【论文翻译】Nonlinear Dimensionality Reduction by Locally Linear Embedding

论文题目：Nonlinear Dimensionality Reduction by Locally Linear Embedding
论文来源：Nonlinear Dimensionality Reduction by Locally Linear Embedding
翻译人：[email protected]实验室

Nonlinear Dimensionality Reduction by Locally Linear Embedding

基于局部线性嵌入的非线性降维方法

Sam T. Roweis and Lawrence K. Saul

Abstract

Many areas of science depend on exploratory data analysis and visualization. The need to analyze large amounts of multivariate data raises the fundamental problem of dimensionality reduction: how to discover compact representations of high-dimensional data. Here, we introduce locally linear embedding (LLE), an unsupervised learning algorithm that computes low-dimensional, neighborhood-preserving embeddings of high-dimensional inputs. Unlike clustering methods for local dimensionality reduction, LLE maps its inputs into a single global coordinate system of lower dimensionality, and its optimizations do not involve local minima. By exploiting the local symmetries of linear reconstructions,LLE is able to learn the global structure of nonlinear manifolds, such as those generated by images of faces or documents of text.

摘要

科学的许多领域都依赖于探索性数据分析和可视化。分析大量多元数据的需求提出了降维的根本问题：如何发现高维数据的紧凑表示。 在这里，我们介绍了局部线性嵌入（LLE），这是一种无监督的学习算法，可计算高维输入的低维，邻域保留嵌入。与用于减少局部维数的聚类方法不同，LLE将其输入映射到较低维的单个全局坐标系中，并且其优化不涉及局部极小值。通过利用线性重构的局部对称性，LLE能够学习非线性流形的整体结构，例如由脸部图像或文本文档生成的那些。

正文

How do we judge similarity? Our mental representations of the world are formed by processing large numbers of sensory inputs—including, for example, the pixel intensities of images, the power spectra of sounds, and the joint angles of articulated bodies. While complex stimuli of this form can be represented by points in a high-dimensional vector space, they typically have a much more compact description. Coherent structure in the world leads to strong correlations between inputs (such as between neighboring pixels in images), generating observations that lie on or close to a smooth low-dimensional manifold. To compare and classify such observations—in effect, to reason about the world—depends crucially on modeling the nonlinear geometry of these low-dimensional manifolds.

我们如何判断相似性？ 我们对世界的心理表征是通过处理大量的感觉输入而形成的，例如，包括图像的像素强度，声音的功率谱以及关节的关节角度。 虽然这种形式的复杂刺激可以由高维向量空间中的点表示，但它们通常具有更为紧凑的描述。世界上的连贯结构导致输入之间（例如图像中相邻像素之间）的强相关性，从而产生位于平滑低维流形上或附近的观测值。 要对这些观察结果进行比较和分类（实际上是为了推理世界），关键在于对这些低维流形的非线性几何进行建模。

Scientists interested in exploratory analysis or visualization of multivariate data (1) face a similar problem in dimensionality reduction. The problem, as illustrated in Fig. 1, involves mapping high-dimensional inputs into a low-dimensional “description” space with as many coordinates as observed modes of variability. Previous approaches to this problem, based on multidimensional scaling (MDS) (2), have computed embeddings that attempt to preserve pairwise distances [or generalized disparities (3)] between data points; these distances are measured along straight lines or, in more sophisticated usages of MDS such as Isomap (4), along shortest paths confined to the manifold of observed inputs. Here, we take a different approach, called locally linear embedding (LLE), that eliminates the need to estimate pairwise distances between widely separated data points. Unlike previous methods, LLE recovers global nonlinear structure from locally linear fits.

对多变量数据（1）进行探索性分析或可视化的科学家在降维方面面临着类似的问题。如图1所示，该问题涉及将高维输入映射到低维“描述”空间，该空间的坐标与观察到的可变性模式一样多。对于降维问题，以前的方法是基于多维缩放（MDS）（2），计算试图保留数据点之间的成对距离[或广义差异（3）]的嵌入。这些距离是沿着直线测量的，或者在MDS的更复杂用法（例如Isomap（4））中，沿着限制在观察到的输入流形上的最短路径。在这里，我们采用了另一种称为局部线性嵌入（LLE）的方法，该方法不需要估计广泛分离的数据点之间的成对距离。与以前的方法不同，LLE从局部线性拟合中恢复全局非线性结构。

Fig. 1. The problem of nonlinear dimensionality reduction, as illustrated (10) for three-dimensional data (B) sampled from a two- dimensional manifold (A). An unsupervised learning algorithm must discover the global internal coordinates of the manifold without signals that explicitly indicate how the data should be embedded in two dimensions. The color coding illustrates the neighborhood-preserving mapping discovered by LLE; black outlines in (B) and © show the neighborhood of a single point. Unlike LLE, projections of the data by principal component analysis (PCA) (28) or classical MDS (2) map faraway data points to nearby points in the plane, failing to identify the underlying structure of the manifold. Note that mixture models for local dimensionality reduction (29), which cluster the data and perform PCA within each cluster, do not address the problem considered here: namely, how to map high-dimensional data into a single global coordinate system of lower dimensionality.

