73em -0.46em); position: absolute; padding-bottom: 0px; border-right-width: 0px; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-left: 0px; padding-right: 0px; border-top-width: 0px; border-bottom-width: 0px; vertical-align: 0px; border-left-width: 0px; top: -2.56em; padding-top: 0px; left: 0em; border-image: initial; "><span id="MathJax-Span-455" class="mrow" style="padding-bottom: 0px; border-right-width: 0px; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-left: 0px; padding-right: 0px; display: inline; border-top-width: 0px; border-bottom-width: 0px; vertical-align: 0px; border-left-width: 0px; padding-top: 0px; border-image: initial; "><span id="MathJax-Span-456" class="mi" style="font-family:MathJax_Math;padding-bottom: 0px; border-right-width: 0px; font-style: italic; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-left: 0px; padding-right: 0px; display: inline; border-top-width: 0px; border-bottom-width: 0px; vertical-align: 0px; border-left-width: 0px; padding-top: 0px; border-image: initial; ">w<span style="padding-bottom: 0px; border-right-width: 0px; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-left: 0px; width: 0px; padding-right: 0px; display: inline-block; border-top-width: 0px; border-bottom-width: 0px; height: 2.56em; vertical-align: 0px; border-left-width: 0px; padding-top: 0px; border-image: initial; ">和<nobr style="padding-bottom: 0px; border-right-width: 0px; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-left: 0px; padding-right: 0px; max-width: none; border-top-width: 0px; border-bottom-width: 0px; max-height: none; vertical-align: 0px; border-left-width: 0px; padding-top: 0px; border-image: initial; "><span id="MathJax-Span-457" class="math" style="padding-bottom: 0px; border-right-width: 0px; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-left: 0px; padding-right: 0px; display: inline; border-top-width: 0px; border-bottom-width: 0px; vertical-align: 0px; border-left-width: 0px; padding-top: 0px; border-image: initial; "><span style="position: relative; padding-bottom: 0px; border-right-width: 0px; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-left: 0px; width: 0.26em; padding-right: 0px; display: inline-block; font-family: MathJax_Main, MathJax_Size1, MathJax_AMS; border-top-width: 0px; border-bottom-width: 0px; height: 0px; font-size: 19px; vertical-align: 0px; border-left-width: 0px; padding-top: 0px; border-image: initial; "><span style="clip: rect(1.7em 1000em 2.73em -0.44em); position: absolute; padding-bottom: 0px; border-right-width: 0px; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-left: 0px; padding-right: 0px; border-top-width: 0px; border-bottom-width: 0px; vertical-align: 0px; border-left-width: 0px; top: -2.56em; padding-top: 0px; left: 0em; border-image: initial; "><span id="MathJax-Span-458" class="mrow" style="padding-bottom: 0px; border-right-width: 0px; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-left: 0px; padding-right: 0px; display: inline; border-top-width: 0px; border-bottom-width: 0px; vertical-align: 0px; border-left-width: 0px; padding-top: 0px; border-image: initial; "><span id="MathJax-Span-459" class="mi" style="font-family:MathJax_Math;padding-bottom: 0px; border-right-width: 0px; font-style: italic; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-left: 0px; padding-right: 0px; display: inline; border-top-width: 0px; border-bottom-width: 0px; vertical-align: 0px; border-left-width: 0px; padding-top: 0px; border-image: initial; ">b<span style="padding-bottom: 0px; border-right-width: 0px; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-left: 0px; width: 0px; padding-right: 0px; display: inline-block; border-top-width: 0px; border-bottom-width: 0px; height: 2.56em; vertical-align: 0px; border-left-width: 0px; padding-top: 0px; border-image: initial; ">最小化，然后求对α的极大，最后利用SMO算法求解对偶因子。

（1）、首先固定α，要让<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-449" class="math"><span style="CLIP: rect(1.74em 1000em 2.79em -0.45em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.61em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-450" class="mrow"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-451" class="texatom"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-452" class="mrow">L<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.61em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">关于<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-454" class="math"><span style="CLIP: rect(1.95em 1000em 2.73em -0.46em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-455" class="mrow">w<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">和<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-457" class="math"><span style="CLIP: rect(1.7em 1000em 2.73em -0.44em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-458" class="mrow">b<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">最小化，我们分别对w，b求偏导数，即令<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-460" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 2.58em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.86em 1000em 3.18em -0.44em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.77em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-461" class="mrow"><span id="MathJax-Span-462" class="mi" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">∂<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-463" class="texatom"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-464" class="mrow"><span id="MathJax-Span-465" class="mi" style="font-family:MathJax_Caligraphic;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">L<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-466" class="texatom"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-467" class="mrow"><span id="MathJax-Span-468" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">/<span id="MathJax-Span-469" class="mi" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">∂<span id="MathJax-Span-470" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">w<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.77em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">和<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-471" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 2.58em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.86em 1000em 3.18em -0.44em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.77em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-472" class="mrow"><span id="MathJax-Span-473" class="mi" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">∂<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-474" class="texatom"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-475" class="mrow"><span id="MathJax-Span-476" class="mi" style="font-family:MathJax_Caligraphic;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">L<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-477" class="texatom"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-478" class="mrow"><span id="MathJax-Span-479" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">/<span id="MathJax-Span-480" class="mi" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">∂<span id="MathJax-Span-481" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">b<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.77em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">等于零（对w求导结果的解释请看本文评论下第45楼回复~~，貌似博客系统故障，原来评论在第45楼的回复现在第45+81楼，即第126楼显示~~）：

带回上述的<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-558" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0.75em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.74em 1000em 2.79em -0.45em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.61em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-559" class="mrow"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-560" class="texatom"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-561" class="mrow"><span id="MathJax-Span-562" class="mi" style="font-family:MathJax_Caligraphic;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">L：<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.61em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="font-family:MathJax_Main, MathJax_Size1, MathJax_AMS;BORDER-LEFT: 0em solid; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; OVERFLOW-X: hidden; OVERFLOW-Y: hidden; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0.93em; VERTICAL-ALIGN: -0.02em; PADDING-TOP: 0px; border-image: initial"> 支持向量机通俗导论（理解SVM的三层境界），得到：

提醒：有读者可能会问上述推导过程如何而来？说实话，其具体推导过程是比较复杂的，如下图所示：支持向量机通俗导论（理解SVM的三层境界）

最后，得到：

如 jerrylead所说：“倒数第4步”推导到“倒数第3步”使用了线性代数的转置运算，由于ai和yi都是实数，因此转置后与自身一样。“倒数第3步”推导到“倒数第2步”使用了(a+b+c+…)(a+b+c+…)=aa+ab+ac+ba+bb+bc+…的乘法运算法则。最后一步是上一步的顺序调整。

从上面的最后一个式子，我们可以看出，此时的拉格朗日函数只包含了一个变量，那就是ai，然后下文的第2步，求出了ai便能求出w，和b，由此可见，上文第1.2节提出来的核心问题：分类函数w^T + b 也就可以轻而易举的求出来了。

由此看出，使用拉格朗日定理解凸最优化问题可以使用一个对偶变量表示，转换为对偶问题后，通常比原问题更容易处理，因为直接处理不等式约束是困难的，而对偶问题通过引入拉格朗日乘子(又称为对偶变量)来解。

（2）、求对α的极大，即是关于对偶变量dual variable<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-732" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0.7em; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.95em 1000em 2.73em -0.45em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-733" class="mrow"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-734" class="mi">α<span style="font-family:MathJax_Main, MathJax_Size1, MathJax_AMS;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="font-family:MathJax_Main, MathJax_Size1, MathJax_AMS;BORDER-LEFT: 0em solid; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; OVERFLOW-X: hidden; OVERFLOW-Y: hidden; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0.58em; VERTICAL-ALIGN: -0.01em; PADDING-TOP: 0px; border-image: initial">（下文将一直用粗体+下划线表示）的优化问题，从上面的式子得到：

(不得不提醒下读者：经过上面第一个步骤的求w和b，得到的拉格朗日函数式子已经没有了变量w，b，只有a，而反过来，求得的a将能导出w，b的解，最终得出分离超平面和分类决策函数。为何呢？因为如果求出了ai，根据支持向量机通俗导论（理解SVM的三层境界），即可求出w。然后通过，即可求出b)

如前面所说，这个问题有更加高效的优化算法，即我们常说的SMO算法。

（3）、序列最小最优化SMO算法。细心的读者读至上节末尾处，怎么求对偶变量α的值可能依然心存疑惑。实际上，关于a的求解过程即是我们常说的SMO算法，这里简要简单介绍下。

OK，当：

要解决的是在参数

上求最大值W的问题，至于支持向量机通俗导论（理解SVM的三层境界）

和

都是已知数（其中<img alt="" src="http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/141/0043.png" style="border: none; max-width: 100%; width: 15px; height: 14px; margin-right: -0.04em;">是一个参数，用于控制目标函数中两项（“寻找 margin 最大的超平面”和“保证数据点偏差量最小”）之间的权重。和上文最后的式子对比一下，可以看到唯一的区别就是现在 dual variable<img alt="" src="http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/141/03B1.png" style="border: none; max-width: 100%; width: 12px; height: 9px; margin-right: 0.04em;">多了一个上限<img alt="" src="http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/141/0043.png" style="border: none; max-width: 100%; width: 15px; height: 14px; margin-right: -0.04em;">，关于支持向量机通俗导论（理解SVM的三层境界）

的具体由来请查看下文第2.3节）。

按照坐标上升的思路，我们首先固定除支持向量机通俗导论（理解SVM的三层境界）

以外的所有参数，然后在支持向量机通俗导论（理解SVM的三层境界）

上求极值。等一下，这个思路有问题，因为如果固定支持向量机通俗导论（理解SVM的三层境界）

以外的所有参数，那么将不再是变量（可以由其他值推出），由上面式子中的最后一个条件可以得出：

因此，我们需要一次选取两个参数做优化，比如支持向量机通俗导论（理解SVM的三层境界）

和

，此时

可以由

和其他参数表示出来。这样回带到W中，W就只是关于支持向量机通俗导论（理解SVM的三层境界）

的函数了，可解。

这样，SMO的主要步骤如下：

意思是，

第一步选取一对和，选取方法使用启发式方法；
第二步，固定除和之外的其他参数，确定W极值条件下的，由表示。

SMO算法每次迭代只选出两个分量ai和aj进行调整，其它分量则保持固定不变，在得到解ai和aj之后，再用ai和aj改进其它分量。与通常的分解算法比较，尽管它可能需要更多的迭代次数，但每次迭代的计算量比较小，所以该算法表现出整理的快速收敛性，且不需要存储核矩阵，也没有矩阵运算。

