網址：http://www.cs.toronto.edu/~hinton/MatlabForSciencePaper.html

這個代碼主要是在mnist上做手寫數字識別的代碼，貼出來的目的主要是想研究一下在迭代過程中能量的變化情況。

1. 標準能量函數

標準的能量函數的表達式為：

那么就將這個表達式放在每一批次迭代的末尾，然后將所有批次迭代一次的結果累加就是當前迭代次數的能量值。

我寫的表達式如下：

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)));具體位置截圖如下

【注】肯能還有更高效簡潔的寫法，只要保重RBM對應的二部圖中的每一條連接線都有計算入能量函數中去就行。

第一層迭代了五十次的結果如下

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 -799861904.731226可以有兩點發現：

1、能量是負值

2、能量是不斷減少，或者更形象點是逐漸降低的，也就是說絕對值是不斷增大的，越來越遠離0

這個也就說明了能量越低，模型越穩定

【注】其實我剛開始一直是以為能量值不斷趨近于0的，實驗結果證明了我的想法是錯的。讀者有任何想法也可以在評論區討論討論。有其他比較有價值的規律謝謝大家一起來完善

為了進一步驗證這個結論，下面列出第二層的能量值變化

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 可以發現能量值依舊遵循上面兩條規律。

2. 自由能量函數

如果看RBM相關論文，經常會提到一個東東叫做自由能量函數(free energy)

這個是一篇關于用玻爾茲曼機做分類的文章中的一個式子，可以去我的CSDN資源中下載

這里面的softplus也是一種函數類型，類似于ReLu、sigmoid等，Softplus函數是Logistic-Sigmoid函數原函數。

按照這個公式，可以抽取一下普通的RBM中的能量函數，在一位作者的博客中有相關解釋，這里搬過來

其實上式在機器之心的某篇文章,戳這里也有提到, 有興趣可以瞅瞅

當隱藏層單元值為二值情況時候，很容易證明出

代碼書寫如下：

energy=energy-sum(sum(negdata.*repmat(visbiases,size(negdata,1),1)))-...sum(sum( log(1+exp(negdata*vishid+repmat(hidbiases,size(poshidstates,1),1))) )); 迭代第一次的結果：

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 迭代第二次的結果

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

回顧一下便能發現，判斷玻爾茲曼機的參數設置是否合理或者訓練是否朝著正常方向的兩個可能的判別標準就是：

1、重構誤差：一般是均方差，是正值、曲線向X軸逼近

2、能量值或者自由能量值：負值、遠離X軸方向收斂

當然如果是判別式的RBM，可以利用CNN的判別方法，觀察驗證集的分類錯誤率。

文章代碼下載地址：http://download.csdn.net/detail/zb1165048017/9750247

數據集即為：鏈接：http://pan.baidu.com/s/1pK9YZIb 密碼：qw3w

【注】博文是博主自己研究的，可能有錯誤，如若有任何問題，請在評論區標注，大家一起研究。

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