图1.非线性降维问题，如图（10）所示，它是从二维流形（A）采样的三维数据（B）。一种无监督的学习算法必须发现流形的全局内部坐标，而没有明确的信号表明如何将数据嵌入二维。颜色编码说明了LLE发现的保留邻域的映射。 （B）和（C）中的黑色轮廓显示了单个点的邻域。与LLE不同，通过主成分分析（PCA）（28）或经典MDS（2）进行的数据投影将遥远的数据点映射到平面中的附近点，从而无法识别流形的基础结构。注意，用于局部降维的混合模型（29）将数据聚类并在每个聚类中执行PCA，但并未解决此处考虑的问题：即，如何将高维数据映射到一个较低维的单一全局坐标系中。

The LLE algorithm, summarized in Fig.2, is based on simple geometric intuitions. Suppose the data consist of N real-valued vectors $\vec X_i$Xi​, each of dimensionality D, sampled from some underlying manifold. Provided there is sufficient data (such that the manifold is well-sampled), we expect each data point and its neighbors to lie on or close to a locally linear patch of the manifold. We characterize the local geometry of these patches by linear coefficients that reconstruct each data point from its neighbors. Reconstruction errors are measured by the cost function $\epsilon(W)=\sum_i|\vec X_i-\sum_jW_{ij}\vec X_j|^2\quad(1)$ϵ(W)=∑i​∣Xi​−∑j​Wij​Xj​∣2(1).which adds up the squared distances between all the data points and their reconstructions. The weights $W_{ij}$Wij​ summarize the contribution of the jth data point to the ith reconstruction. To compute the weights $w_{ij}$wij​, we minimize the cost function subject to two constraints: first, that each data point $\vec X_i$Xi​ is reconstructed only from its neighbors (5), enforcing $W_{ij}=0$Wij​=0 if $\vec X_j$Xj​ does not belong to the set of neighbors of $\vec X_i$Xi​; second, that the rows of the weight matrix sum to one: $\sum_jW_{ij}=1$∑j​Wij​=1. The optimal weights $W_{ij}$Wij​subject to these constraints (6) are found by solving a least-squares problem (7).

如图2所示，LLE算法基于简单的几何直觉。假设数据由N个实值向量$\vec X_i$Xi​组成，每个向量的维数为D，从底层流形中采样。如果有足够的数据（如此流形就可以很好的采样），我们期望每个数据点及其邻域都位于流形的局部线性面片上或附近。我们用线性系数来描述这些斑块的局部几何特征，这些系数从相邻的数据点重建每个数据点。重建误差由代价函数$\epsilon(W)=\sum_i|\vec X_i-\sum_jW_{ij}\vec X_j|^2\quad(1)$ϵ(W)=∑i​∣Xi​−∑j​Wij​Xj​∣2(1)测量，该函数将所有数据点及其重建之间的平方距离相加。权值$W_{ij}$Wij​表示第j个数据点对第i个数据点重建的贡献。为了计算权重$W_{ij}$Wij​，我们在两个约束条件下最小化代价函数：第一，每个数据点$\vec X_i$Xi​仅从其邻域（5）重构，如果$\vec X_j$Xj​不属于$\vec X_i$Xi​的邻域集合，则强制执行$W_{ij}=0$Wij​=0；第二，权重矩阵的行和为$\sum_jW_{ij}=1$∑j​Wij​=1。通过求解一个最小二乘问题（7），得到了这些约束条件（6）下的最优权重$W_{ij}$Wij​。

Fig. 2. Steps of locally linear embedding: (1) Assign neighbors to each data point $\vec X_i$Xi​ (for example by using the K nearest neighbors). (2) Compute the weights $W_{ij}$Wij​ that best linearly reconstruct $\vec X_i$Xi​ from its neighbors, solving the constrained least-squares problem in Eq. 1. (3) Compute the low-dimensional embedding vectors $\vec Y_i$Yi​ best reconstructed by $W_{ij}$Wij​, minimizing Eq. 2 by finding the smallest eigenmodes of the sparse symmetric matrix in Eq. 3. Although the weights $W_{ij}$Wij​ and vectors $Y_i$Yi​ are computed by methods in linear algebra, the constraint that points are only reconstructed from neighbors can result in highly nonlinear embeddings.