更多请参看文末参考文献和推荐阅读中的条目6《支持向量机--算法、理论和扩展》和条目11《统计学习方法》的相关章节，或跳至下文3.4节。

2.1.2、线性不可分的情况

OK，为过渡到下节2.2节所介绍的核函数，让我们再来看看上述推导过程中得到的一些有趣的形式。首先就是关于我们的 hyper plane ，对于一个数据点<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-839" class="math"><span style="position: relative; padding: 0px; border-width: 0px; margin: 0px; width: 0.26em; display: inline-block; font-family: MathJax_Main, MathJax_Size1, MathJax_AMS; height: 0px; font-size: 19px; vertical-align: 0px; "><span style="CLIP: rect(1.95em 1000em 2.73em -0.45em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; 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BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span id="MathJax-Span-855" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">x<span id="MathJax-Span-856" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.22em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">+<span id="MathJax-Span-857" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0.22em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">b<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.77em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"> 算出结果然后根据其正负号来进行类别划分的。而前面的推导中我们得到

<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-866" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 6.4em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.82em 1000em 3.23em -0.46em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.77em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" class="mrow"><span class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"> <span style="CLIP: rect(1.82em 1000em 3.23em -0.46em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.77em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" class="mrow"><span class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"> <span style="CLIP: rect(1.82em 1000em 3.23em -0.46em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.77em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-867" class="mrow"><span id="MathJax-Span-868" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"> w<span id="MathJax-Span-869" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.27em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">=<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.27em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-870" class="texatom"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-871" class="mrow"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-872" class="munderover"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 2.46em; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.86em 1000em 3.18em -0.42em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.77em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span id="MathJax-Span-873" class="mo" style="font-family:MathJax_Size1;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0em; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">∑<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.77em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.54em 1000em 2.19em -0.47em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.5em; PADDING-TOP: 0px; LEFT: 1.2em; border-image: initial"><span id="MathJax-Span-874" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; FONT-SIZE: 13px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">n<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.02em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.55em 1000em 2.35em -0.47em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -1.89em; PADDING-TOP: 0px; LEFT: 1.2em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-875" class="texatom"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-876" class="mrow"><span id="MathJax-Span-877" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; FONT-SIZE: 13px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">i<span id="MathJax-Span-878" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; FONT-SIZE: 13px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">=<span id="MathJax-Span-879" class="mn" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; FONT-SIZE: 13px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">1<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.18em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-880" class="msubsup"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 1.11em; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.95em 1000em 2.73em -0.45em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span id="MathJax-Span-881" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">α<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -1.87em; PADDING-TOP: 0px; LEFT: 0.77em; border-image: initial"><span id="MathJax-Span-882" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; FONT-SIZE: 13px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">i<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.02em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-883" class="msubsup"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0.68em; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.95em 1000em 2.92em -0.46em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span id="MathJax-Span-884" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">y<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -1.78em; PADDING-TOP: 0px; LEFT: 0.34em; border-image: initial"><span id="MathJax-Span-885" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; FONT-SIZE: 13px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">i<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.02em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-886" class="msubsup"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0.68em; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.95em 1000em 2.73em -0.45em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span id="MathJax-Span-887" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">x<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -1.87em; PADDING-TOP: 0px; LEFT: 0.34em; border-image: initial"><span id="MathJax-Span-888" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; FONT-SIZE: 13px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">i<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.02em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.77em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">，

因此分类函数为：

这里的形式的有趣之处在于，对于新点x的预测，只需要计算它与训练数据点的内积即可（⋅,⋅表示向量内积），这一点至关重要，是之后使用 Kernel 进行非线性推广的基本前提。此外，所谓 Supporting Vector 也在这里显示出来——事实上，所有非 Supporting Vector 所对应的系数α都是等于零的，因此对于新点的内积计算实际上只要针对少量的“支持向量”而不是所有的训练数据即可。

为什么非支持向量对应的<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-1008" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0.7em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.95em 1000em 2.73em -0.45em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-1009" class="mrow"><span id="MathJax-Span-1010" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">α<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">等于零呢？直观上来理解的话，就是这些“后方”的点——正如我们之前分析过的一样，对超平面是没有影响的，由于分类完全有超平面决定，所以这些无关的点并不会参与分类问题的计算，因而也就不会产生任何影响了。

回忆一下我们2.1.1节中通过 Lagrange multiplier得到的目标函数：

注意到如果<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-1092" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0.64em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(0.39em 1000em 1.31em -0.45em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -1em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-1093" class="mrow"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-1094" class="msubsup"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0.68em; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.95em 1000em 2.73em -0.45em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span id="MathJax-Span-1095" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">x<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -1.87em; PADDING-TOP: 0px; LEFT: 0.34em; border-image: initial"><span id="MathJax-Span-1096" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; FONT-SIZE: 13px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">i<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.02em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 1em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">是支持向量的话，上式中红颜色的部分是等于 0 的（因为支持向量的 functional margin 等于 1 ），而对于非支持向量来说，functional margin 会大于 1 ，因此红颜色部分是大于零的，而<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-1097" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 1.07em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(0.39em 1000em 1.31em -0.45em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -1em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-1098" class="mrow"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-1099" class="msubsup"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 1.11em; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.95em 1000em 2.73em -0.45em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span id="MathJax-Span-1100" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">α<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -1.87em; PADDING-TOP: 0px; LEFT: 0.77em; border-image: initial"><span id="MathJax-Span-1101" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; FONT-SIZE: 13px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">i<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.02em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 1em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">又是非负的，为了满足最大化，<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-1102" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 1.07em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(0.39em 1000em 1.31em -0.45em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -1em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-1103" class="mrow"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-1104" class="msubsup"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 1.11em; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.95em 1000em 2.73em -0.45em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span id="MathJax-Span-1105" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">α<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -1.87em; PADDING-TOP: 0px; LEFT: 0.77em; border-image: initial"><span id="MathJax-Span-1106" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; FONT-SIZE: 13px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">i<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.02em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 1em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">必须等于 0 。这也就是这些非 Supporting Vector 的点的局限性。

从1.5节到上述所有这些东西，便得到了一个maximum margin hyper plane classifier，这就是所谓的支持向量机（Support Vector Machine）。当然，到目前为止，我们的 SVM 还比较弱，只能处理线性的情况，不过，在得到了对偶dual 形式之后，通过Kernel 推广到非线性的情况就变成了一件非常容易的事情了(相信，你还记得本节开头所说的：通过求解对偶问题得到最优解，这就是线性可分条件下支持向量机的对偶算法，这样做的优点在于：一者对偶问题往往更容易求解；二者可以自然的引入核函数，进而推广到非线性分类问题)。

2.2、核函数Kernel

咱们首先给出核函数的来头：

在上文中，我们已经了解到了SVM处理线性可分的情况，而对于非线性的情况，SVM 的处理方法是选择一个核函数<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-1071" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 2.42em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.86em 1000em 3.19em -0.43em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.77em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-1072" class="mrow"><span id="MathJax-Span-1073" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">κ<span id="MathJax-Span-1074" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">(<span id="MathJax-Span-1075" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">⋅<span id="MathJax-Span-1076" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">,<span id="MathJax-Span-1077" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.16em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">⋅<span id="MathJax-Span-1078" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">)<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.77em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="font-family:MathJax_Main, MathJax_Size1, MathJax_AMS;BORDER-LEFT: 0em solid; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; OVERFLOW-X: hidden; OVERFLOW-Y: hidden; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 1.29em; VERTICAL-ALIGN: -0.32em; PADDING-TOP: 0px; border-image: initial">，通过将数据映射到高维空间，来解决在原始空间中线性不可分的问题。由于核函数的优良品质，这样的非线性扩展在计算量上并没有比原来复杂多少，这一点是非常难得的。当然，这要归功于核方法——除了 SVM 之外，任何将计算表示为数据点的内积的方法，都可以使用核方法进行非线性扩展。

也就是说，Minsky和Papert早就在20世纪60年代就已经明确指出线性学习器计算能力有限。为什么呢？因为总体上来讲，现实世界复杂的应用需要有比线性函数更富有表达能力的假设空间，也就是说，目标概念通常不能由给定属性的简单线性函数组合产生，而是应该一般地寻找待研究数据的更为一般化的抽象特征。

而下文我们将具体介绍的核函数则提供了此种问题的解决途径，从下文你将看到，核函数通过把数据映射到高维空间来增加第一节所述的线性学习器的能力，使得线性学习器对偶空间的表达方式让分类操作更具灵活性和可操作性。我们知道，训练样例一般是不会独立出现的，它们总是以成对样例的内积形式出现，而用对偶形式表示学习器的优势在为在该表示中可调参数的个数不依赖输入属性的个数，通过使用恰当的核函数来替代内积，可以隐式得将非线性的训练数据映射到高维空间，而不增加可调参数的个数(当然，前提是核函数能够计算对应着两个输入特征向量的内积)。

1、简而言之：在线性不可分的情况下，支持向量机通过某种事先选择的非线性映射(核函数)将输入变量映射到一个高维特征空间，在这个空间中构造最优分类超平面。我们使用SVM进行数据集分类工作的过程首先是同预先选定的一些非线性映射将输入空间映射到高维特征空间(下图很清晰的表达了通过映射到高维特征空间，而把平面上本身不好分的非线性数据分了开来)：

使得在高维属性空间中有可能最训练数据实现超平面的分割，避免了在原输入空间中进行非线性曲面分割计算。SVM数据集形成的分类函数具有这样的性质：它是一组以支持向量为参数的非线性函数的线性组合，因此分类函数的表达式仅和支持向量的数量有关，而独立于空间的维度，在处理高维输入空间的分类时，这种方法尤其有效，其工作原理如下图所示：

2、具体点说：在我们遇到核函数之前，如果用原始的方法，那么在用线性学习器学习一个非线性关系，需要选择一个非线性特征集，并且将数据写成新的表达形式，这等价于应用一个固定的非线性映射，将数据映射到特征空间，在特征空间中使用线性学习器，因此，考虑的假设集是这种类型的函数：

这里ϕ：X->F是从输入空间到某个特征空间的映射，这意味着建立非线性学习器分为两步：

首先使用一个非线性映射将数据变换到一个特征空间F，
然后在特征空间使用线性学习器分类。

在上文我提到过对偶形式，而这个对偶形式就是线性学习器的一个重要性质，这意味着假设可以表达为训练点的线性组合，因此决策规则可以用测试点和训练点的内积来表示：

如果有一种方式可以在特征空间中直接计算内积〈φ(x_i·φ(x)〉，就像在原始输入点的函数中一样，就有可能将两个步骤融合到一起建立一个非线性的学习器，这样直接计算法的方法称为核函数方法，于是，核函数便横空出世了。

这里我直接给出一个定义：核是一个函数K，对所有x，z(-X，满足支持向量机通俗导论（理解SVM的三层境界）

，这里φ是从X到内积特征空间F的映射。

3、总而言之，举个简单直接点的例子，则是如果不是用核技术，就会先计算线性映射phy(x1)和phy(x2),然后计算这两个特征的内积，使用了核技术之后，先把phy(x1)和phy(x2)的通用表达式子：< phy(x1)，phy(x2) >=k( <x1,x2> )计算出来，注意到这里的< ， >表示内积，k( , )就是对应的核函数，这个表达往往非常简单，所以计算非常方便。

....

