限制波尔兹曼机(Restricted Boltzmann Machines)

能量模型的概念从统计力学中得来，它描述着整个系统的某种状态，系统越有序，系统能量波动越小，趋近于平衡状态，系统越无序，能量波动越大。例如：一个孤立的物体，其内部各处的温度不尽相同，那么热就从温度较高的地方流向温度较低的地方，最后达到各处温度都相同的状态，也就是热平衡的状态。在统计力学中，系统处于某个状态的相对概率为，即玻尔兹曼因子，其中T表示温度，是玻尔兹曼常数，是状态的能量。玻尔兹曼因子本身并不是一个概率，因为它还没有归一化。为了把玻尔兹曼因子归一化，使其成为一个概率，我们把它除以系统所有可能的状态的玻尔兹曼因子之和Z，称为配分函数(partition function)。这便给出了玻尔兹曼分布。

玻尔兹曼机（Boltzmann Machine，BM）是一种特殊形式的对数线性的马尔科夫随机场（Markov Random Field，MRF），即能量函数是自由变量的线性函数。 通过引入隐含单元，我们可以提升模型的表达能力，表示非常复杂的概率分布。限制性玻尔兹曼机（RBM）进一步加一些约束，在RBM中不存在可见单元与可见单元的链接，也不存在隐含单元与隐含单元的链接，如下图所示

能量函数在限制玻尔兹曼机中定义为，b,c,W为模型的参数，b,c分别为可见层和隐含层的偏置，W为可见层与隐含层的链接权重

有了上述三个公式我们可以使用最大似然估计来求解模型的参数：设 。把概率p(x)改写为 。

由于可见单元V和不可见单元h条件独立，利用这一性质，我们可以得到：

logistic回归估计v与h取一的概率：

有了以上条件，我们可以推导出参数变化的梯度值：

使用基于马尔可夫链的gibbs抽样，对于一个d维的随机向量x=(x1,x2,…xd)，假设我们无法求得x的联合概率分布p(x)，但我们知道给定x的其他分量是其第i个分量xi的条件分布，即p(xi|xi-),xi-=(x1,x2,…xi-1,xi+1…xd)。那么，我们可以从x的一个任意状态（如(x1(0),x2(0),…,xd(0))）开始，利用条件分布p(xi|xi-)，迭代地对这状态的每个分量进行抽样，随着抽样次数n的增加，随机变量(x1(n),x2(n),…,xd(n))的概率分布将以n的几何级数的速度收敛到x的联合概率分布p(v)。

基于RBM模型的对称结构，以及其中节点的条件独立行，我们可以使用Gibbs抽样方法得到服从RBM定义的分布的随机样本。在RBM中进行k步Gibbs抽样的具体算法为：用一个训练样本（或者可视节点的一个随机初始状态）初始化可视节点的状态v0，交替进行下面的抽样：

理论上，参数的每次更新需要让上面的链条图形遍历一次，这样带来的性能损耗毫无疑问是不能承受的。

Hinton教授提出一种改进方法叫做对比分歧（Contrastive Divergence）,即CD-K。他指出CD没有必要等待链收敛，样本可以通过k步 的gibbs抽样完成，仅需要较少的抽样步数（实验中使用一步）就可以得到足够好的效果。

下面给出RBM用到的CD-K算法伪代码。

关于deeplearning的c++实现放到了github上，由于时间关系只是实现了大致框架，细节方面有待改善，也欢迎大家的参与：https://github.com/loujiayu/deeplearning

　　下面附上Geoff Hinton提供的关于RBM的matlab代码

% Version 1.000
%
% Code provided by Geoff Hinton and Ruslan Salakhutdinov
%
% Permission is granted for anyone to copy, use, modify, or distribute this
% program and accompanying programs and documents for any purpose, provided
% this copyright notice is retained and prominently displayed, along with
% a note saying that the original programs are available from our
% web page.
% The programs and documents are distributed without any warranty, express or
% implied.  As the programs were written for research purposes only, they have
% not been tested to the degree that would be advisable in any important
% application.  All use of these programs is entirely at the user's own risk.

% This program trains Restricted Boltzmann Machine in which
% visible, binary, stochastic pixels are connected to
% hidden, binary, stochastic feature detectors using symmetrically
% weighted connections. Learning is done with -step Contrastive Divergence.
% The program assumes that the following variables are set externally:
% maxepoch  -- maximum number of epochs
% numhid    -- number of hidden units
% batchdata -- the data that is divided into batches (numcases numdims numbatches)
% restart   -- set to  if learning starts from beginning 

epsilonw      = 0.1;   % Learning rate for weights
epsilonvb     = 0.1;   % Learning rate for biases of visible units
epsilonhb     = 0.1;   % Learning rate for biases of hidden units
weightcost  = 0.0002;
initialmomentum  = 0.5;
finalmomentum    = 0.9;

[numcases numdims numbatches]=size(batchdata);

if restart ==,
  restart=;
  epoch=;

% Initializing symmetric weights and biases.
  vishid     = 0.1*randn(numdims, numhid);
  hidbiases  = zeros(,numhid);
  visbiases  = zeros(,numdims);

  poshidprobs = zeros(numcases,numhid);
  neghidprobs = zeros(numcases,numhid);
  posprods    = zeros(numdims,numhid);
  negprods    = zeros(numdims,numhid);
  vishidinc  = zeros(numdims,numhid);
  hidbiasinc = zeros(,numhid);
  visbiasinc = zeros(,numdims);
  batchposhidprobs=zeros(numcases,numhid,numbatches);
end

for epoch = epoch:maxepoch,
 fprintf(,'epoch %d\r',epoch);
 errsum=;
 for batch = :numbatches,
 fprintf(,'epoch %d batch %d\r',epoch,batch); 

%%%%%%%%% START POSITIVE PHASE %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  data = batchdata(:,:,batch);
  poshidprobs = ./( + exp(-data*vishid - repmat(hidbiases,numcases,)));
  batchposhidprobs(:,:,batch)=poshidprobs;
  posprods    = data' * poshidprobs;
  poshidact   = sum(poshidprobs);
  posvisact = sum(data);

%%%%%%%%% END OF POSITIVE PHASE  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  poshidstates = poshidprobs > rand(numcases,numhid);

%%%%%%%%% START NEGATIVE PHASE  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  negdata = ./( + exp(-poshidstates*vishid' - repmat(visbiases,numcases,1)));
  neghidprobs = ./( + exp(-negdata*vishid - repmat(hidbiases,numcases,)));
  negprods  = negdata'*neghidprobs;
  neghidact = sum(neghidprobs);
  negvisact = sum(negdata); 

%%%%%%%%% END OF NEGATIVE PHASE %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  err= sum(sum( (data-negdata).^ ));
  errsum = err + errsum;

   if epoch>,
     momentum=finalmomentum;
   else
     momentum=initialmomentum;
   end;

%%%%%%%%% UPDATE WEIGHTS AND BIASES %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    vishidinc = momentum*vishidinc + ...
                epsilonw*( (posprods-negprods)/numcases - weightcost*vishid);
    visbiasinc = momentum*visbiasinc + (epsilonvb/numcases)*(posvisact-negvisact);
    hidbiasinc = momentum*hidbiasinc + (epsilonhb/numcases)*(poshidact-neghidact);

    vishid = vishid + vishidinc;
    visbiases = visbiases + visbiasinc;
    hidbiases = hidbiases + hidbiasinc;

%%%%%%%%%%%%%%%% END OF UPDATES %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

  end
  fprintf(, 'epoch %4i error %6.1f  \n', epoch, errsum);
end;