图2.局部线性嵌入的步骤：（1）为每个数据点$\vec X_i$Xi​指定邻域（例如，使用K个最近邻域）。 （2）计算权值$W_{ij}$Wij​，以最佳方式从其邻域线性重构$\vec X_i$Xi​，从而解决等式（1）中的约束最小二乘问题。（3）计算由$W_{ij}$Wij​最佳重构的低维嵌入向量$\vec Y_i$Yi​，通过求出等式3中稀疏对称矩阵的最小本征模，最小化等式（2）。 3.尽管权值$W_{ij}$Wij​和向量$Y_i$Yi​是通过线性代数中的方法计算的，但仅从邻域重构点的约束会导致高度非线性嵌入。

The constrained weights that minimize these reconstruction errors obey an important symmetry: for any particular data point, they are invariant to rotations, rescalings, and translations of that data point and its neighbors. By symmetry, it follows that the reconstruction weights characterize intrinsic geometric properties of each neighborhood, as opposed to properties that depend on a particular frame of reference (8). Note that the invariance to translations is specifically enforced by the sum-to-one constraint on the rows of the weight matrix.

使这些重建误差最小化的约束权重需遵循一个重要的对称性：对于任何特定的数据点，它们对该数据点及其相邻点的旋转、重定位和平移都是不变的。根据对称性，重构权值表征了每个邻域的固有几何特性，而不是依赖于特定参考系的特性（8）。注意，对平移的不变性是通过权重矩阵行上的和为1这个约束强制实现的。

Suppose the data lie on or near a smooth nonlinear manifold of lower dimensionality d << D. To a good approximation then, there exists a linear mapping—consisting of a translation, rotation, and rescaling—that maps the high-dimensional coordinates of each neighborhood to global internal coordinates on the manifold. By design, the reconstruction weights $W_{ij}$Wij​ reflect intrinsic geometric properties of the data that are invariant to exactly such transformations. We therefore expect their characterization of local geometry in the original data space to be equally valid for local patches on the manifold. In particular, the same weights $W_{ij}$Wij​ that reconstruct the ith data point in D dimensions should also reconstruct its embedded manifold coordinates in d dimensions.

假设数据位于较低维d<<D的平滑非线性流形上或附近。要得到一个很好的近似值，存在一个线性映射——由平移，旋转和重新缩放组成,把每个邻域的高维坐标映射到流形上全局内部坐标。通过设计，重建权重$W_{ij}$Wij​反映了数据的固有几何属性，这些属性对于精确的此类转换始终不变。因此，我们希望它们在原始数据空间中的局部几何图形的表征对于流形上的局部补丁同样有效。特别地，在D维中重建第i个数据点的相同权重$W_{ij}$Wij​也应在d维中重建其嵌入式流形坐标。

LLE constructs a neighborhood-preserving mapping based on the above idea. In the final step of the algorithm, each high-dimensional observation $\vec X_i$Xi​ is mapped to a low-dimensional vector $\vec Y_i$Yi​ representing global internal coordinates on the manifold. This is done by choosing d-dimensional coordinates $\vec Y_i$Yi​ to minimize the embedding cost function $\Phi(Y)=\sum_i|\vec Y_i-\sum_jW_{ij}\vec Y_j|^2\ (2)$Φ(Y)=∑i​∣Yi​−∑j​Wij​Yj​∣2 (2).This cost function, like the previous one, is based on locally linear reconstruction errors, but here we fix the weights $ W_{ij}$ while optimizing the coordinates $\vec Y_i$Yi​. The embedding cost in Eq. 2 defines a quadratic form in the vectors $\vec Y_i$Yi​. Subject to constraints that make the problem well-posed, it can be minimized by solving a sparse $N\times N$N×N eigenvalue problem (9), whose bottom d nonzero eigenvectors provide an ordered set of orthogonal coordinates centered on the origin.

