图片上两点之间的距离和两组图片之间的差异的关系
制作两个矩阵A和D
用神经网络分类A和D,让x和y都是(0,1)之间的小数,则A与D的第三点A:x[2]=0,D:x[2]=a(x-y)的距离为
所以如果变化a对分类A和D有什么影响?
实验过程
制作一个4*4*2的网络分别向两边输入矩阵A和D。并让4*4*2部分的权重共享,前面大量实验表明这种效果相当于将两个弹性系数为k1,k2的弹簧并联成一个弹性系数为k的弹簧,并且让k1=k2=k/2的过程。
将这个网络简写成
d2(A,D)-4-4-2-(2*k),k∈{0,1}
具体进样顺序 |
|||
进样顺序 |
迭代次数 |
||
δ=0.5 |
|||
A |
1 |
判断是否达到收敛 |
|
D |
2 |
判断是否达到收敛 |
|
梯度下降 |
|||
A |
3 |
判断是否达到收敛 |
|
D |
4 |
判断是否达到收敛 |
|
梯度下降 |
|||
…… |
|||
达到收敛标准测量准确率,记录迭代次数n,将这个过程重复199次 |
|||
δ=0.4 |
|||
… |
|||
δ=1e-6 |
这个网络的收敛标准是
if (Math.abs(f2[0]-y[0])< δ && Math.abs(f2[1]-y[1])< δ )
因为对应每个收敛标准δ都有一个特征的迭代次数n与之对应因此可以用迭代次数曲线n(δ)来评价网络性能。
当网络小于收敛标准δ记录迭代次数n
矩阵A和矩阵D的构造方式
矩阵A是吸引子
Random rand1 =new Random();
int ti1=rand1.nextInt(99)+1;
x[0]=sigmoid((double)ti1/100);
Random rand2 =new Random();
int ti2=rand2.nextInt(99)+1;
x[3]=sigmoid((double)ti2/100);
x[1]=0;
x[2]=0;
D的输入
x[0]=sigmoid((double)ti1/100);
x[1]=0;
x[2]=sigmoid(sigmoid((double)ti1/100)- sigmoid((double)ti2/100) );
x[3]=sigmoid((double)ti2/100);
本文测量了a从0.5到1e-6共34组值观察a对分类性能的影响
δ |
1.00E-03 |
1.00E-02 |
0.1 |
0.2 |
0.3 |
0.4 |
0.5 |
0.6 |
0.7 |
0.8 |
0.9 |
1.1 |
1.2 |
1.3 |
1.4 |
1.5 |
1.6 |
1.7 |
1.8 |
1.9 |
2 |
2.1 |
2.2 |
2.3 |
2.5 |
3 |
5 |
0.5 |
128.4623116 |
129.2160804 |
129.86935 |
127.9196 |
130.46231 |
130.0603 |
131.67839 |
132.0201 |
132.71357 |
129.49749 |
131.4472362 |
130.2915 |
132.7337 |
132.3216 |
128.4523 |
129.5879 |
132.3216 |
131.9698 |
134.5327 |
130.9548 |
130.4121 |
132.3719 |
129.7588 |
131.9899 |
130.1407 |
131.4673 |
129.5176 |
0.4 |
1211.788945 |
1198.422111 |
1223.5126 |
1212.3417 |
1220.6281 |
1229.7286 |
1200.3518 |
1239.0754 |
1162.794 |
1241.4372 |
1210.733668 |
1208.231 |
1196.141 |
1219.884 |
1202.814 |
1186.121 |
1211.779 |
1169.879 |
1201.879 |
1205.337 |
1195.598 |
1162.704 |
1157.849 |
1155.709 |
1160.332 |
1159.889 |
1128.683 |
0.3 |
1522.201005 |
1511.165829 |
1492.3166 |
1488.0653 |
1509.6633 |
1483.593 |
1532.0905 |
1498.5729 |
1488.0101 |
1508.2864 |
1517.060302 |
1482.704 |
1500.402 |
1494.412 |
1493.397 |
1464.874 |
1469.357 |
1477.854 |
1488.432 |
1466.281 |
1470.322 |
1439.759 |
1451.296 |
1400 |
1441.95 |
1404.06 |
1407.628 |
0.2 |
1764.175879 |
1765.226131 |
1746.3719 |
1761.8844 |
1801.9397 |
1741.3266 |
1742 |
1765.5025 |
1762.3065 |
1764.5025 |
1736.81407 |
1718.623 |
1745.327 |
1737.487 |
1752.271 |
1747.457 |
1727.528 |
1711.799 |
1688.804 |
1705.648 |
1693.518 |
1700.693 |
1660.312 |
1680.714 |
1670.844 |
1658.06 |
1598.533 |
0.1 |
