网址：http://www.cs.toronto.edu/~hinton/MatlabForSciencePaper.html
这个代码主要是在mnist上做手写数字识别的代码，贴出来的目的主要是想研究一下在迭代过程中能量的变化情况。
1. 标准能量函数
标准的能量函数的表达式为：
那么就将这个表达式放在每一批次迭代的末尾，然后将所有批次迭代一次的结果累加就是当前迭代次数的能量值。
我写的表达式如下：
energy=energy-sum(sum(negdata*vishid.*poshidstates))-...sum(sum(negdata.*repmat(visbiases,size(negdata,1),1)))-...sum(sum(poshidstates.*repmat(hidbiases,size(poshidstates,1),1)));
【注】肯能还有更高效简洁的写法，只要保重RBM对应的二部图中的每一条连接线都有计算入能量函数中去就行。
第一层迭代了五十次的结果如下
Pretraining Layer 1 with RBM: 784-500
epoch 1
epoch 1 error 906233.0 energy -11099587.215172
epoch 2
epoch 2 error 557528.5 energy -30047032.045595
epoch 3
epoch 3 error 496027.4 energy -53112598.240393
epoch 4
epoch 4 error 468267.6 energy -78539687.671057
epoch 5
epoch 5 error 452678.9 energy -105299865.601975
epoch 6
epoch 6 error 478165.1 energy -129373309.014318
epoch 7
epoch 7 error 454438.2 energy -152106055.038886
epoch 8
epoch 8 error 438489.1 energy -174308595.910380
epoch 9
epoch 9 error 428747.2 energy -195910590.668272
epoch 10
epoch 10 error 421810.9 energy -217028311.853859
epoch 11
epoch 11 error 416392.9 energy -237610724.670849
epoch 12
epoch 12 error 411178.5 energy -257759919.144502
epoch 13
epoch 13 error 408115.5 energy -277438854.842965
epoch 14
epoch 14 error 405091.1 energy -296657959.425424
epoch 15
epoch 15 error 403533.2 energy -315397871.648862
epoch 16
epoch 16 error 400616.4 energy -333710720.884443
epoch 17
epoch 17 error 398482.7 energy -351563160.004860
epoch 18
epoch 18 error 396461.9 energy -369009936.125511
epoch 19
epoch 19 error 395664.3 energy -386036922.651997
epoch 20
epoch 20 error 394134.4 energy -402656725.716640
epoch 21
epoch 21 error 392316.9 energy -418988195.512326
epoch 22
epoch 22 error 391985.3 energy -435041246.765398
epoch 23
epoch 23 error 390845.3 energy -450774503.774346
epoch 24
epoch 24 error 389345.9 energy -466241495.153597
epoch 25
epoch 25 error 388229.8 energy -481430774.200754
epoch 26
epoch 26 error 387980.9 energy -496349569.785631
epoch 27
epoch 27 error 386886.4 energy -510986252.435888
epoch 28
epoch 28 error 385948.6 energy -525397051.912323
epoch 29
epoch 29 error 385386.0 energy -539599081.300994
epoch 30
epoch 30 error 385348.8 energy -553574564.640536
epoch 31
epoch 31 error 385159.7 energy -567319607.941765
epoch 32
epoch 32 error 384260.7 energy -580846734.110570
epoch 33
epoch 33 error 383174.4 energy -594214877.616236
epoch 34
epoch 34 error 382934.3 energy -607405266.358145
epoch 35
epoch 35 error 382697.1 energy -620430109.937463
epoch 36
epoch 36 error 382095.5 energy -633260002.931164
epoch 37
epoch 37 error 381486.8 energy -645950143.694427
epoch 38
epoch 38 error 380661.3 energy -658521662.217831
epoch 39
epoch 39 error 379594.0 energy -670919982.496346
epoch 40
epoch 40 error 379996.7 energy -683233935.943210
epoch 41
epoch 41 error 379679.9 energy -695420009.453536
epoch 42
epoch 42 error 379632.1 energy -707458130.547251
epoch 43
epoch 43 error 379168.7 energy -719365334.031710
epoch 44
epoch 44 error 378787.6 energy -731171831.263565
epoch 45
epoch 45 error 378450.0 energy -742883647.550836
epoch 46
epoch 46 error 379054.0 energy -754479143.953231
epoch 47
epoch 47 error 377713.5 energy -765939049.101519
epoch 48