LLE基于以上思想构造了一个保留邻域的映射。在算法的最后一步中，每个高维观测值$\vec X_i$Xi​都映射到代表流形上全局内部坐标的低维向量$\vec Y_i$Yi​。这是通过选择d维坐标$\vec Y_i$Yi​来完成的，最小化嵌入成本函数$\Phi(Y)=\sum_i|\vec Y_i-\sum_jW_{ij}\vec Y_j|^2\quad (2)$Φ(Y)=∑i​∣Yi​−∑j​Wij​Yj​∣2(2)。与先前的方法一样，此成本函数是基于局部线性重构误差的，但是这里我们在优化坐标$\vec Y_i$Yi​的同时修正权重$W_{ij}$Wij​。 等式（2）中的嵌入成本定义了向量$\vec Y_i$Yi​二次型。在约束条件下，可以通过解稀疏的$N\times N$N×N特征值问题（9）将其最小化来解决问题，该问题的底部d个非零特征向量提供了以原点为中心的正交坐标的有序集合。

Implementation of the algorithm is straightforward. In our experiments, data points were reconstructed from their K nearest neighbors, as measured by Euclidean distance or normalized dot products. For such implementations of LLE, the algorithm has only one free parameter: the number of neighbors, K. Once neighbors are chosen, the optimal weights $ W_{ij}$ and coordinates $\vec Y_i$Yi​ are computed by standard methods in linear algebra. The algorithm involves a single pass through the three steps in Fig. 2 and finds global minima of the reconstruction and embedding costs in Eqs. 1 and 2.

该算法的实现很简单。在我们的实验中，通过欧几里得距离或归一化的点积测量，从它们的K个近邻重建了数据点。对于LLE的此类实现，该算法仅具有一个*参数：邻域数K。一旦选择了邻域，则通过线性代数中的标准方法计算最佳权重$W_{ij}$Wij​和坐标$\vec Y_i$Yi​。该算法涉及图2中的三个步骤的一次遍历，并在等式（1）和（2）中找到了重建和嵌入成本的全局最小值。

In addition to the example in Fig. 1, for which the true manifold structure was known (10), we also applied LLE to images of faces (11) and vectors of word-document counts (12). Two-dimensional embeddings of faces and words are shown in Figs. 3 and 4. Note how the coordinates of these embedding spaces are related to meaningful attributes, such as the pose and expression of human faces and the semantic associations of words.

除了图1中的示例（已知真正的流形结构（10））之外，我们还将LLE应用于人脸图像（11）和单词文档计数矢量（12）。面部和单词的二维嵌入如图3和4中所示。请注意这些嵌入空间的坐标如何与有意义的属性相关联，例如人脸的姿势和表情以及单词的语义关联。

Fig. 3. Images of faces (11) mapped into the embedding space described by the first two coordinates of LLE. Representative faces are shown next to circled points in different parts of the space. The bottom images correspond to points along the top-right path (linked by solid line), illustrating one particular mode of variability in pose and expression.

图3.映射到由LLE的前两个坐标描述的嵌入空间中的人脸（11）的图像。代表性的脸部显示在空间不同部分的圆圈点旁边。底部图像对应于沿右上路径的点（由实线链接），说明了姿势和表情变化的一种特定模式。

Fig. 4. Arranging words in a continuous semantic space. Each word was initially represented by a high-dimensional vector that counted the number of times it appeared in different encyclopedia articles. LLE was applied to these word-document count vectors (12), resulting in an embedding location for each word. Shown are words from two different bounded regions (A) and (B) of the embedding space discovered by LLE. Each panel shows a two-dimensional projection onto the third and fourth coordinates of LLE; in these two dimensions, the regions (A) and (B) are highly overlapped. The inset in (A) shows a three-dimensional projection onto the third, fourth, and fifth coordinates, revealing an extra dimension along which regions (A) and (B) are more separated. Words that lie in the intersection of both regions are capitalized. Note how LLE co-locates words with similar contexts in this continuous semantic space.

图4.在连续语义空间中排列单词。每个单词最初都由一个高维向量表示，该向量计算了它在不同百科全书文章中出现的次数。将LLE应用于这些单词文档计数向量（12），从而为每个单词生成嵌入位置。显示的是来自LLE发现的嵌入空间的两个不同有界区域（A）和（B）的单词。 每个面板都在LLE的第三和第四坐标上显示了二维投影；在这两个维度上，区域（A）和（B）高度重叠。（A）中的插图显示了在第三，第四和第五坐标上的三维投影，揭示了一个额外的维度，沿着该维度，区域（A）和（B）更加分开。位于两个区域相交处的单词均大写。请注意，LLE如何在此连续语义空间中具有相似上下文的单词共置。

Many popular learning algorithms for nonlinear dimensionality reduction do not share the favorable properties of LLE. Iterative hill-climbing methods for autoencoder neural networks (13, 14), self-organizing maps (15), and latent variable models (16) do not have the same guarantees of global optimality or convergence; they also tend to involve many more free parameters, such as learning rates, convergence criteria, and architectural specifications. Finally, whereas other nonlinear methods rely on deterministic annealing schemes (17) to avoid local minima, the optimizations of LLE are especially tractable.