OK，接下来，咱们就进一步从外到里，来探探这个核函数的真面目。

2.2.1、如何处理非线性数据

在2.1节中我们介绍了线性情况下的支持向量机，它通过寻找一个线性的超平面来达到对数据进行分类的目的。不过，由于是线性方法，所以对非线性的数据就没有办法处理了。举个例子来说，则是如下图所示的两类数据，分别分布为两个圆圈的形状，这样的数据本身就是线性不可分的，你准备如何把这两类数据分开呢(下文将会有一个相应的三维空间图)？

支持向量机通俗导论（理解SVM的三层境界）

上图所述的这个数据集，就是用两个半径不同的圆圈加上了少量的噪音生成得到的，所以，一个理想的分界应该是一个“圆圈”而不是一条线（超平面）。如果用<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-10" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 1.34em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; 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BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">X<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.03em; PADDING-TOP: 0px; LEFT: 0.93em; border-image: initial"><span id="MathJax-Span-19" class="mn" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; FONT-SIZE: 13px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">2<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.18em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 1em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">来表示这个二维平面的两个坐标的话，我们知道一条二次曲线（圆圈是二次曲线的一种特殊情况）的方程可以写作这样的形式：

<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-20" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 22.56em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.73em 1000em 3.18em -0.45em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.77em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.77em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"></span></span></span></span></nobr>

注意上面的形式，如果我们构造另外一个五维的空间，其中五个坐标的值分别为<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-67" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 3.87em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.93em 1000em 3.08em -0.42em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.77em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-68" class="mrow"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-69" class="msubsup"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 1.22em; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.71em 1000em 2.72em -0.42em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span id="MathJax-Span-70" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">Z<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.03em; PADDING-TOP: 0px; LEFT: 0.77em; border-image: initial"><span id="MathJax-Span-71" class="mn" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; FONT-SIZE: 13px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">1<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.18em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span id="MathJax-Span-72" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.27em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">=<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.27em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-73" class="msubsup"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 1.38em; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.71em 1000em 2.72em -0.46em); 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PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 1.22em; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.71em 1000em 2.72em -0.42em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span id="MathJax-Span-79" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">Z<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.03em; PADDING-TOP: 0px; LEFT: 0.77em; border-image: initial"><span id="MathJax-Span-80" class="mn" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; FONT-SIZE: 13px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">2<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.18em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span id="MathJax-Span-81" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.27em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">=<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.27em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-82" class="msubsup"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 1.38em; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.71em 1000em 2.72em -0.46em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span id="MathJax-Span-83" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">X<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.55em 1000em 2.34em -0.45em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.59em; PADDING-TOP: 0px; LEFT: 0.93em; border-image: initial"><span id="MathJax-Span-84" class="mn" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; FONT-SIZE: 13px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">2<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.18em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.55em 1000em 2.34em -0.42em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -1.93em; PADDING-TOP: 0px; LEFT: 0.93em; border-image: initial"><span id="MathJax-Span-85" class="mn" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; FONT-SIZE: 13px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">1<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.18em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.77em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">,<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-86" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 3.87em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.93em 1000em 3.1em -0.42em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.77em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-87" class="mrow"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-88" class="msubsup"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 1.22em; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.71em 1000em 2.72em -0.42em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span id="MathJax-Span-89" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">Z<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.03em; PADDING-TOP: 0px; LEFT: 0.77em; border-image: initial"><span id="MathJax-Span-90" class="mn" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; FONT-SIZE: 13px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">3<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.18em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span id="MathJax-Span-91" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.27em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">=<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.27em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-92" class="msubsup"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 1.38em; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.71em 1000em 2.72em -0.46em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span id="MathJax-Span-93" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">X<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.03em; PADDING-TOP: 0px; LEFT: 0.93em; border-image: initial"><span id="MathJax-Span-94" class="mn" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; FONT-SIZE: 13px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">2<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.18em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.77em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">,<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-95" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 3.87em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.73em 1000em 3.18em -0.42em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.77em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-96" class="mrow"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-97" class="msubsup"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 1.22em; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.71em 1000em 2.72em -0.42em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span id="MathJax-Span-98" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">Z<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.03em; PADDING-TOP: 0px; LEFT: 0.77em; border-image: initial"><span id="MathJax-Span-99" class="mn" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; FONT-SIZE: 13px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">4<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.18em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span id="MathJax-Span-100" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.27em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">=<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.27em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-101" class="msubsup"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 1.38em; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.71em 1000em 2.72em -0.46em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span id="MathJax-Span-102" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">X<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.55em 1000em 2.34em -0.45em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.59em; PADDING-TOP: 0px; LEFT: 0.93em; border-image: initial"><span id="MathJax-Span-103" class="mn" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; FONT-SIZE: 13px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">2<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.18em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.55em 1000em 2.34em -0.45em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -1.93em; PADDING-TOP: 0px; LEFT: 0.93em; border-image: initial"><span id="MathJax-Span-104" class="mn" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; FONT-SIZE: 13px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">2<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.18em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.77em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">,<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-105" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 5.22em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.93em 1000em 3.1em -0.42em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.77em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-106" class="mrow"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-107" class="msubsup"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 1.22em; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.71em 1000em 2.72em -0.42em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span id="MathJax-Span-108" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">Z<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.03em; PADDING-TOP: 0px; LEFT: 0.77em; border-image: initial"><span id="MathJax-Span-109" class="mn" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; FONT-SIZE: 13px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">5<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.18em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span id="MathJax-Span-110" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.27em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">=<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.27em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-111" class="msubsup"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 1.38em; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.71em 1000em 2.72em -0.46em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span id="MathJax-Span-112" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">X<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.03em; PADDING-TOP: 0px; LEFT: 0.93em; border-image: initial"><span id="MathJax-Span-113" class="mn" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; FONT-SIZE: 13px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">1<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.18em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-114" class="msubsup"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 1.38em; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.71em 1000em 2.72em -0.46em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span id="MathJax-Span-115" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">X<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.03em; PADDING-TOP: 0px; LEFT: 0.93em; border-image: initial"><span id="MathJax-Span-116" class="mn" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; FONT-SIZE: 13px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">2<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.18em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.77em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">，那么显然，上面的方程在新的坐标系下可以写作：

关于新的坐标<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-147" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0.7em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.71em 1000em 2.72em -0.42em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-148" class="mrow"><span id="MathJax-Span-149" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">Z<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">，这正是一个 hyper plane 的方程！也就是说，如果我们做一个映射<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-174" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 5.49em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.73em 1000em 3.14em -0.44em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.77em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-175" class="mrow"><span id="MathJax-Span-176" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">ϕ<span id="MathJax-Span-177" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.27em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">:<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.27em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-178" class="msubsup"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 1.28em; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(2.25em 1000em 3.26em -0.46em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -3.1em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-179" class="texatom"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-180" class="mrow"><span id="MathJax-Span-181" class="mi" style="font-family:MathJax_AMS;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">R<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 3.1em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.59em; PADDING-TOP: 0px; LEFT: 0.82em; border-image: initial"><span id="MathJax-Span-182" class="mn" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; FONT-SIZE: 13px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">2<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.18em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span id="MathJax-Span-183" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.27em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">→<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.27em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-184" class="msubsup"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 1.28em; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(2.25em 1000em 3.26em -0.46em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -3.1em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-185" class="texatom"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-186" class="mrow"><span id="MathJax-Span-187" class="mi" style="font-family:MathJax_AMS;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">R<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 3.1em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.59em; PADDING-TOP: 0px; LEFT: 0.82em; border-image: initial"><span id="MathJax-Span-188" class="mn" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; FONT-SIZE: 13px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">5<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.18em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.77em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">，将<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-189" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0.86em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.71em 1000em 2.72em -0.46em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-190" class="mrow"><span id="MathJax-Span-191" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">X<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">按照上面的规则映射为<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-192" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0.7em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.71em 1000em 2.72em -0.42em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-193" class="mrow"><span id="MathJax-Span-194" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">Z<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">，那么在新的空间中原来的数据将变成线性可分的，从而使用之前我们推导的线性分类算法就可以进行处理了。这正是 Kernel 方法处理非线性问题的基本思想。

2.2.2、特征空间的隐式映射：核函数

再进一步描述 Kernel 的细节之前，不妨再来看看这个例子映射过后的直观例子。当然，你我可能无法把 5 维空间画出来，不过由于我这里生成数据的时候就是用了特殊的情形，具体来说，我这里的超平面实际的方程是这个样子（圆心在<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-195" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 1.34em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(0.15em 1000em 1.31em -0.46em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -1em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-196" class="mrow"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-197" class="msubsup"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 1.38em; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.71em 1000em 2.72em -0.46em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span id="MathJax-Span-198" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">X<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.03em; PADDING-TOP: 0px; LEFT: 0.93em; border-image: initial"><span id="MathJax-Span-199" class="mn" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; FONT-SIZE: 13px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">2<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.18em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 1em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">轴上的一个正圆）：