epoch 48 error 377721.1 energy -777334241.922596
epoch 49
epoch 49 error 376576.0 energy -788647805.279271
epoch 50
epoch 50 error 377725.3 energy -799861904.7312261、能量是负值
2、能量是不断减少，或者更形象点是逐渐降低的，也就是说绝对值是不断增大的，越来越远离0
这个也就说明了能量越低，模型越稳定
【注】其实我刚开始一直是以为能量值不断趋近于0的，实验结果证明了我的想法是错的。读者有任何想法也可以在评论区讨论讨论。有其他比较有价值的规律谢谢大家一起来完善
为了进一步验证这个结论，下面列出第二层的能量值变化
Pretraining Layer 2 with RBM: 500-500
epoch 1
epoch 1 error 987086.0 energy -9062710.051405
epoch 2
epoch 2 error 665834.8 energy -22633241.117738
epoch 3
epoch 3 error 610662.3 energy -36830053.019285
epoch 4
epoch 4 error 572091.6 energy -51281490.520769
epoch 5
epoch 5 error 540614.1 energy -65915030.623658
epoch 6
epoch 6 error 508557.3 energy -79431130.648043
epoch 7
epoch 7 error 434258.1 energy -92467710.762025
epoch 8
epoch 8 error 399902.2 energy -105430274.786403
epoch 9
epoch 9 error 383352.3 energy -118255093.037551
epoch 10
epoch 10 error 374529.1 energy -130904210.951242
epoch 11
epoch 11 error 368514.4 energy -143354512.397248
epoch 12
epoch 12 error 363563.4 energy -155627227.699558
epoch 13
epoch 13 error 360383.7 energy -167717570.110705
epoch 14
epoch 14 error 358861.5 energy -179615072.973138
epoch 15
epoch 15 error 356927.2 energy -191293665.350483
epoch 16
epoch 16 error 354678.5 energy -202753482.269224
epoch 17
epoch 17 error 353843.7 energy -214037964.189715
epoch 18
epoch 18 error 352791.3 energy -225155932.564075
epoch 19
epoch 19 error 351835.7 energy -236104963.909265
epoch 20
epoch 20 error 351010.4 energy -246873022.282188
epoch 21
epoch 21 error 350847.1 energy -257470705.809567
epoch 22
epoch 22 error 349911.6 energy -267900561.925812
epoch 23
epoch 23 error 349589.7 energy -278184302.605534
epoch 24
epoch 24 error 349400.3 energy -288317945.532343
epoch 25
epoch 25 error 349051.0 energy -298312328.865682
epoch 26
epoch 26 error 348249.4 energy -308184434.905750
epoch 27
epoch 27 error 348383.1 energy -317909079.840016
epoch 28
epoch 28 error 347618.3 energy -327508125.959673
epoch 29
epoch 29 error 347152.3 energy -336984710.426838
epoch 30
epoch 30 error 346707.3 energy -346366295.669933
epoch 31
epoch 31 error 347245.8 energy -355642353.726434
epoch 32
epoch 32 error 346502.2 energy -364798534.538484
epoch 33
epoch 33 error 346450.0 energy -373854688.256938
epoch 34
epoch 34 error 346358.1 energy -382818178.531560
epoch 35
epoch 35 error 346985.5 energy -391695106.532281
epoch 36
epoch 36 error 346818.4 energy -400478029.118356
epoch 37
epoch 37 error 346446.7 energy -409183885.047261
epoch 38
epoch 38 error 346409.3 energy -417793622.658515
epoch 39
epoch 39 error 346342.5 energy -426331454.717778
epoch 40
epoch 40 error 345685.5 energy -434801072.747726
epoch 41
epoch 41 error 345374.3 energy -443199522.909101
epoch 42
epoch 42 error 345590.6 energy -451522224.505664
epoch 43
epoch 43 error 345611.4 energy -459769819.711364
epoch 44
epoch 44 error 345459.2 energy -467949310.324651
epoch 45
epoch 45 error 345785.0 energy -476052215.833106
epoch 46
epoch 46 error 345640.1 energy -484099406.063227
epoch 47
epoch 47 error 344962.8 energy -492075533.907423
epoch 48
epoch 48 error 344727.7 energy -500000744.350654
epoch 49
epoch 49 error 345307.3 energy -507865250.165273
epoch 50
epoch 50 error 345707.6 energy -515675253.128295 2. 自由能量函数