2170.81407 |
2179.025126 |
2197.6181 |
2177.1759 |
2163.608 |
2175.3367 |
2185.3065 |
2215.1859 |
2160.5276 |
2145.8894 |
2173.809045 |
2137.543 |
2114.201 |
2133.975 |
2118.231 |
2108.704 |
2068.945 |
2053.729 |
2071.106 |
2021.709 |
2058.724 |
2038.603 |
2051.176 |
2017.025 |
2000.915 |
1970.492 |
1906.07 |
0.01 |
5430.522613 |
5434.246231 |
5452.8995 |
5362.9548 |
5422.3065 |
5458.6533 |
5370.3266 |
5419.0553 |
5325.8442 |
5316.7437 |
5179.552764 |
4985.98 |
4891.035 |
4869.035 |
4726.281 |
4708.874 |
4571.005 |
4547.719 |
4473.93 |
4369.065 |
4293.698 |
4206.945 |
4208.312 |
4109.487 |
4027.99 |
3830.583 |
3355.065 |
0.001 |
24738.25126 |
24839.61809 |
24559.769 |
24541.628 |
24068.583 |
24239.085 |
24575.442 |
23943.151 |
23665.08 |
23708.633 |
23321.49749 |
21804.97 |
20796.04 |
20370.24 |
19360.34 |
18595.84 |
18009.57 |
17497.97 |
16652.58 |
16766.2 |
15971.77 |
15091.03 |
14781.51 |
14551.64 |
13570.41 |
12585.38 |
8767.719 |
9.00E-04 |
26642.90452 |
26930.52261 |
26590.638 |
26465.508 |
26887.874 |
26539.447 |
26522.513 |
26502.382 |
26135.784 |
25919.829 |
25564.26131 |
23629.83 |
22386.26 |
22237.74 |
20886.27 |
20211.73 |
19409.95 |
18879.27 |
18284.55 |
17839.83 |
17258.27 |
16868.11 |
16428.08 |
15459.84 |
15074.01 |
13276.57 |
9948.221 |
8.00E-04 |
29364.40704 |
29703.26633 |
29132.452 |
29290.392 |
29735.141 |
28914.452 |
28868.935 |
29384.201 |
28551.186 |
29267.769 |
28221.34673 |
26097.72 |
25675.75 |
23912.77 |
22969.87 |
22334.02 |
21586.86 |
20445.58 |
20152.49 |
18717.98 |
18534.89 |
17508.76 |
17231.01 |
17299.2 |
16500.4 |
14173.36 |
10063.67 |
7.00E-04 |
31861.86432 |
32648.58291 |
32101.327 |
32929.698 |
32162.784 |
32281.015 |
32406.417 |
31786.05 |
32227.015 |
31906.497 |
30858.68844 |
28459.33 |
27266.76 |
26460.01 |
25735.64 |
24532.55 |
23166.4 |
23268.58 |
21573.72 |
20888.87 |
20611.61 |
20048.53 |
19159.39 |
19219.29 |
17178.8 |
15705.96 |
11450.41 |
6.00E-04 |
36693.41206 |
36409.9799 |
37046.894 |
36249.492 |
36524.276 |
35904.749 |
36202.97 |
36148.578 |
35844.643 |
36004.251 |
35515.87437 |
31952.59 |
30864.45 |
30212.48 |
28276.7 |
27592.36 |
26319.37 |
25800.18 |
24886.17 |
22961.31 |
22994.16 |
22223.64 |
21596.41 |
20504.57 |
19578.9 |
17255.83 |
11987.95 |
5.00E-04 |
41929.60302 |
42666.30151 |
41844.975 |
42295 |
42593.503 |
42051.176 |
42107.04 |
42515.94 |
41554.372 |
41531.899 |
40135.1407 |
36980.14 |
35416.02 |
34078.33 |
34266.76 |
31532.55 |
30882.48 |