因此我只需要把它映射到<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-230" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 3.87em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.73em 1000em 3.18em -0.42em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.77em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-231" class="mrow"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-232" class="msubsup"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 1.22em; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.71em 1000em 2.72em -0.42em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span id="MathJax-Span-233" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">Z<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.03em; PADDING-TOP: 0px; LEFT: 0.77em; border-image: initial"><span id="MathJax-Span-234" class="mn" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; FONT-SIZE: 13px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">1<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.18em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span id="MathJax-Span-235" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.27em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">=<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.27em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-236" class="msubsup"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 1.38em; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.71em 1000em 2.72em -0.46em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span id="MathJax-Span-237" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">X<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.55em 1000em 2.34em -0.45em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.59em; PADDING-TOP: 0px; LEFT: 0.93em; border-image: initial"><span id="MathJax-Span-238" class="mn" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; FONT-SIZE: 13px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">2<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.18em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.55em 1000em 2.34em -0.42em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -1.93em; PADDING-TOP: 0px; LEFT: 0.93em; border-image: initial"><span id="MathJax-Span-239" class="mn" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; FONT-SIZE: 13px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">1<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.18em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.77em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">,<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-240" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 3.87em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.73em 1000em 3.18em -0.42em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.77em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-241" class="mrow"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-242" class="msubsup"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 1.22em; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.71em 1000em 2.72em -0.42em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span id="MathJax-Span-243" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">Z<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.03em; PADDING-TOP: 0px; LEFT: 0.77em; border-image: initial"><span id="MathJax-Span-244" class="mn" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; FONT-SIZE: 13px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">2<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.18em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span id="MathJax-Span-245" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.27em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">=<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.27em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-246" class="msubsup"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 1.38em; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.71em 1000em 2.72em -0.46em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span id="MathJax-Span-247" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">X<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.55em 1000em 2.34em -0.45em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.59em; PADDING-TOP: 0px; LEFT: 0.93em; border-image: initial"><span id="MathJax-Span-248" class="mn" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; FONT-SIZE: 13px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">2<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.18em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.55em 1000em 2.34em -0.45em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -1.93em; PADDING-TOP: 0px; LEFT: 0.93em; border-image: initial"><span id="MathJax-Span-249" class="mn" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; FONT-SIZE: 13px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">2<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.18em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.77em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">,<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-250" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 3.87em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.93em 1000em 3.1em -0.42em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.77em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-251" class="mrow"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-252" class="msubsup"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 1.22em; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.71em 1000em 2.72em -0.42em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span id="MathJax-Span-253" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">Z<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.03em; PADDING-TOP: 0px; LEFT: 0.77em; border-image: initial"><span id="MathJax-Span-254" class="mn" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; FONT-SIZE: 13px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">3<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.18em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span id="MathJax-Span-255" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.27em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">=<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0.27em; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-256" class="msubsup"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 1.38em; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.71em 1000em 2.72em -0.46em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span id="MathJax-Span-257" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">X<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.03em; PADDING-TOP: 0px; LEFT: 0.93em; border-image: initial"><span id="MathJax-Span-258" class="mn" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; FONT-SIZE: 13px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">2<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.18em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.77em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">这样一个三维空间中即可，下图即是映射之后的结果，将坐标轴经过适当的旋转，就可以很明显地看出，数据是可以通过一个平面来分开的(pluskid：下面的gif 动画，先用 Matlab 画出一张张图片，再用 Imagemagick 拼贴成)：

现在让我们再回到 SVM 的情形，假设原始的数据时非线性的，我们通过一个映射<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-259" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 1.72em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.86em 1000em 3.19em -0.44em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.77em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-260" class="mrow"><span id="MathJax-Span-261" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">ϕ<span id="MathJax-Span-262" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">(<span id="MathJax-Span-263" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">⋅<span id="MathJax-Span-264" class="mo" style="font-family:MathJax_Main;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">)<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.77em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">将其映射到一个高维空间中，数据变得线性可分了，这个时候，我们就可以使用原来的推导来进行计算，只是所有的推导现在是在新的空间，而不是原始空间中进行。当然，推导过程也并不是可以简单地直接类比的，例如，原本我们要求超平面的法向量<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-265" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0.75em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.95em 1000em 2.73em -0.46em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-266" class="mrow"><span id="MathJax-Span-267" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">w<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">，但是如果映射之后得到的新空间的维度是无穷维的（确实会出现这样的情况，比如后面会提到的高斯核Gaussian Kernel ），要表示一个无穷维的向量描述起来就比较麻烦。于是我们不妨先忽略过这些细节，直接从最终的结论来分析，回忆一下，我们上一次2.1节中得到的最终的分类函数是这样的：

现在则是在映射过后的空间，即：

而其中的<nobr style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; MAX-WIDTH: none; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; MAX-HEIGHT: none; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-338" class="math"><span style="POSITION: relative; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0.7em; PADDING-RIGHT: 0px; DISPLAY: inline-block; FONT-FAMILY: MathJax_Main, MathJax_Size1, MathJax_AMS; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 0px; FONT-SIZE: 19px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial"><span style="CLIP: rect(1.95em 1000em 2.73em -0.45em); POSITION: absolute; PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; TOP: -2.56em; PADDING-TOP: 0px; LEFT: 0em; border-image: initial"><span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial" id="MathJax-Span-339" class="mrow"><span id="MathJax-Span-340" class="mi" style="font-family:MathJax_Math;PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; FONT-STYLE: italic; MARGIN: 0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; DISPLAY: inline; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">α<span style="PADDING-BOTTOM: 0px; BORDER-RIGHT-WIDTH: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 0px; PADDING-RIGHT: 0px; DISPLAY: inline-block; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; HEIGHT: 2.56em; VERTICAL-ALIGN: 0px; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px; border-image: initial">也是通过求解如下 dual 问题而得到的：

这样一来问题就解决了吗？似乎是的：拿到非线性数据，就找一个映射 <img style="WIDTH: 12px; HEIGHT: 18px; VERTICAL-ALIGN: -4px; MARGIN-RIGHT: 0.01em" alt="" src="http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/141/03D5.png"><img style="WIDTH: 7px; HEIGHT: 21px; VERTICAL-ALIGN: -5px; MARGIN-RIGHT: 0.06em" alt="" src="http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/141/0028.png"><img style="WIDTH: 4px; HEIGHT: 4px; VERTICAL-ALIGN: 3px; MARGIN-RIGHT: 0.08em" alt="" src="http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/141/22C5.png"><img style="WIDTH: 6px; HEIGHT: 21px; VERTICAL-ALIGN: -5px; MARGIN-RIGHT: 0.1em" alt="" src="http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/141/0029.png"> ，然后一股脑把原来的数据映射到新空间中，再做线性 SVM 即可。不过事实上没有这么简单！其实刚才的方法稍想一下就会发现有问题：在最初的例子里，我们对一个二维空间做映射，选择的新空间是原始空间的所有一阶和二阶的组合，得到了五个维度；如果原始空间是三维，那么我们会得到 19 维的新空间，这个数目是呈爆炸性增长的，这给 <img style="WIDTH: 12px; HEIGHT: 18px; VERTICAL-ALIGN: -4px; MARGIN-RIGHT: 0.01em" alt="" src="http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/141/03D5.png"><img style="WIDTH: 7px; HEIGHT: 21px; VERTICAL-ALIGN: -5px; MARGIN-RIGHT: 0.06em" alt="" src="http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/141/0028.png"><img style="WIDTH: 4px; HEIGHT: 4px; VERTICAL-ALIGN: 3px; MARGIN-RIGHT: 0.08em" alt="" src="http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/141/22C5.png"><img style="WIDTH: 6px; HEIGHT: 21px; VERTICAL-ALIGN: -5px; MARGIN-RIGHT: 0.1em" alt="" src="http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/141/0029.png"> 的计算带来了非常大的困难，而且如果遇到无穷维的情况，就根本无从计算了。所以就需要 Kernel 出马了。

不妨还是从最开始的简单例子出发，设两个向量支持向量机通俗导论（理解SVM的三层境界）和，而即是到前面2.2.1节说的五维空间的映射，因此映射过后的内积为：

（公式说明：上面的这两个推导过程中，所说的前面的五维空间的映射，这里说的前面便是文中2.2.1节的所述的映射方式，仔细看下2.2.1节的映射规则，再看那第一个推导，其实就是计算x1，x2各自的内积，然后相乘相加即可，第二个推导则是直接平方，去掉括号，也很容易推出来）

另外，我们又注意到：

二者有很多相似的地方，实际上，我们只要把某几个维度线性缩放一下，然后再加上一个常数维度，具体来说，上面这个式子的计算结果实际上和映射

之后的内积支持向量机通俗导论（理解SVM的三层境界）的结果是相等的，那么区别在于什么地方呢？

一个是映射到高维空间中，然后再根据内积的公式进行计算；
而另一个则直接在原来的低维空间中进行计算，而不需要显式地写出映射后的结果。

（公式说明：上面之中，最后的两个式子，第一个算式，是带内积的完全平方式，可以拆开，然后，通过凑一个得到，第二个算式，也是根据第一个算式凑出来的）

回忆刚才提到的映射的维度爆炸，在前一种方法已经无法计算的情况下，后一种方法却依旧能从容处理，甚至是无穷维度的情况也没有问题。

我们把这里的计算两个向量在隐式映射过后的空间中的内积的函数叫做核函数 (Kernel Function) ，例如，在刚才的例子中，我们的核函数为：

核函数能简化映射空间中的内积运算——刚好“碰巧”的是，在我们的 SVM 里需要计算的地方数据向量总是以内积的形式出现的。对比刚才我们上面写出来的式子，现在我们的分类函数为：

其中 <img style="WIDTH: 12px; HEIGHT: 9px; MARGIN-RIGHT: 0.04em" alt="" src="http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/141/03B1.png"> 由如下 dual 问题计算而得：

这样一来计算的问题就算解决了，避开了直接在高维空间中进行计算，而结果却是等价的！当然，因为我们这里的例子非常简单，所以我可以手工构造出对应于支持向量机通俗导论（理解SVM的三层境界）的核函数出来，如果对于任意一个映射，想要构造出对应的核函数就很困难了。

2.2.3、几个核函数

最理想的情况下，我们希望知道数据的具体形状和分布，从而得到一个刚好可以将数据映射成线性可分的支持向量机通俗导论（理解SVM的三层境界），然后通过这个得出对应的进行内积计算。然而，第二步通常是非常困难甚至完全没法做的。不过，由于第一步也是几乎无法做到，因为对于任意的数据分析其形状找到合适的映射本身就不是什么容易的事情，所以，人们通常都是“胡乱”选择映射的，所以，根本没有必要精确地找出对应于映射的那个核函数，而只需要“胡乱”选择一个核函数即可——我们知道它对应了某个映射，虽然我们不知道这个映射具体是什么。由于我们的计算只需要核函数即可，所以我们也并不关心也没有必要求出所对应的映射的具体形式。

当然，也并不是任意的二元函数都可以作为核函数，所以除非某些特殊的应用中可能会构造一些特殊的核（例如用于文本分析的文本核，注意其实使用了 Kernel 进行计算之后，其实完全可以去掉原始空间是一个向量空间的假设了，只要核函数支持，原始数据可以是任意的“对象”——比如文本字符串），通常人们会从一些常用的核函数中选择（根据问题和数据的不同，选择不同的参数，实际上就是得到了不同的核函数），例如：

