[UFLDL] Exercise 1C：Softmax Regression

痛定思痛。 2023-02-26 14:27 27阅读 0赞

> 代码来源：**[ UFLDL-Tutorial-notes-code][UFLDL-Tutorial-notes-code]**，感谢其分享！
> 
> 本人工作只是对代码进行讲解，如有改动会单独指出。

### 文章目录 ###

*   *   *  任务
         *  代码
         *  思路
         *  解析
         *   *  计算目标函数  J ( θ ) J(\\theta) J(θ)
             *  计算梯度

### 任务 ###

完成多分类softmax的目标函数 J ( θ ) J(\\theta) J(θ)和梯度 ∇ θ J ( θ ) \\nabla\_\{\\theta\}J(\\theta) ∇θJ(θ)的计算。

### 代码 ###

softmax\_regression\_vec.m 代码如下：

function [f,g] = softmax_regression(theta, X,y)
      %
      % Arguments:
      %   theta - A vector containing the parameter values to optimize.
      %       In minFunc, theta is reshaped to a long vector.  So we need to
      %       resize it to an n-by-(num_classes-1) matrix.
      %       Recall that we assume theta(:,num_classes) = 0.
      %
      %   X - The examples stored in a matrix.  
      %       X(i,j) is the i'th coordinate of the j'th example.
      %   y - The label for each example.  y(j) is the j'th example's label.
      %
      m=size(X,2);
      n=size(X,1);
    
      % theta is a vector;  need to reshape to n x num_classes.
      theta=reshape(theta, n, []);
      num_classes=size(theta,2)+1;
      
      % initialize objective value and gradient.
      f = 0;
      g = zeros(size(theta));
    
      %
      % TODO:  Compute the softmax objective function and gradient using vectorized code.
      %        Store the objective function value in 'f', and the gradient in 'g'.
      %        Before returning g, make sure you form it back into a vector with g=g(:);
      %
    %%% YOUR CODE HERE %%%
    
      A = exp([theta' * X; zeros(1,m)]);        % 计算θ^T * X
      B = bsxfun(@rdivide, A, sum(A));          % 对θ^T * X 归一化
      C = log(B);                               % 取对数
      I = sub2ind(size(C),y,1:size(C,2));       % 获取label对应的位置的值
      f = (-1) * sum(C(I));                     % 计算目标函数
      
      % calculate g 
      Y = repmat(y',1,num_classes);
      for i=1:num_classes
          Y(Y(:,i)~=i,i) = 0;       % 第i列中，仅标签为i的样本对应的位置不为0。
      end
      Y(Y~=0)=1;                % 第i列非零元所在的位置，表示标签为i的元素的索引
      disp(size(Y(:,1:(size(Y,2)-1))))
      disp(size(B(1:(size(B,1)-1),:)'))
      g = (-1) * X * (Y(:,1:(size(Y,2)-1))-B(1:(size(B,1)-1),:)');
      
      %%% 别人的写法，两种写法效果一样,主要是稀疏矩阵生成不一样一点，他的速度略快%%
      %%% 因为这里num_classes还很小，我耗时0.014272秒，他的耗时0.014249秒 %%%
    %   h = theta'*X;%h(k,i)第k个theta，第i个样本
    %   a = exp(h);
    %   a = [a;ones(1,size(a,2))];%加1行
    %   p = bsxfun(@rdivide,a,sum(a));
    %   c = log2(p);
    %   i = sub2ind(size(c), y,[1:size(c,2)]);
    %   values = c(i);
    %   f = -sum(values);
    
    %   d = full(sparse(1:m,y,1));
    %   d = d(:,1:(size(d,2)-1));%减1列
    %   p = p(1:(size(p,1)-1),:);%减1行
    %   g = X*(p'.-d);  
    