如果看RBM相关论文，经常会提到一个东东叫做自由能量函数(free energy)
这个是一篇关于用玻尔兹曼机做分类的文章中的一个式子，可以去我的CSDN资源中下载
这里面的softplus也是一种函数类型，类似于ReLu、sigmoid等，Softplus函数是Logistic-Sigmoid函数原函数。
按照这个公式，可以抽取一下普通的RBM中的能量函数，在一位作者的博客中有相关解释，这里搬过来
其实上式在机器之心的某篇文章,戳这里也有提到, 有兴趣可以瞅瞅
当隐藏层单元值为二值情况时候，很容易证明出
代码书写如下：
energy=energy-sum(sum(negdata.*repmat(visbiases,size(negdata,1),1)))-...sum(sum( log(1+exp(negdata*vishid+repmat(hidbiases,size(poshidstates,1),1))) ));
Pretraining Layer 1 with RBM: 784-500
epoch 1
epoch 1 error 905276.0 energy -19881397.779961
epoch 2
epoch 2 error 558779.6 energy -47693109.210766
epoch 3
epoch 3 error 495834.6 energy -79830190.885475
epoch 4
epoch 4 error 468697.6 energy -114398712.137404
epoch 5
epoch 5 error 453039.3 energy -150310200.560388
epoch 6
epoch 6 error 478686.3 energy -182605142.786704
epoch 7
epoch 7 error 451919.2 energy -213180793.202729
epoch 8
epoch 8 error 438010.1 energy -242816234.242592
epoch 9
epoch 9 error 427760.4 energy -271590223.327619
epoch 10
epoch 10 error 420847.2 energy -299660032.178816
epoch 11
epoch 11 error 415645.8 energy -326954499.696240
epoch 12
epoch 12 error 410386.1 energy -353688087.921110
epoch 13
epoch 13 error 408491.9 energy -379773439.302978
epoch 14
epoch 14 error 404256.8 energy -405219040.048392
epoch 15
epoch 15 error 402607.1 energy -430136766.920731
epoch 16
epoch 16 error 398868.0 energy -454524480.565422
epoch 17
epoch 17 error 398208.7 energy -478411816.895757
epoch 18
epoch 18 error 396493.4 energy -501819653.242841
epoch 19
epoch 19 error 394605.5 energy -524762199.874034
epoch 20
epoch 20 error 392981.9 energy -547265960.648713
epoch 21
epoch 21 error 392338.4 energy -569340099.227584
epoch 22
epoch 22 error 390601.4 energy -591085154.512036
epoch 23
epoch 23 error 389879.6 energy -612461539.974077
epoch 24
epoch 24 error 388782.1 energy -633468451.054830
epoch 25
epoch 25 error 388364.5 energy -654125425.131981
epoch 26
epoch 26 error 386876.1 energy -674518443.101429
epoch 27
epoch 27 error 386544.7 energy -694619635.858860
epoch 28
epoch 28 error 385006.0 energy -714467061.612437
epoch 29
epoch 29 error 384898.6 energy -734014091.415583
epoch 30
epoch 30 error 384137.1 energy -753300736.046747
epoch 31
epoch 31 error 383938.5 energy -772303792.788062
epoch 32
epoch 32 error 383642.5 energy -791090169.476486
epoch 33
epoch 33 error 382299.7 energy -809661333.174433
epoch 34
epoch 34 error 381459.4 energy -828056903.650036
epoch 35
epoch 35 error 382378.4 energy -846229339.194885
epoch 36
epoch 36 error 381321.8 energy -864236457.069615
epoch 37
epoch 37 error 380576.9 energy -882130475.692142
epoch 38
epoch 38 error 380819.1 energy -899805881.559751
epoch 39
epoch 39 error 380575.8 energy -917291008.281778
epoch 40
epoch 40 error 380441.0 energy -934597287.769714
epoch 41
epoch 41 error 378924.6 energy -951799519.728817
epoch 42
epoch 42 error 379242.5 energy -968837996.416408
epoch 43
epoch 43 error 378084.0 energy -985763999.393868
epoch 44
epoch 44 error 377760.6 energy -1002485955.994587
epoch 45
epoch 45 error 378065.3 energy -1019135408.410493
epoch 46
epoch 46 error 378113.7 energy -1035609968.466859
epoch 47
epoch 47 error 378137.6 energy -1051977206.619751
epoch 48
epoch 48 error 377804.7 energy -1068256355.969882