多项式核，显然刚才我们举的例子是这里多项式核的一个特例（R = 1，d = 2）。虽然比较麻烦，而且没有必要，不过这个核所对应的映射实际上是可以写出来的，该空间的维度是，其中 <img style="WIDTH: 17px; HEIGHT: 9px; MARGIN-RIGHT: 0.02em" alt="" src="http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/141/006D.png"> 是原始空间的维度。
高斯核，这个核就是最开始提到过的会将原始空间映射为无穷维空间的那个家伙。不过，如果选得很大的话，高次特征上的权重实际上衰减得非常快，所以实际上（数值上近似一下）相当于一个低维的子空间；反过来，如果选得很小，则可以将任意的数据映射为线性可分——当然，这并不一定是好事，因为随之而来的可能是非常严重的过拟合问题。不过，总的来说，通过调控参数，高斯核实际上具有相当高的灵活性，也是使用最广泛的核函数之一。下图所示的例子便是把低维线性不可分的数据通过高斯核函数映射到了高维空间：
线性核，这实际上就是原始空间中的内积。这个核存在的主要目的是使得“映射后空间中的问题”和“映射前空间中的问题”两者在形式上统一起来了(意思是说，咱们有的时候，写代码，或写公式的时候，只要写个模板或通用表达式，然后再代入不同的核，便可以了，于此，便在形式上统一了起来，不用再分别写一个线性的，和一个非线性的)。

2.2.4、核函数的本质

上面说了这么一大堆，读者可能还是没明白核函数到底是个什么东西？我再简要概括下，即以下三点：

实际中，我们会经常遇到线性不可分的样例，此时，我们的常用做法是把样例特征映射到高维空间中去(如上文2.2节最开始的那幅图所示，映射到高维空间后，相关特征便被分开了，也就达到了分类的目的)；
但进一步，如果凡是遇到线性不可分的样例，一律映射到高维空间，那么这个维度大小是会高到可怕的(如上文中19维乃至无穷维的例子)。那咋办呢？
此时，核函数就隆重登场了，核函数的价值在于它虽然也是讲特征进行从低维到高维的转换，但核函数绝就绝在它事先在低维上进行计算，而将实质上的分类效果表现在了高维上，也就如上文所说的避免了直接在高维空间中的复杂计算。

此外，读者来信，向我提出了一个问题：

In a SVM without using slack variables, if we remove one of the support vectors from the training set, what will happen to the maximal margin? List all the possibilities and give a sample for each possible situation, i.e., generate a training set, indicate which point is to be removed and clarify the change of the maximal margin.

大意是：在没有松弛变量的svm中，如果我们移去训练集中的一个支持向量，那最大margin会怎么变化呢？列举出所有的可能，每种情况给出一个例子。也就是，举出一个训练集，指出移去哪个点，并指明最大margin怎么变。

解答：你可能会说最终的maximal margin会变大，会变小，或不变？但一切依据呢？与其胡乱猜测，不如实际推导.计算.证明！接下来，咱们回顾下上文「所有图片截取自上文」：

maximummargin就是
支持向量机通俗导论（理解SVM的三层境界）

remove后(w,b)变了，w一变，maximummargin自然也就会变了，至于如何变，请读者继续计算看具体结果。此外，还可以看看这里的分析：http://www.cs.berkeley.edu/~russell/classes/cs194/f11/assignments/a2/a2-solution.pdf。

与此同时，读者自会发现到：上文中很大一部分篇幅都是在阐述怎么求及优化这个最大间隔分类超平面，包括后面的
支持向量机通俗导论（理解SVM的三层境界）

下面，就是这两个步骤，第一步：

第二步、对α求极大如下所示：

再到后来，有了核函数，解决高维空间计算的复杂性问题，再到对对偶因子α的求解：SMO算法，通篇都不过是为了寻找一个最大间隔分类超平面，也即寻找一个分类函数，然后对这个分类函数各个因子的求解，如w，b，对偶因子α..
支持向量机通俗导论（理解SVM的三层境界）

自此，你看到，上文中各个知识点是可以联系起来的，每一个步骤也都是一步一步推导下来的。

2.3、使用松弛变量处理 outliers 方法

在本文第一节最开始讨论支持向量机的时候，我们就假定，数据是线性可分的，亦即我们可以找到一个可行的超平面将数据完全分开。后来为了处理非线性数据，在上文2.2节使用 Kernel 方法对原来的线性 SVM 进行了推广，使得非线性的的情况也能处理。虽然通过映射 <img style="WIDTH: 12px; HEIGHT: 18px; VERTICAL-ALIGN: -4px; MARGIN-RIGHT: 0.01em" alt="" src="http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/141/03D5.png"><img style="WIDTH: 7px; HEIGHT: 21px; VERTICAL-ALIGN: -5px; MARGIN-RIGHT: 0.06em" alt="" src="http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/141/0028.png"><img style="WIDTH: 4px; HEIGHT: 4px; VERTICAL-ALIGN: 3px; MARGIN-RIGHT: 0.08em" alt="" src="http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/141/22C5.png"><img style="WIDTH: 6px; HEIGHT: 21px; VERTICAL-ALIGN: -5px; MARGIN-RIGHT: 0.1em" alt="" src="http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/141/0029.png"> 将原始数据映射到高维空间之后，能够线性分隔的概率大大增加，但是对于某些情况还是很难处理。例如可能并不是因为数据本身是非线性结构的，而只是因为数据有噪音。对于这种偏离正常位置很远的数据点，我们称之为 outlier ，在我们原来的 SVM 模型里，outlier 的存在有可能造成很大的影响，因为超平面本身就是只有少数几个 support vector 组成的，如果这些 support vector 里又存在 outlier 的话，其影响就很大了。例如下图：

支持向量机通俗导论（理解SVM的三层境界）

用黑圈圈起来的那个蓝点是一个 outlier ，它偏离了自己原本所应该在的那个半空间，如果直接忽略掉它的话，原来的分隔超平面还是挺好的，但是由于这个 outlier 的出现，导致分隔超平面不得不被挤歪了，变成途中黑色虚线所示（这只是一个示意图，并没有严格计算精确坐标），同时 margin 也相应变小了。当然，更严重的情况是，如果这个 outlier 再往右上移动一些距离的话，我们将无法构造出能将数据分开的超平面来。

为了处理这种情况，SVM 允许数据点在一定程度上偏离一下超平面。例如上图中，黑色实线所对应的距离，就是该 outlier 偏离的距离，如果把它移动回来，就刚好落在原来的超平面上，而不会使得超平面发生变形了。具体来说，原来的约束条件

<nobr><span id="MathJax-Span-7" class="math"><span style="POSITION: relative; WIDTH: 14.79em; DISPLAY: inline-block; HEIGHT: 0pt; FONT-SIZE: 129%"><span style="CLIP: rect(1.26em 1000em 2.77em -0.46em); POSITION: absolute; TOP: -2.34em; LEFT: 0em"><span id="MathJax-Span-8" class="mrow"><span style="PADDING-LEFT: 0.16em" id="MathJax-Span-32" class="mi"></span></span><span style="WIDTH: 0pt; DISPLAY: inline-block; HEIGHT: 2.34em"></span></span></span><span style="BORDER-LEFT: 0em solid; WIDTH: 0pt; DISPLAY: inline-block; HEIGHT: 1.54em; VERTICAL-ALIGN: -0.34em; OVERFLOW: hidden"></span></span></nobr>

现在变成

<nobr><span id="MathJax-Span-33" class="math"><span style="position: relative; width: 17em; display: inline-block; height: 0pt; font-size: 129%;"><span style="CLIP: rect(1.26em 1000em 2.77em -0.46em); POSITION: absolute; TOP: -2.34em; LEFT: 0em"><span style="WIDTH: 0pt; DISPLAY: inline-block; HEIGHT: 2.34em"></span></span></span><span style="BORDER-LEFT: 0em solid; WIDTH: 0pt; DISPLAY: inline-block; HEIGHT: 1.54em; VERTICAL-ALIGN: -0.34em; OVERFLOW: hidden"></span></span></nobr>

其中支持向量机通俗导论（理解SVM的三层境界）称为松弛变量 (slack variable) ，对应数据点允许偏离的 functional margin 的量。当然，如果我们运行任意大的话，那任意的超平面都是符合条件的了。所以，我们在原来的目标函数后面加上一项，使得这些支持向量机通俗导论（理解SVM的三层境界）的总和也要最小：

其中 <img style="WIDTH: 15px; HEIGHT: 14px; MARGIN-RIGHT: -0.04em" alt="" src="http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/141/0043.png"> 是一个参数，用于控制目标函数中两项（“寻找 margin 最大的超平面”和“保证数据点偏差量最小”）之间的权重。注意，其中 <img style="WIDTH: 9px; HEIGHT: 18px; VERTICAL-ALIGN: -4px; MARGIN-RIGHT: 0em" alt="" src="http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/141/03BE.png"> 是需要优化的变量（之一），而 <img style="WIDTH: 15px; HEIGHT: 14px; MARGIN-RIGHT: -0.04em" alt="" src="http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/141/0043.png"> 是一个事先确定好的常量。完整地写出来是这个样子：

用之前的方法将限制或约束条件加入到目标函数中，得到新的拉格朗日函数，如下所示：

分析方法和前面一样，转换为另一个问题之后，我们先让支持向量机通俗导论（理解SVM的三层境界）针对、和最小化：

将<img style="WIDTH: 14px; HEIGHT: 9px; MARGIN-RIGHT: 0.02em" alt="" src="http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/141/0077.png"> 带回 <img style="WIDTH: 13px; HEIGHT: 14px; MARGIN-RIGHT: 0.03em" alt="" src="http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Caligraphic/Regular/141/004C.png"> 并化简，得到和原来一样的目标函数：

不过，由于我们得到支持向量机通俗导论（理解SVM的三层境界）而又有（作为 Lagrange multiplier 的条件），因此有，所以整个 dual 问题现在写作：

把前后的结果对比一下：

可以看到唯一的区别就是现在 dual variable 支持向量机通俗导论（理解SVM的三层境界）

多了一个上限 <img style="WIDTH: 15px; HEIGHT: 14px; MARGIN-RIGHT: -0.04em" alt="" src="http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/141/0043.png"> 。而 Kernel 化的非线性形式也是一样的，只要把支持向量机通俗导论（理解SVM的三层境界）

