用神经网络分类两条夹角为θ的直线
分类两条直线y=0和y=x*tanθ，
设r为0到1之间的随机数，两个训练集为
A:[[r][r*tanθ]
B:[r][0]
训练集有5000个，测试集初始化方式相同，有1000个。
网络结构为
收敛标准从0.5-1e-4,共16个收敛标准，每个收敛标准收敛199次，统计平均值。θ从10到350共有32个值。共收敛了32*16*199次。
观察网络的迭代次数是如何随着θ的改变而改变的。
数据表格
|
10 |
20 |
30 |
40 |
50 |
60 |
70 |
80 |
100 |
110 |
120 |
130 |
140 |
150 |
160 |
170 |
190 |
200 |
210 |
220 |
230 |
240 |
250 |
260 |
280 |
290 |
300 |
310 |
320 |
330 |
340 |
350 |
|
|
δ |
迭代次数n |
迭代次数n |
迭代次数n |
迭代次数n |
迭代次数n |
迭代次数n |
迭代次数n |
迭代次数n |
迭代次数n |
迭代次数n |
迭代次数n |
迭代次数n |
迭代次数n |
迭代次数n |
迭代次数n |
迭代次数n |
迭代次数n |
迭代次数n |
迭代次数n |
迭代次数n |
迭代次数n |
迭代次数n |
迭代次数n |
迭代次数n |
迭代次数n |
迭代次数n |
迭代次数n |
迭代次数n |
迭代次数n |
迭代次数n |
迭代次数n |
迭代次数n |
|
0.5 |
195.3317 |
198.0754 |
157.6432 |
135.6131 |
109.7889 |
97.42714 |
72.41709 |
36.61307 |
31.77889 |
75.9196 |
103.3518 |
91.22613 |
118.407 |
141.9296 |
180.8945 |
220.9548 |
175.3317 |
169.603 |
151.7286 |
132.2111 |
80.53769 |
85.11558 |
76.45226 |
43.9799 |
43.38693 |
60.45729 |
93.25628 |
115.0653 |
142.7638 |
141.6131 |
198.7337 |
201.7437 |
|
0.4 |
13914.29 |
6238.05 |
4033.643 |
2535.312 |
1726.704 |
1186.769 |
719.3216 |
352.6985 |
326.6533 |
712.3015 |
1183.181 |
1713.543 |
2665.688 |
3767.573 |
6406.779 |
14131.71 |
14513.17 |
6469.362 |
3712.377 |
2558.819 |
1774.955 |
1177.97 |
738.7136 |
360.8794 |
345.1508 |
726.6935 |
1170.754 |
1801.859 |
2534.94 |
3807.482 |
6263.241 |
13953.42 |
|
0.3 |
15479.9 |
6880.447 |
4314.97 |
2875.08 |
1991.296 |
1433.704 |
943.6281 |
550.6734 |
537.206 |
927.4472 |
1436.005 |
2005.688 |
2904.95 |
4326.799 |
7092.92 |
15352.95 |
15828.34 |
7207.407 |
4280.477 |
2905.291 |
1995.302 |
1365.608 |
926.2161 |
550.2261 |
551.6533 |
926.1106 |
1374.543 |
1969.181 |
2854.638 |
4348.296 |
7014.372 |
15623.51 |
|
0.2 |
16439.69 |
7453.307 |
4676.276 |
3160.387 |
2190.97 |
1625.216 |
1172.698 |
829.4322 |
792.809 |
1161.095 |
1610.412 |
2313.342 |
3237.151 |
4708.729 |
7503.362 |
16668.35 |
17014.31 |
7809.156 |
4654.045 |
3265.834 |
2267.121 |
1598.603 |
1189.804 |
786.3769 |
810.1608 |
1168.784 |
1611.04 |
2278.844 |
3176.864 |
4650.327 |
7539.683 |
16186.92 |
|
0.1 |
17919.78 |
8359.814 |
5291.221 |
3667.03 |
2638.603 |
2058.513 |
1536.347 |
1112.829 |
1116.719 |
1504.347 |
2001.141 |
2615.673 |
3714.839 |
5351.764 |