29294.53 |
28915.74 |
26966.13 |
26796.87 |
25875.29 |
24484.54 |
23996.92 |
22520.47 |
19399.82 |
13751.09 |
4.00E-04 |
50075.44221 |
52496.91457 |
51649.894 |
51591.653 |
50380.101 |
51137.347 |
48885.427 |
50134.437 |
50458.387 |
49843.598 |
49125 |
45857.86 |
43013.91 |
40814.19 |
39371.25 |
38653.73 |
36465.36 |
34859.58 |
32004.82 |
31674.71 |
31793.43 |
30142.8 |
29257.31 |
28639.16 |
25722.36 |
23299.63 |
15515.53 |
3.00E-04 |
65622.76382 |
64610.66834 |
64340.859 |
64711.709 |
64427.05 |
65007.975 |
64739.698 |
63350.99 |
64549.256 |
62802.729 |
61459.51759 |
57201.58 |
54014.03 |
53236.75 |
49049.79 |
47075.13 |
46305.52 |
44722.85 |
42200.51 |
40795.64 |
38953.5 |
38256.84 |
37046.75 |
34738.03 |
33814.12 |
28378.35 |
19756.32 |
2.00E-04 |
92569.18593 |
92765.70854 |
89084.417 |
91586.317 |
91576.628 |
88964.839 |
90940.884 |
89206.623 |
91700.568 |
89190.387 |
86401.88945 |
80964.61 |
73677.25 |
72979.15 |
68488.01 |
67381.05 |
64121.69 |
61448.08 |
58004.96 |
56123.57 |
54096.58 |
52133.9 |
50391.83 |
50934.25 |
46758 |
38999.75 |
26043.12 |
1.00E-04 |
160643.8241 |
165371.8543 |
166297.55 |
165634.01 |
160929.24 |
161838.12 |
162103.39 |
162535.3 |
165745.27 |
163626.24 |
156234.0905 |
144866.1 |
145502 |
134964.8 |
126139.6 |
117150.5 |
114140.2 |
106811.3 |
106610.5 |
101497 |
95615.24 |
95267.46 |
91044.01 |
84126.96 |
79696 |
71294.61 |
42615.69 |
9.00E-05 |
186286.4623 |
184232.9146 |
180495.68 |
185461.3 |
182664.79 |
176149.11 |
179780.03 |
178448.05 |
183129.95 |
174383.59 |
172895 |
157584 |
155452.6 |
147611.6 |
140324.4 |
131875.2 |
123775.8 |
120813.2 |
113232 |
112913.1 |
107182.8 |
103395.1 |
94619.81 |
95078.86 |
85908.8 |
74179.51 |
54020.35 |
8.00E-05 |
200154.8995 |
202144.2261 |
203010.25 |
202924.11 |
200561.32 |
202181.71 |
194458.53 |
201402.91 |
195427.11 |
196630.34 |
197028.4271 |
179416.9 |
165193.7 |
158696.6 |
159417 |
149042.9 |
138704 |
135488.4 |
123261.1 |
126976.4 |
114853 |
114598.8 |
110847.4 |
104954.4 |
98976.92 |
81168.17 |
51728.38 |
7.00E-05 |
220892.9246 |
230403.402 |
223449.67 |
229844.96 |
227188.46 |
223330.78 |
219958.65 |
221808.62 |
222717.76 |
215797.87 |
215569.4724 |
202120 |
184584.8 |
184618 |
172709.1 |
168747.7 |
158308.4 |
152088.3 |
147196.8 |
137857.2 |
136608.2 |
127470 |
124260.5 |
117198.9 |
111331.7 |
91813.31 |
56356.86 |
6.00E-05 |
259019.9146 |
252816.6332 |
251246.68 |
256015.75 |
258797.01 |
264843.67 |
256515.32 |
252709.67 |