换成

即可。这样一来，一个完整的，可以处理线性和非线性并能容忍噪音和 outliers 的支持向量机才终于介绍完毕了。

2.4、小结

综上所述，对于一个线性可分的两类问题，我们可以通过g(x)=w^T*x+w0=0确定一个分类面把这两类分开。我们的目的是为了建立一个这样的分类面，事实上这样的分类面不是唯一确定的。在感知器里我们通过梯度下降法优化出一个分类器初始值的选择、迭代步长等的不同得到的分类器也不同。那么我们需要在这些分离器里找一个最好的。

如图所示，我们可以认为direction2的分类效果要比direction1的分类效果要好，因为direction2的裕量比direction1大。我们需要在这各个分类器中选择一个最优的。SVM是根据统计学习理论依照结构风险最小化的原则提出的，要求实现两个目的：1）两类问题能够分开（经验风险最小）2）margin最大化（风险上界最小）既是在保证风险最小的子集中选择经验风险最小的函数。

把样本到分类面的距离支持向量机通俗导论（理解SVM的三层境界）

进行归一化处理后我们得到里分类面最近的样本g(x)=1。

这样我们就有边界margin：支持向量机通俗导论（理解SVM的三层境界），这里满足这样条件的样本点就是我们所谓的支持向量。

这样我们就转化为一个优化问题SergiosTheodoridis：

建立拉格朗日方程并引入KKT条件得到：

由上式得到判别函数式：

对于不是线性可分的问题，我们可以通过加入松弛子C来解决：

由以上讨论我们得到判别函数只与向量的內积有关，因此我们可以选择一个非线性变换将x映射到高维空间，在低维空间不可分的问题映射到高维空间后就有可能是线性可分的。这里我们不需要知道是什么形式的只需要关注內积运算即可。由此，可以通过构造核函数实现：

这里的核函数的选择没有特别的方式，在Chih-WeiHsu中推荐使用径向基函数。

故不准确的说，SVM它本质上即是一个分类方法，用w^T+b定义分类函数，于是求w、b，为寻最大间隔，引出1/2||w||^2，继而引入拉格朗日因子，化为对单一因数对偶变量a的求解（求解过程中会涉及到一系列最优化或凸二次规划等问题），如此，求w.b与求a等价，而求a的解法即为SMO，至于核函数，是为处理非线性情况，若直接映射到高维计算恐维度爆炸，故在低维计算，等效高维表现。

OK，理解到这第二层，已经能满足绝大部分人一窥SVM原理的好奇心，然对于那些想在证明层面理解SVM的则还很不够，但进入第三层理解境界之前，你必须要有比较好的数理基础和逻辑证明能力，不然你会跟我一样，吃不少苦头的。

第三层、证明SVM

说实话，凡是涉及到要证明的东西.理论，便一般不是怎么好惹的东西。绝大部分时候，看懂一个东西不难，但证明一个东西则需要点数学功底，进一步，证明一个东西也不是特别难，难的是从零开始发明创造这个东西的时候，则显艰难(因为任何时代，大部分人的研究所得都不过是基于前人的研究成果，前人所做的是开创性工作，而这往往是最艰难最有价值的，他们被称为真正的先驱。牛顿也曾说过，他不过是站在巨人的肩上。你，我则更是如此)。

正如陈希孺院士在他的著作「数理统计学简史」的第4章、最小二乘法中所讲：在科研上诸多观念的革新和突破是有着很多的不易的，或许某个定理在某个时期由某个人点破了，现在的我们看来一切都是理所当然，但在一切没有发现之前，可能许许多多的*学者毕其功于一役，耗尽一生，努力了几十年最终也是无功而返。

OK，以下内容基本属于自己在看支持向量机导论一书的理解，包括自己对一些证明的理解，还是读书笔记。

本部分导述

3.1节线性学习器一部分中，主要阐述3个东西，感知机算法，松弛变量，及最小二乘理论，同时，基本上是贴的用相机拍的照片（为什么？懒）；
3.2节、核函数特征空间；
3.3节、SMO算法；
3.4节、简略谈谈SVM的应用。

3.1、线性学习器

3.1.1、感知机算法

这个感知机算法是1956年提出的，年代久远，依然影响着当今，当然，可以肯定的是，此算法亦非最优，后续会有更详尽阐述。不过，有一点，你必须清楚，这个算法是为了干嘛的：不断的训练试错以期寻找一个合适的超平面(是的，就这么简单)。

下图算是读书笔记，若看不清，可以点击查看：大图（后面一大堆的证明不过是为了证明上述感知机算法在线性条件下收敛，说白了，为了得到一个界，不至于无穷循环下去）。

3.1.2、松弛变量

（点击查看大图：左图，右图）

针对上面左图的说明：我们知道，当分类出现了误差，要么就是被误分，要么就是没有以正常的间隔被分开：

被误分。如上面左图所示，如果一切正常的话，那么xi出现的位置不该是那里，而是该左图中左边那个箭头所示，被“拉回去”，而既然出现在了这个不正常的位置，那么有什么后果内，这就导致了所谓的被误分，使得最终的松弛变量>0；同理，oj也不该出现在那个位置，而应该被“拉回去”。
没有以正常的间隔被分开。还是如上面所示左图，xk显然没有被以正常的间隔分开，而是过于靠近超平面。

也就是说，对于正常的间隔大于r的点，这个值为0，而对于误分点，松弛变量大于r。同时，数据噪声使得单个点的间隔变小以致成为负值，因此，松弛变量可以作为数据噪声的度量，所以说，从松弛变量得来的方法适合处理噪声数据。

还是不懂？别忘了，本文第一部分1.4节中，有：“这样一来，我们的 maximum margin classifier 的目标函数可以定义为：

当然，还需要满足一些条件，根据 margin 的定义，我们有

3.2、最小二乘法

既然本节开始之前提到了最小二乘法，那么下面稍微简单阐述下。

我们口头中经常说：一般来说，平均来说。如平均来说，不吸烟的健康优于吸烟者，之所以要加“平均”二字，是因为凡事皆有例外，总存在某个特别的人他吸烟但由于经常锻炼所以他的健康状况可能会优于他身边不吸烟的朋友。而最小二乘法的一个最简单的例子便是算术平均。

最小二乘法（又称最方法）是一种数学优化技术。它通过最小化误差的平方和寻找数据的最佳函数匹配。利用最小二乘法可以简便地求得未知的数据，并使得这些求得的数据与实际数据之间误差的平方和为最小。用函数表示为：

使误差「所谓误差，当然是观察值与实际真实值的差量」平方和达到最小以寻求估计值的方法，就叫做最小二乘法，用最小二乘法得到的估计，叫做最小二乘估计。当然，取平方和作为目标函数只是众多可取的方法之一。

最小二乘法的一般形式可表示为：

有效的最小二乘法是勒让德在 1805 年发表的，基本思想就是认为测量中有误差，所以所有方程的累积误差为

我们求解出导致累积误差最小的参数即可。

勒让德在论文中对最小二乘法的优良性做了几点说明：

最小二乘使得误差平方和最小，并在各个方程的误差之间建立了一种平衡，从而防止某一个极端误差取得支配地位
计算中只要求偏导后求解线性方程组，计算过程明确便捷
最小二乘可以导出算术平均值作为估计值

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background-repeat: initial initial; ">为n次测量值, 每次测量的误差为<nobr style="border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; max-width: 5000em; max-height: 5000em; vertical-align: 0px; line-height: normal; text-decoration: none; white-space: nowrap !important; "><span class="math" id="MathJax-Span-320" style="background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: transparent; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; vertical-align: 0px; display: inline; position: static; border-style: initial; border-color: initial; line-height: normal; text-decoration: none; background-position: initial initial; background-repeat: initial initial; "><span style="background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: transparent; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; vertical-align: 0px; display: inline-block; position: relative; border-style: initial; border-color: initial; line-height: normal; text-decoration: none; height: 0px; font-size: 19px; width: 4.688em; background-position: initial initial; background-repeat: initial initial; "><span style="background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: transparent; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; vertical-align: 0px; display: inline; position: absolute; border-style: initial; border-color: initial; line-height: normal; text-decoration: none; clip: rect(1.426em 1000em 2.613em -0.446em); top: -2.293em; left: 0em; background-position: initial initial; background-repeat: initial initial; "><span class="mrow" id="MathJax-Span-321" style="background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: transparent; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; vertical-align: 0px; display: inline; position: static; border-style: initial; border-color: initial; line-height: normal; text-decoration: none; background-position: initial initial; background-repeat: initial initial; "><span class="msubsup" id="MathJax-Span-322" style="background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: transparent; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; vertical-align: 0px; display: inline; position: static; border-style: initial; border-color: initial; line-height: normal; text-decoration: none; background-position: initial initial; background-repeat: initial initial; "><span style="background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: transparent; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; vertical-align: 0px; display: inline-block; position: relative; border-style: initial; border-color: initial; line-height: normal; text-decoration: none; height: 0px; width: 0.775em; background-position: initial initial; background-repeat: initial initial; "><span style="background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: transparent; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; vertical-align: 0px; display: inline; position: absolute; border-style: initial; border-color: initial; line-height: normal; text-decoration: none; clip: rect(1.636em 1000em 2.412em -0.446em); top: -2.239em; left: 0em; background-position: initial initial; background-repeat: initial initial; "><span class="mi" id="MathJax-Span-323" style="background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: transparent; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; vertical-align: 0px; display: inline; position: static; border-style: initial; border-color: initial; line-height: normal; text-decoration: none; font-family: MathJax_Math; font-style: italic; background-position: initial initial; background-repeat: initial initial; ">e<span style="background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: transparent; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; vertical-align: 0px; display: inline-block; position: static; border-style: initial; border-color: initial; line-height: normal; text-decoration: none; width: 0px; height: 2.239em; background-position: initial initial; background-repeat: initial initial; "><span style="background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: transparent; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; vertical-align: 0px; display: inline; position: absolute; border-style: initial; border-color: initial; line-height: normal; text-decoration: none; top: -2.089em; left: 0.485em; background-position: initial initial; background-repeat: initial initial; "><span class="mi" id="MathJax-Span-324" style="background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: transparent; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; vertical-align: 0px; display: inline; position: static; border-style: initial; border-color: initial; line-height: normal; text-decoration: none; font-size: 13px; font-family: MathJax_Math; font-style: italic; background-position: initial initial; background-repeat: initial initial; ">i<span style="background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: transparent; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; vertical-align: 0px; display: inline-block; position: static; border-style: initial; border-color: initial; line-height: normal; text-decoration: none; width: 0px; height: 2.239em; background-position: initial initial; background-repeat: initial initial; "><span class="mo" id="MathJax-Span-325" style="background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: transparent; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0.278em; vertical-align: 0px; display: inline; position: static; border-style: initial; border-color: initial; line-height: normal; text-decoration: none; font-family: MathJax_Main; background-position: initial initial; background-repeat: initial initial; ">=<span class="msubsup" id="MathJax-Span-326" style="background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: transparent; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0.278em; vertical-align: 0px; display: inline; position: static; border-style: initial; border-color: initial; line-height: normal; text-decoration: none; background-position: initial initial; background-repeat: initial initial; "><span style="background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: transparent; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; vertical-align: 0px; display: inline-block; position: relative; border-style: initial; border-color: initial; line-height: normal; text-decoration: none; height: 0px; width: 0.883em; background-position: initial initial; background-repeat: initial initial; "><span style="background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: transparent; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; vertical-align: 0px; display: inline; position: absolute; border-style: initial; border-color: initial; line-height: normal; text-decoration: none; clip: rect(1.636em 1000em 2.412em -0.45em); top: -2.239em; left: 0em; background-position: initial initial; background-repeat: initial initial; "><span class="mi" id="MathJax-Span-327" style="background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: transparent; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; vertical-align: 0px; display: inline; position: static; border-style: initial; border-color: initial; line-height: normal; text-decoration: none; font-family: MathJax_Math; font-style: italic; background-position: initial initial; background-repeat: initial initial; ">x<span style="background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: transparent; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; vertical-align: 0px; display: inline-block; position: static; border-style: initial; border-color: initial; line-height: normal; text-decoration: none; width: 0px; height: 2.239em; background-position: initial initial; background-repeat: initial initial; "><span style="background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: transparent; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; vertical-align: 0px; display: inline; position: absolute; border-style: initial; border-color: initial; line-height: normal; text-decoration: none; top: -2.089em; left: 0.593em; background-position: initial initial; background-repeat: initial initial; "><span class="mi" id="MathJax-Span-328" style="background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: transparent; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; vertical-align: 0px; display: inline; position: static; border-style: initial; border-color: initial; line-height: normal; text-decoration: none; font-size: 13px; font-family: MathJax_Math; font-style: italic; background-position: initial initial; background-repeat: initial initial; ">i<span style="background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: transparent; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; vertical-align: 0px; display: inline-block; position: static; border-style: initial; border-color: initial; line-height: normal; text-decoration: none; width: 0px; height: 2.239em; background-position: initial initial; background-repeat: initial initial; "><span class="mo" id="MathJax-Span-329" style="background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: transparent; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0.222em; vertical-align: 0px; display: inline; position: static; border-style: initial; border-color: initial; line-height: normal; text-decoration: none; font-family: MathJax_Main; background-position: initial initial; background-repeat: initial initial; ">−<span class="mi" id="MathJax-Span-330" style="background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: transparent; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0.222em; vertical-align: 0px; display: inline; position: static; border-style: initial; border-color: initial; line-height: normal; text-decoration: none; font-family: MathJax_Math; font-style: italic; background-position: initial initial; background-repeat: initial initial; ">θ<span style="background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: transparent; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; vertical-align: 0px; display: inline-block; position: static; border-style: initial; border-color: initial; line-height: normal; text-decoration: none; width: 0px; height: 2.293em; background-position: initial initial; background-repeat: initial initial; "><span style="background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: transparent; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0em; border-style: initial; border-color: initial; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; vertical-align: -0.246em; display: inline-block; position: static; border-style: initial; border-color: initial; line-height: normal; text-decoration: none; border-left-style: solid; border-left-color: initial; overflow-x: hidden; overflow-y: hidden; width: 0px; height: 1.126em; background-position: initial initial; background-repeat: initial initial; ">，按最小二乘法，误差累积为