8178.975 |
17654.81 |
18440.28 |
8570.09 |
5176.899 |
3672.869 |
2699.02 |
1934.538 |
1498.281 |
1179.724 |
1219.583 |
1566.337 |
1950.065 |
2628.94 |
3601.472 |
5212.307 |
8356.849 |
18137.86 |
|
0.01 |
22466.49 |
10807.97 |
7469.513 |
5596.93 |
4627.839 |
3859.342 |
3361.714 |
3373.261 |
3130.035 |
3677.377 |
3937.111 |
4609.623 |
5599.352 |
7404.98 |
10909.85 |
22333.64 |
22566.98 |
11233.77 |
7381.844 |
5608.055 |
4598.362 |
3856.548 |
4436.608 |
4165.668 |
3228.794 |
3790.955 |
3726.744 |
4560.789 |
5529.422 |
7406.422 |
10770.9 |
22136.5 |
|
0.001 |
29045.94 |
16087.35 |
12214.04 |
10616.67 |
9757.819 |
10020.35 |
11161.16 |
52953.74 |
30886.15 |
10725.99 |
9588.814 |
10233.06 |
10915.11 |
12154.33 |
15966.08 |
29443.16 |
29662.3 |
16204.87 |
12123.7 |
10728.71 |
10144.03 |
9265.98 |
9767.955 |
40860.69 |
27934.85 |
10644.34 |
9200.06 |
10061.53 |
10669.82 |
12118.55 |
16012.09 |
28558.85 |
|
9.00E-04 |
29906.22 |
16205.63 |
12414.62 |
11048.55 |
10249.3 |
9974.377 |
12840.99 |
32423.63 |
69727.28 |
11143.14 |
10130.93 |
10492.75 |
11330.62 |
12524.45 |
16315.51 |
29979.6 |
30114.91 |
16524.46 |
12456.86 |
11204.39 |
10473.9 |
9534.101 |
11885.01 |
29902.56 |
42065.91 |
11187.51 |
9696.09 |
10372.28 |
11021.08 |
12422.62 |
16464.74 |
29512.95 |
|
8.00E-04 |
30505.62 |
16730.83 |
12846.1 |
11500.53 |
10718.98 |
10449.65 |
10274.39 |
50047.21 |
22749.25 |
12309.07 |
10513.88 |
10991.35 |
11740.57 |
12941.65 |
16602.16 |
30174.19 |
30631.14 |
16917.49 |
12932.58 |
11518 |
10773.39 |
9895.116 |
11344.74 |
40052.26 |
51881.49 |
9800.09 |
10291.73 |
10812.21 |
11393.32 |
12991.64 |
16670.48 |
29615.91 |
|
7.00E-04 |
30878.16 |
17098.47 |
13142.76 |
12078.96 |
11400.84 |
11165.35 |
11746.1 |
46881.28 |
29219.01 |
11548.01 |
11167.45 |
11310.91 |
12349.44 |
13389.07 |
17058.72 |
30827.22 |
31302.73 |
17200.23 |
13269.82 |
12124.67 |
11100.07 |
10716.31 |
11559.13 |
34973.5 |
25655.78 |
12788.38 |
11415.21 |
11380.51 |
11897.51 |
13437.53 |
17166.15 |
30572.86 |
|
6.00E-04 |
31735.77 |
17696.11 |
13751.94 |
12778.49 |
12138.33 |
11835.68 |
13506.68 |
43265.07 |
58553.6 |
12512.39 |
11808.45 |
11886.3 |
12950.76 |
13877.09 |
17620.75 |
31522.76 |
31610.18 |
17922.39 |
13852.37 |
12693.26 |
11883.21 |
12214.49 |
11677.46 |