许多流行的用于非线性降维的学习算法并不具有LLE的良好特性。自动编码器神经网络（13、14），自组织映射（15）和潜变量模型（16）的迭代爬山方法不能保证全局最优或收敛。它们还倾向于包含更多的*参数，例如学习率，收敛标准和体系结构规范。 最后，尽管其他非线性方法依靠确定性退火方案（17）来避免局部极小值，但LLE的优化特别容易处理。

LLE scales well with the intrinsic manifold dimensionality, d, and does not require a discretized gridding of the embedding space. As more dimensions are added to the embedding space, the existing ones do not change, so that LLE does not have to be rerun to compute higher dimensional embeddings. Unlike methods such as principal curves and surfaces (18) or additive component models (19), LLE is not limited in practice to manifolds of extremely low dimensionality or codimensionality. Also, the intrinsic value of d can itself be estimated by analyzing a reciprocal cost function, in which reconstruction weights derived from the embedding vectors $\vec Y_i$Yi​ are applied to the data points $\vec X_i$Xi​.

LLE可以很好地适应内在流形维数d，并且不需要嵌入空间的离散化网络。当嵌入空间中加入更多的维度时，现有维度不变，因此不必重新运行LLE来计算高维嵌入。与主曲线和曲面（18）或加性分量模型（19）等方法不同，LLE在实际中并不局限于极低维或余维的流形。此外，d的内在值可以通过分析一个互成本函数来估计，其中由嵌入向量$\vec Y_i$Yi​导出的重建权重被应用于数据点$\vec X_i$Xi​。

LLE illustrates a general principle of manifold learning, elucidated by Martinetz and Schulten (20) and Tenenbaum (4), that overlapping local neighborhoods—collectively analyzed—can provide information about global geometry. Many virtues of LLE are shared by Tenenbaum’s algorithm, Isomap, which has been successfully applied to similar problems in nonlinear dimensionality reduction. Isomap’s embeddings, however, are optimized to preserve geodesic distances between general pairs of data points, which can only be estimated by computing shortest paths through large sublattices of data. LLE takes a different approach, analyzing local symmetries, linear coefficients, and reconstruction errors instead of global constraints,pairwise distances, and stress functions. It thus avoids the need to solve large dynamic programming problems, and it also tends to accumulate very sparse matrices,whose structure can be exploited for savings in time and space.

LLE阐明了流形学习的一般原理，Martinetz和Schulten（20）和Tenenbaum（4）阐明了这一点，即重叠的局部邻域（共同分析）可以提供有关全局几何的信息。Tenenbaum的算法Isomap具有LLE的许多优点，该算法已成功应用于非线性降维中的类似问题。但是，优化了Isomap的嵌入以保留一般数据点对之间的测地距离，这只能通过计算通过大数据子格的最短路径来估算。LLE采用不同的方法，分析局部对称性，线性系数和重构误差，而不是全局约束，成对距离和应力函数。 因此，它避免了解决大型动态编程问题的需要，并且还倾向于累积非常稀疏的矩阵，可以利用这些矩阵的结构节省时间和空间。

LLE is likely to be even more useful in combination with other methods in data analysis and statistical learning. For example, a parametric mapping between the observation and embedding spaces could be learned by supervised neural networks (21) whose target values are generated by LLE. LLE can also be generalized to harder settings, such as the case of disjoint data manifolds (22), and specialized to simpler ones, such as the case of time-ordered observations (23).

在数据分析和统计学习中，LLE与其他方法相结合可能会更加有用。例如，观察空间和嵌入空间之间的参数映射可以由监督神经网络（21）学习，其目标值由LLE生成。LLE还可以推广到更难的设置，例如不相交的数据流形（22），并专门用于更简单的设置，例如时序观测的情况（23）。

Perhaps the greatest potential lies in applying LLE to diverse problems beyond those considered here. Given the broad appeal of traditional methods, such as PCA and MDS, the algorithm should find widespread use in many areas of science.

也许最大的潜力在于将LLE应用于此处未考虑的各种问题。 鉴于PCA和MDS等传统方法的广泛吸引力，该算法应在许多科学领域中得到广泛使用。