求解<nobr style="border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; max-width: 5000em; max-height: 5000em; vertical-align: 0px; line-height: normal; text-decoration: none; white-space: nowrap !important; "><span class="math" id="MathJax-Span-368" style="background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: transparent; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; vertical-align: 0px; display: inline; position: static; border-style: initial; border-color: initial; line-height: normal; text-decoration: none; background-position: initial initial; background-repeat: initial initial; "><span style="background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: transparent; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; vertical-align: 0px; display: inline-block; position: relative; border-style: initial; border-color: initial; line-height: normal; text-decoration: none; height: 0px; font-size: 19px; width: 0.485em; background-position: initial initial; background-repeat: initial initial; "><span style="background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: transparent; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; vertical-align: 0px; display: inline; position: absolute; border-style: initial; border-color: initial; line-height: normal; text-decoration: none; clip: rect(1.373em 1000em 2.411em -0.45em); top: -2.239em; left: 0em; background-position: initial initial; background-repeat: initial initial; "><span class="mrow" id="MathJax-Span-369" style="background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: transparent; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; vertical-align: 0px; display: inline; position: static; border-style: initial; border-color: initial; line-height: normal; text-decoration: none; background-position: initial initial; background-repeat: initial initial; "><span class="mi" id="MathJax-Span-370" style="background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: transparent; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; vertical-align: 0px; display: inline; position: static; border-style: initial; border-color: initial; line-height: normal; text-decoration: none; font-family: MathJax_Math; font-style: italic; background-position: initial initial; background-repeat: initial initial; ">θ<span style="background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: transparent; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; vertical-align: 0px; display: inline-block; position: static; border-style: initial; border-color: initial; line-height: normal; text-decoration: none; width: 0px; height: 2.239em; background-position: initial initial; background-repeat: initial initial; "><span style="background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: transparent; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0em; border-style: initial; border-color: initial; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; vertical-align: -0.074em; display: inline-block; position: static; border-style: initial; border-color: initial; line-height: normal; text-decoration: none; border-left-style: solid; border-left-color: initial; overflow-x: hidden; overflow-y: hidden; width: 0px; height: 0.954em; background-position: initial initial; background-repeat: initial initial; ">使得<nobr style="border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; max-width: 5000em; max-height: 5000em; vertical-align: 0px; line-height: normal; text-decoration: none; white-space: nowrap !important; "><span class="math" id="MathJax-Span-371" style="background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: transparent; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; vertical-align: 0px; display: inline; position: static; border-style: initial; border-color: initial; line-height: normal; text-decoration: none; background-position: initial initial; background-repeat: initial initial; "><span style="background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: transparent; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; vertical-align: 0px; display: inline-block; position: relative; border-style: initial; border-color: initial; line-height: normal; text-decoration: none; height: 0px; font-size: 19px; width: 1.94em; background-position: initial initial; background-repeat: initial initial; "><span style="background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: transparent; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; vertical-align: 0px; display: inline; position: absolute; border-style: initial; border-color: initial; line-height: normal; text-decoration: none; clip: rect(1.381em 1000em 2.705em -0.453em); top: -2.293em; left: 0em; background-position: initial initial; background-repeat: initial initial; "><span class="mrow" id="MathJax-Span-372" style="background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: transparent; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; vertical-align: 0px; display: inline; position: static; border-style: initial; border-color: initial; line-height: normal; text-decoration: none; background-position: initial initial; background-repeat: initial initial; "><span class="mi" id="MathJax-Span-373" style="background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: transparent; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; vertical-align: 0px; display: inline; position: static; border-style: initial; border-color: initial; line-height: normal; text-decoration: none; font-family: MathJax_Math; font-style: italic; background-position: initial initial; background-repeat: initial initial; ">L<span class="mo" id="MathJax-Span-374" style="background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: transparent; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; vertical-align: 0px; display: inline; position: static; border-style: initial; border-color: initial; line-height: normal; text-decoration: none; font-family: MathJax_Main; background-position: initial initial; background-repeat: initial initial; ">(<span class="mi" id="MathJax-Span-375" style="background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: transparent; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; vertical-align: 0px; display: inline; position: static; border-style: initial; border-color: initial; line-height: normal; text-decoration: none; font-family: MathJax_Math; font-style: italic; background-position: initial initial; background-repeat: initial initial; ">θ<span class="mo" id="MathJax-Span-376" style="background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: transparent; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; vertical-align: 0px; display: inline; position: static; border-style: initial; border-color: initial; line-height: normal; text-decoration: none; font-family: MathJax_Main; background-position: initial initial; background-repeat: initial initial; ">)<span style="background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: transparent; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; vertical-align: 0px; display: inline-block; position: static; border-style: initial; border-color: initial; line-height: normal; text-decoration: none; width: 0px; height: 2.293em; background-position: initial initial; background-repeat: initial initial; "><span style="background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: transparent; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0em; border-style: initial; border-color: initial; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; vertical-align: -0.352em; display: inline-block; position: static; border-style: initial; border-color: initial; line-height: normal; text-decoration: none; border-left-style: solid; border-left-color: initial; overflow-x: hidden; overflow-y: hidden; width: 0px; height: 1.285em; background-position: initial initial; background-repeat: initial initial; ">达到最小，正好是算术平均<nobr style="border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; max-width: 5000em; max-height: 5000em; vertical-align: 0px; line-height: normal; text-decoration: none; white-space: nowrap !important; "><span class="math" id="MathJax-Span-377" style="background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: transparent; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; vertical-align: 0px; display: inline; position: static; border-style: initial; border-color: initial; line-height: normal; text-decoration: none; background-position: initial initial; background-repeat: initial initial; "><span style="background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: transparent; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; vertical-align: 0px; display: inline-block; position: relative; border-style: initial; border-color: initial; line-height: normal; text-decoration: none; height: 0px; font-size: 19px; width: 4.634em; background-position: initial initial; background-repeat: initial initial; "><span style="background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: transparent; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; vertical-align: 0px; display: inline; position: absolute; border-style: initial; border-color: initial; line-height: normal; text-decoration: none; clip: rect(0.955em 1000em 2.807em -0.45em); top: -2.293em; left: 0em; background-position: initial initial; background-repeat: initial initial; "><span class="mrow" id="MathJax-Span-378" style="background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: transparent; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; vertical-align: 0px; display: inline; position: static; border-style: initial; border-color: initial; line-height: normal; text-decoration: none; background-position: initial initial; background-repeat: initial initial; "><span class="texatom" id="MathJax-Span-379" style="background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: transparent; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; vertical-align: 0px; display: inline; position: static; border-style: initial; border-color: initial; line-height: normal; text-decoration: none; background-position: initial initial; background-repeat: initial initial; "><span class="mrow" id="MathJax-Span-380" style="background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: transparent; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; vertical-align: 0px; display: inline; position: static; border-style: initial; border-color: initial; line-height: normal; text-decoration: none; background-position: initial initial; background-repeat: initial initial; "><span class="munderover" id="MathJax-Span-381" style="background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: transparent; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; vertical-align: 0px; display: inline; position: static; border-style: initial; border-color: initial; line-height: normal; text-decoration: none; background-position: initial initial; background-repeat: initial initial; "><span style="background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: transparent; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; vertical-align: 0px; display: inline-block; position: relative; border-style: initial; border-color: initial; line-height: normal; text-decoration: none; height: 0px; width: 0.594em; background-position: initial initial; background-repeat: initial initial; "><span style="background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: transparent; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; vertical-align: 0px; display: inline; position: absolute; border-style: initial; border-color: initial; line-height: normal; text-decoration: none; clip: rect(1.636em 1000em 2.412em -0.45em); top: -2.239em; left: 0em; background-position: initial initial; background-repeat: initial initial; "><span class="mi" id="MathJax-Span-382" style="background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: transparent; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; vertical-align: 0px; display: inline; position: static; border-style: initial; border-color: initial; line-height: normal; text-decoration: none; font-family: MathJax_Math; font-style: italic; background-position: initial initial; background-repeat: initial initial; ">x<span style="background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: transparent; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; vertical-align: 0px; display: inline-block; position: static; border-style: initial; border-color: initial; line-height: normal; text-decoration: none; width: 0px; height: 2.239em; background-position: initial initial; background-repeat: initial initial; "><span style="background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: transparent; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; vertical-align: 0px; display: inline; position: absolute; border-style: initial; border-color: initial; line-height: normal; text-decoration: none; clip: rect(1.541em 1000em 1.911em -0.416em); top: -2.326em; left: 0.055em; background-position: initial initial; background-repeat: initial initial; "><span class="mo" id="MathJax-Span-383" style="background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: transparent; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; vertical-align: 0px; display: inline; position: static; border-style: initial; border-color: initial; line-height: normal; text-decoration: none; font-family: MathJax_Main; background-position: initial initial; background-repeat: initial initial; ">ˉ<span style="background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: transparent; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; vertical-align: 0px; display: inline-block; position: static; border-style: initial; border-color: initial; line-height: normal; text-decoration: none; width: 0px; height: 2.293em; background-position: initial initial; background-repeat: initial initial; "><span class="mo" id="MathJax-Span-384" style="background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: transparent; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0.278em; vertical-align: 0px; display: inline; position: static; border-style: initial; border-color: initial; line-height: normal; text-decoration: none; font-family: MathJax_Main; background-position: initial initial; background-repeat: initial initial; ">=<span class="mfrac" id="MathJax-Span-385" style="background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: transparent; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0.12em; padding-bottom: 0px; padding-left: 0.398em; vertical-align: 0px; display: inline; position: static; border-style: initial; border-color: initial; line-height: normal; text-decoration: none; background-position: initial initial; background-repeat: initial initial; "><span style="background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: transparent; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; vertical-align: 0px; display: inline-block; position: relative; border-style: initial; border-color: initial; line-height: normal; text-decoration: none; height: 0px; width: 2.494em; background-position: initial initial; background-repeat: initial initial; "><span style="background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: transparent; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: -1.187em; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; vertical-align: 0px; display: inline; position: absolute; border-style: initial; border-color: initial; line-height: normal; text-decoration: none; clip: rect(3.28em 1000em 4.369em -0.445em); top: -4.618em; left: 50%; background-position: initial initial; background-repeat: initial initial; "><span class="mrow" id="MathJax-Span-386" style="background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: transparent; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; vertical-align: 0px; display: inline; position: static; border-style: initial; border-color: initial; line-height: normal; text-decoration: none; background-position: initial initial; background-repeat: initial initial; "><span class="munderover" id="MathJax-Span-387" style="background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: transparent; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; vertical-align: 0px; display: inline; position: static; border-style: initial; border-color: initial; line-height: normal; text-decoration: none; background-position: initial initial; background-repeat: initial initial; "><span style="background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: transparent; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; vertical-align: 0px; display: inline-block; position: relative; border-style: initial; border-color: initial; line-height: normal; text-decoration: none; height: 0px; width: 1.616em; background-position: initial initial; background-repeat: initial initial; "><span style="background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: transparent; border-top-width: 0px; 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由于算术平均是一个历经考验的方法，而以上的推理说明，算术平均是最小二乘的一个特例，所以从另一个角度说明了最小二乘方法的优良性，使我们对最小二乘法更加有信心。