35671.03 |
64251.91 |
11230.14 |
12521.11 |
12228.15 |
12669.77 |
13910.53 |
17700.87 |
30899.94 |
|
5.00E-04 |
32248.65 |
18160.16 |
14352.14 |
13474.12 |
12806.03 |
12512.8 |
13329.56 |
19513.03 |
59224.43 |
16901.82 |
12651.84 |
12745.24 |
13859.97 |
14401.74 |
18307.13 |
32465.1 |
32552.29 |
18456.9 |
14614.96 |
13634.92 |
12887.25 |
13418.06 |
13367.04 |
38435.47 |
42394.26 |
14905.44 |
13005.3 |
13377.39 |
13589.15 |
14619.36 |
18203.93 |
32070.51 |
|
4.00E-04 |
33464.13 |
19170.51 |
15337.98 |
14424.92 |
14200.39 |
13589.12 |
14980.47 |
45571.48 |
60102.32 |
17157.57 |
14247.02 |
14223.89 |
14849.99 |
15113.84 |
18904.38 |
33212.31 |
33518.07 |
19342.46 |
15453.51 |
14744.36 |
14233.24 |
14479.19 |
15331.63 |
45132.32 |
39034.05 |
14730.45 |
14746.82 |
14687.08 |
14433.19 |
15425.19 |
19210.96 |
33230.69 |
|
3.00E-04 |
34532.6 |
20339.17 |
16654.45 |
15992.67 |
15477.73 |
16568.63 |
17935.51 |
32716.51 |
38947.18 |
15841.03 |
15278.81 |
15971.04 |
16196.7 |
16659.85 |
20064.71 |
34419.91 |
35243.98 |
20352.41 |
16645.13 |
15725.6 |
15932.52 |
16824.29 |
18251.16 |
50232.28 |
65534.15 |
16970.77 |
15538.96 |
16528.97 |
16241.62 |
16750.31 |
20295.76 |
34232.78 |
|
2.00E-04 |
37157.44 |
22060.26 |
18760.1 |
18567.65 |
18526.91 |
19728.33 |
21393.43 |
85402.25 |
41534.36 |
21412.18 |
20275.08 |
18604.46 |
18489.37 |
18997.44 |
21990.85 |
35925.2 |
36928.21 |
22214.2 |
18753.98 |
17449.03 |
19393.23 |
19266.34 |
21286.21 |
44564.16 |
47518.88 |
19542.49 |
18753.08 |
19352.96 |
18648.54 |
18746.47 |
21862.37 |
36171.3 |
|
1.00E-04 |
40897.64 |
25703.87 |
22880.25 |
23550.47 |
24741.73 |
26411.01 |
26570.07 |
78260.95 |
78524.66 |
29832.78 |
25279.78 |
24705.55 |
23927.27 |
23034.28 |
25414.78 |
40711.15 |
41047.39 |
25718.63 |
23009.71 |
23444.37 |
24825.76 |
27288.6 |
32539.15 |
58313.76 |
75453.45 |
27253.39 |
26135.43 |
26387.03 |
23973.38 |
22846.3 |
25862.5 |
39897.13 |
将迭代次数曲线画成图
可以看到明显的对称关系，有两个小峰位于0和180位置，有两个大峰位于90，270.
由假设1：完全相同的两个对象无法被分成两类，与之对应的分类迭代次数为无穷大，分类准确率是50%，50%。相等收敛标准下迭代次数越大表明二者差异越小。
两个对象之间的差异越小迭代次数越大，这个假设可以很好的解释0和180的两个小峰，因为θ越小，两条直线的夹角越小，越相似。
但如何解释位于90和270的两个大峰？如果θ接近90或270将与y=0垂直，直观上这应该是两条直线差异最大的情况，按照假设1应该迭代次数变小才对。
但想象两条垂直的直线只有1个交点，而两条重合的直线有无数的交点，1个点和1条线之间的差异小于两条线之间的差异似乎也是合理的。由此两个90和270的大峰也得以解释。