258744.22 |
257857.52 |
247695 |
228819.8 |
207249.5 |
203361.4 |
196486.3 |
188452.8 |
179651.1 |
171733.9 |
169391.7 |
162893 |
142626.2 |
140041.6 |
141043.8 |
129736.8 |
130254.7 |
105280.8 |
66726.5 |
5.00E-05 |
305939.4724 |
310143.9095 |
307924.59 |
298638.32 |
303183.49 |
298232.88 |
306459.75 |
303921.39 |
294887.71 |
294445.02 |
291326.3618 |
276902 |
254669.4 |
244408.4 |
237249 |
224955.4 |
215313.7 |
198229.6 |
198676.9 |
199729.9 |
176320.6 |
175077.5 |
169756.3 |
157093 |
145877.3 |
131440.7 |
76321.85 |
4.00E-05 |
381694.3869 |
371664.9598 |
371113.86 |
373714.59 |
367964.23 |
363150.83 |
383585.05 |
364084.41 |
364202.05 |
351471.58 |
362103.0302 |
341771.6 |
310986.9 |
293993.3 |
280626.6 |
268127.4 |
267391.7 |
245890.6 |
240082.1 |
234875.3 |
213823.5 |
196491.5 |
205850.3 |
199963.6 |
175481.9 |
150042.8 |
91308.68 |
3.00E-05 |
475350.1859 |
476796.9799 |
474007.34 |
481228.25 |
472738.95 |
474138.89 |
468082.81 |
476884.02 |
468669.11 |
451195.26 |
463830.6935 |
431640.6 |
406311.8 |
391581.9 |
360372.6 |
363322.5 |
345725.6 |
311048.3 |
297976.8 |
282152.4 |
272986.5 |
270379.9 |
260505.5 |
237141.4 |
241500.1 |
206000.8 |
130432.1 |
2.00E-05 |
690683.6533 |
708650.5226 |
715184.95 |
688840.96 |
700356.32 |
686737.82 |
675954.92 |
652367.38 |
680538.43 |
673054.95 |
661376.1658 |
629661.9 |
602247.6 |
553782.3 |
554477.2 |
505081.1 |
503119.8 |
480955.1 |
437419.4 |
445859.1 |
397096.1 |
383377.8 |
405126.3 |
357890.5 |
329485.4 |
282542 |
159564.7 |
1.00E-05 |
1319909.06 |
1304310.573 |
1280054.6 |
1281523.2 |
1282234.7 |
1279703.3 |
1316592.7 |
1317496.3 |
1290298.5 |
1246012.7 |
1240805.477 |
1155889 |
1073770 |
1057344 |
1027268 |
965041.3 |
901240.1 |
927876.4 |
825169.8 |
799342.1 |
764294.4 |
737732 |
744803.6 |
703801.3 |
629568.7 |
532123.3 |
310428.7 |
9.00E-06 |
1415643.116 |
1462645.09 |
1486968.8 |
1409226.9 |
1420476.5 |
1451016.5 |
1391674.4 |
1422415 |
1384179.1 |
1379721.7 |
1405242.91 |
1269484 |
1242273 |
1131755 |
1131837 |
1065732 |
1008012 |
909285.5 |
951330.1 |
907500.7 |
831663.4 |
846394.1 |
772981.5 |
774477.6 |
749237.7 |
578167.7 |
355695.3 |
8.00E-06 |
1560931.899 |
1623360.683 |
1583651.1 |
1571575.7 |
1567481.5 |
1580216.3 |
1582918.5 |
1555602.9 |
1574942.8 |
1559014.3 |
1495475.874 |
1419133 |
1384587 |
1306935 |
1215699 |
1218470 |
1098315 |
1077431 |
1053434 |
1025781 |
966703.6 |
915698.6 |
889028 |
878534.6 |
812715.5 |
650479.3 |
415233.8 |