最小二乘法发表之后很快得到了大家的认可接受，并迅速的在数据分析实践中被广泛使用。不过历史上又有人把最小二乘法的发明归功于高斯，这又是怎么一回事呢。高斯在1809年也发表了最小二乘法，并且声称自己已经使用这个方法多年。高斯发明了小行星定位的数学方法，并在数据分析中使用最小二乘方法进行计算，准确的预测了谷神星的位置。

OK，更多请参看陈希孺院士的「数理统计学简史」的第4章、最小二乘法。

3.3、核函数特征空间

经过前面第一、二部分，我们已经知道，当把内积支持向量机通俗导论（理解SVM的三层境界）就变成之后，求将有两种方法：

1、先找到这种映射，然后将输入空间中的样本映射到新的空间中，最后在新空间中去求内积支持向量机通俗导论（理解SVM的三层境界）。以多项式为例，对其进行变换，，，，，得到：，也就是说通过把输入空间从二维向四维映射后，样本由线性不可分变成了线性可分，但是这种转化带来的直接问题是维度变高了，这意味着，首先可能导致后续计算变复杂，其次可能出现维度之咒，对于学习器而言就是：特征空间维数可能最终无法计算，而它的泛化能力(学习器对训练样本以外数据的适应性)会随着维度的增长而大大降低，这也违反了“奥坎姆的剃刀”，最终可能会使得内积支持向量机通俗导论（理解SVM的三层境界）无法求出，于是也就失去了这种转化的优势了；

2、或者是找到某种方法，它不需要显式的将输入空间中的样本映射到新的空间中而能够在输入空间中直接计算出内积支持向量机通俗导论（理解SVM的三层境界）。它其实是对输入空间向高维空间的一种隐式映射，它不需要显式的给出那个映射，在输入空间就可以计算，这就是传说中的核函数方法。

OK，不再做过多介绍了，对核函数有进一步兴趣的，还可以看看此文：http://www.cnblogs.com/vivounicorn/archive/2010/12/13/1904720.html。

3.4、SMO算法

SMO算法是<<Minimal Optimization:A Fast Algorithm for Training Support Vector Machines》一文中提出的，作者信息：http://research.microsoft.com/en-us/people/jplatt/，其基本思想是将Vapnik在1982年提出的Chunking方法推到极致，即：通过将原问题分解为一系列小规模凸二次规划问题而获得原问题解的方法，每次迭代只优化由2个点组成的工作集，SMO算法每次启发式地选择两个因子同时固定其它因子来找到这两个因子的最优值，直到达到停止条件。

上文2.1.1节已经提到过，SMO算法不过是为了解决对偶问题中对偶因子α的求解问题，这两篇文章：http://www.cnblogs.com/jerrylead/archive/2011/03/18/1988419.html，http://www.cnblogs.com/vivounicorn/archive/2011/06/01/2067496.html，已经介绍的相当详尽，包括算法思想及其实现，此处不再赘述。

3.5、SVM的应用

或许我们已经听到过，SVM在很多诸如文本分类，图像分类，生物序列分析和生物数据挖掘，手写字符识别等领域有很多的应用，但或许你并没强烈的意识到，SVM可以成功应用的领域远远超出现在已经在开发应用了的领域。

3.5.1、文本分类

一个文本分类系统不仅是一个自然语言处理系统，也是一个典型的模式识别系统，系统的输入是需要进行分类处理的文本，系统的输出则是与文本关联的类别。由于篇幅所限，其它更具体内容本文将不再详述。

OK，本节虽取标题为证明SVM，但聪明的读者们想必早已看出，其实本部分并无多少证明部分（特此致歉），怎么办呢？可以参阅支持向量机导论一书，此书精简而有趣。

最后，非常感谢pluskid及诸多朋友&牛人们的文章及著作，让我有机会在其基础上总结、深入，本文基本成型。谢谢。

参考文献及推荐阅读

《支持向量机导论》，[美] Nello Cristianini / John Shawe-Taylor 著；
支持向量机导论一书的支持网站：http://www.support-vector.net/；
《数据挖掘导论》，[美] Pang-Ning Tan / Michael Steinbach / Vipin Kumar 著；
《数据挖掘：概念与技术》，(加)Jiawei Han;Micheline Kamber 著；
《数据挖掘中的新方法：支持向量机》，邓乃扬田英杰著；
《支持向量机--理论、算法和扩展》，邓乃扬田英杰著；
支持向量机系列，pluskid：http://blog.pluskid.org/?page_id=683；
http://www.360doc.com/content/07/0716/23/11966_615252.shtml；
数据挖掘十大经典算法初探；
《模式识别支持向量机指南》，C.J.C Burges 著；
《统计学习方法》，李航著(第7章有不少内容参考自支持向量机导论一书，不过，可以翻翻看看)；
《统计自然语言处理》，宗成庆编著，第十二章、文本分类；
SVM入门系列，Jasper：http://www.blogjava.net/zhenandaci/category/31868.html；
最近邻决策和SVM数字识别的实现和比较，作者不详；
斯坦福大学机器学习课程原始讲义：http://www.cnblogs.com/jerrylead/archive/2012/05/08/2489725.html；
斯坦福机器学习课程笔记：http://www.cnblogs.com/jerrylead/tag/Machine%20Learning/；
http://www.cnblogs.com/jerrylead/archive/2011/03/13/1982639.html；
http://www.cnblogs.com/jerrylead/archive/2011/03/18/1988419.html；
数据挖掘掘中所需的概率论与数理统计知识、上；
关于机器学习方面的文章，可以读读：http://www.cnblogs.com/vivounicorn/category/289453.html；
数学系教材推荐：http://blog.sina.com.cn/s/blog_5e638d950100dswh.html；
《神经网络与机器学习(原书第三版)》，[加] Simon Haykin 著；
正态分布的前世今生：http://t.cn/zlH3Ygc；
《数理统计学简史》，陈希孺院士著；
《最优化理论与算法(第2版)》，陈宝林编著；
A Gentle Introduction to Support Vector Machines in Biomedicine：http://www.nyuinformatics.org/downloads/supplements/SVM_Tutorial_2010/Final_WB.pdf，此PPT很赞，除了对引入拉格朗日对偶变量后的凸二次规划问题的深入度不够之外，其它都挺好，配图很精彩，本文有几张图便引自此PPT中；
来自卡梅隆大学的讲解SVM的PPT：http://www.autonlab.org/tutorials/svm15.pdf；

支持向量机通俗导论（理解SVM的三层境界）