epoch 49
epoch 49 error 377078.1 energy -1084361351.832399
epoch 50
epoch 50 error 376941.1 energy -1100352833.515859 Pretraining Layer 2 with RBM: 500-500
epoch 1
epoch 1 error 993889.2 energy -15205761.985310
epoch 2
epoch 2 error 676743.8 energy -34794600.316635
epoch 3
epoch 3 error 622237.7 energy -54860642.213566
epoch 4
epoch 4 error 585614.3 energy -75025441.023491
epoch 5
epoch 5 error 555836.9 energy -95270049.773320
epoch 6
epoch 6 error 521063.7 energy -113064624.687646
epoch 7
epoch 7 error 442315.3 energy -129533185.056863
epoch 8
epoch 8 error 407058.7 energy -145427948.901373
epoch 9
epoch 9 error 389512.7 energy -160926615.310918
epoch 10
epoch 10 error 379349.7 energy -176038677.575432
epoch 11
epoch 11 error 372263.0 energy -190819909.913752
epoch 12
epoch 12 error 368580.7 energy -205272520.010386
epoch 13
epoch 13 error 365315.1 energy -219429209.242861
epoch 14
epoch 14 error 363198.0 energy -233282955.187790
epoch 15
epoch 15 error 360865.5 energy -246872028.136193
epoch 16
epoch 16 error 360440.2 energy -260191385.979074
epoch 17
epoch 17 error 358262.1 energy -273277908.363102
epoch 18
epoch 18 error 356881.8 energy -286135114.134431
epoch 19
epoch 19 error 356049.6 energy -298750619.512386
epoch 20
epoch 20 error 356009.3 energy -311137943.591194
epoch 21
epoch 21 error 355319.0 energy -323321281.109947
epoch 22
epoch 22 error 354783.1 energy -335312699.884763
epoch 23
epoch 23 error 354477.3 energy -347119931.104351
epoch 24
epoch 24 error 354285.7 energy -358745715.833463
epoch 25
epoch 25 error 353366.3 energy -370214880.549267
epoch 26
epoch 26 error 352662.9 energy -381532009.670373
epoch 27
epoch 27 error 353588.7 energy -392675673.518439
epoch 28
epoch 28 error 352470.9 energy -403697030.809657
epoch 29
epoch 29 error 353054.5 energy -414589450.592768
epoch 30
epoch 30 error 352482.0 energy -425354947.125350
epoch 31
epoch 31 error 351719.4 energy -435996803.022372
epoch 32
epoch 32 error 351546.8 energy -446524221.283293
epoch 33
epoch 33 error 351439.6 energy -456927518.928294
epoch 34
epoch 34 error 351518.5 energy -467237727.772437
epoch 35
epoch 35 error 351350.9 energy -477451966.520637
epoch 36
epoch 36 error 350986.0 energy -487572943.439880
epoch 37
epoch 37 error 350939.2 energy -497598679.794740
epoch 38
epoch 38 error 350509.6 energy -507538226.850344
epoch 39
epoch 39 error 350470.0 energy -517375214.715247
epoch 40
epoch 40 error 351177.7 energy -527125526.454195
epoch 41
epoch 41 error 350052.4 energy -536812334.692637
epoch 42
epoch 42 error 350338.3 energy -546408162.241688
epoch 43
epoch 43 error 349897.6 energy -555922938.066808
epoch 44
epoch 44 error 350848.5 energy -565355955.788417
epoch 45
epoch 45 error 350459.8 energy -574712019.150728
epoch 46
epoch 46 error 349903.4 energy -583997810.590286
epoch 47
epoch 47 error 350130.9 energy -593219406.465314
epoch 48
epoch 48 error 349957.5 energy -602379132.214209 回顾一下便能发现，判断玻尔兹曼机的参数设置是否合理或者训练是否朝着正常方向的两个可能的判别标准就是：
1、重构误差：一般是均方差，是正值、曲线向X轴逼近
2、能量值或者自由能量值：负值、远离X轴方向收敛
当然如果是判别式的RBM，可以利用CNN的判别方法，观察验证集的分类错误率。
文章代码下载地址：http://download.csdn.net/detail/zb1165048017/9750247
数据集即为：链接：http://pan.baidu.com/s/1pK9YZIb 密码：qw3w
【注】博文是博主自己研究的，可能有错误，如若有任何问题，请在评论区标注，大家一起研究。