7.00E-06 |
1705373.94 |
1805524.246 |
1825951.2 |
1785149.1 |
1799023.2 |
1752176.6 |
1783813.6 |
1762994.3 |
1817556.9 |
1732249.6 |
1735545.658 |
1596230 |
1510009 |
1437189 |
1389737 |
1295544 |
1279528 |
1215496 |
1150113 |
1154580 |
1103819 |
1020556 |
938728.7 |
926912.2 |
870120.1 |
757956.3 |
468636.6 |
6.00E-06 |
2054988.437 |
2147323.995 |
2089832.8 |
2026900.6 |
2008255.5 |
2065680.3 |
2063529.6 |
2012895.3 |
2018646.3 |
2019512.1 |
1997271.337 |
1882814 |
1710636 |
1695008 |
1572382 |
1598904 |
1503469 |
1482494 |
1424434 |
1308605 |
1234652 |
1152154 |
1071785 |
1100709 |
1027444 |
800914.8 |
540448.7 |
5.00E-06 |
2404500.905 |
2426074.146 |
2456192.3 |
2362463.1 |
2459894.8 |
2444007.4 |
2380623.6 |
2401251.3 |
2461559.7 |
2423819.7 |
2339349.191 |
2188884 |
2121634 |
1936639 |
1870625 |
1808831 |
1778483 |
1741619 |
1644482 |
1494393 |
1477053 |
1454789 |
1377725 |
1301081 |
1204413 |
1049618 |
666964.1 |
4.00E-06 |
3026583.497 |
3027668.02 |
3005676.5 |
2934050.4 |
3007859 |
3063148.8 |
3009375.5 |
2868062.9 |
2924405.5 |
2931314.3 |
2946858.236 |
2721951 |
2573131 |
2420087 |
2214564 |
2177407 |
2298046 |
2058596 |
1953759 |
1840770 |
1847457 |
1725486 |
1644338 |
1476870 |
1480641 |
1157032 |
858850.4 |
3.00E-06 |
3964188.784 |
3836020.995 |
3835702.9 |
4003851.2 |
3864993.8 |
3944114.5 |
3930845.1 |
3721951 |
3738882.1 |
3867369.4 |
3830464.97 |
3498057 |
3285130 |
3227860 |
3108969 |
3009554 |
2762537 |
2712967 |
2597385 |
2557251 |
2471113 |
2286074 |
2135585 |
2069868 |
2003542 |
1611491 |
1140339 |
2.00E-06 |
5658638.352 |
5780181.055 |
5928295.7 |
5644884.1 |
5544114.3 |
5678374.2 |
5509905.8 |
5583132.2 |
5496451.6 |
5608446.1 |
5400579.005 |
5150186 |
4761965 |
4679598 |
4515210 |
4243803 |
4342979 |
3870394 |
3885246 |
3480037 |
3583468 |
3282269 |
3176682 |
3160933 |
2867858 |
2300469 |
1379227 |
1.00E-06 |
1.07E+07 |
1.07E+07 |
1.10E+07 |
1.08E+07 |
1.11E+07 |
1.05E+07 |
1.05E+07 |
1.05E+07 |
1.06E+07 |
1.03E+07 |
1.05E+07 |
9875165 |
9404626 |
8896366 |
8483615 |
8262433 |
7651135 |
7394275 |
6909363 |
6434851 |
6843746 |
6618061 |
6088555 |
6143251 |
5332034 |
5114519 |
2726750 |
d2(A,D)-4-4-2-(2*k),k∈{0,1}
当a>1时,随着a的增加网络的迭代次数开始减小,完全相同的对象不能被分为两类,与之对应的迭代次数将是无限大。也就是迭代次数越大表明两个对象差别越小。因此a越大使得A和D的差异越大。
所以从表格很容易的看到a越大则越容易被分类,对这道题
这个表达式的当a>1时,a越大两个对象差异越大,a越小A和D的差异越小。
实验参数
学习率 0.1 |
权重初始化方式 |
Random rand1 =new Random(); |
int ti1=rand1.nextInt(98)+1; |
tw[a][b]=xx*((double)ti1/100); |