      g=g(:); % make gradient a vector for minFunc

### 思路 ###

主要需要的就是这两个公式：

*  损失函数  J ( θ ) J(\\theta) J(θ)  
     J ( θ ) = − \[ ∑ i = 1 m ∑ k = 1 K 1 \{ y ( i ) = k \} log ⁡ exp ⁡ ( θ ( k ) ⊤ x ( i ) ) ∑ j = 1 K exp ⁡ ( θ ( j ) ⊤ x ( i ) ) \] = − \[ ∑ i = 1 m ( 1 − y ( i ) ) log ⁡ ( 1 − h θ ( x ( i ) ) ) + y ( i ) log ⁡ h θ ( x ( i ) ) \] = − \[ ∑ i = 1 m ∑ k = 0 1 1 \{ y ( i ) = k \} log ⁡ P ( y ( i ) = k ∣ x ( i ) ; θ ) \] \\begin\{aligned\} J(\\theta) &= - \\left\[ \\sum\_\{i=1\}^\{m\} \\sum\_\{k=1\}^\{K\} 1\\left\\\{y^\{(i)\} = k\\right\\\} \\log \\frac\{\\exp(\\theta^\{(k)\\top\} x^\{(i)\})\}\{\\sum\_\{j=1\}^K \\exp(\\theta^\{(j)\\top\} x^\{(i)\})\}\\right\] \\\\ &= - \\left\[ \\sum\_\{i=1\}^m (1-y^\{(i)\}) \\log (1-h\_\\theta(x^\{(i)\})) + y^\{(i)\} \\log h\_\\theta(x^\{(i)\}) \\right\] \\\\ &= - \\left\[ \\sum\_\{i=1\}^\{m\} \\sum\_\{k=0\}^\{1\} 1\\left\\\{y^\{(i)\} = k\\right\\\} \\log P(y^\{(i)\} = k | x^\{(i)\} ; \\theta) \\right\] \\end\{aligned\} J(θ)=−\[i=1∑mk=1∑K1\{ y(i)=k\}log∑j=1Kexp(θ(j)⊤x(i))exp(θ(k)⊤x(i))\]=−\[i=1∑m(1−y(i))log(1−hθ(x(i)))\+y(i)loghθ(x(i))\]=−\[i=1∑mk=0∑11\{ y(i)=k\}logP(y(i)=k∣x(i);θ)\]
 *  梯度  ∇ θ J ( θ ) \\nabla\_\{\\theta\}J(\\theta) ∇θJ(θ)  
     ∇ θ ( k ) J ( θ ) = − ∑ i = 1 m \[ x ( i ) ( 1 \{ y ( i ) = k \} − P ( y ( i ) = k ∣ x ( i ) ; θ ) ) \] \\begin\{aligned\} \\nabla\_\{\\theta^\{(k)\}\} J(\\theta) = - \\sum\_\{i=1\}^\{m\}\{ \\left\[ x^\{(i)\} \\left( 1\\\{ y^\{(i)\} = k\\\} - P(y^\{(i)\} = k | x^\{(i)\}; \\theta) \\right) \\right\] \} \\end\{aligned\} ∇θ(k)J(θ)=−i=1∑m\[x(i)(1\{ y(i)=k\}−P(y(i)=k∣x(i);θ))\]

**但是有一点奇怪的地方：**

在所给的框架下， θ \\theta θ 的维度是  n × ( K − 1 ) n×(K-1) n×(K−1)，其中  K K K 为类别数，此处为10。

这样，我们的预测值  θ T X \\theta^TX θTX 的维度就变成了  ( K − 1 ) × m (K-1)×m (K−1)×m，而不是  K × m K×m K×m，我个人认为还是挺奇怪的。

解决办法就是增加一个与  X X X 无关的偏置项（同样也不可优化），附加在  θ T X \\theta^TX θTX 之后，从而将维度变为  K × m K×m K×m。

这样  n × ( K − 1 ) n×(K-1) n×(K−1) 维的参数确实也可以完成  K K K 类的分类，只不过有点难以理解。因为当  θ \\theta θ 的每个元素都很小时，则偏置项相对较大，则被分类为第  K K K 类。

理解了这个之后，下面就很容易懂了。

### 解析 ###

#### 计算目标函数  J ( θ ) J(\\theta) J(θ) ####

A = exp([theta' * X; zeros(1,m)]);        % 计算θ^T * X
    B = bsxfun(@rdivide, A, sum(A));          % 对θ^T * X 归一化
    C = log(B);                               % 取对数
    I = sub2ind(size(C),y,1:size(C,2));       % 获取label对应的位置的值
    f = (-1) * sum(C(I));                     % 计算目标函数

这部分对应目标函数的公式：  
 J ( θ ) = − \[ ∑ i = 1 m ∑ k = 1 K 1 \{ y ( i ) = k \} log ⁡ exp ⁡ ( θ ( k ) ⊤ x ( i ) ) ∑ j = 1 K exp ⁡ ( θ ( j ) ⊤ x ( i ) ) \] = − \[ ∑ i = 1 m ( 1 − y ( i ) ) log ⁡ ( 1 − h θ ( x ( i ) ) ) + y ( i ) log ⁡ h θ ( x ( i ) ) \] = − \[ ∑ i = 1 m ∑ k = 0 1 1 \{ y ( i ) = k \} log ⁡ P ( y ( i ) = k ∣ x ( i ) ; θ ) \] \\begin\{aligned\} J(\\theta) &= - \\left\[ \\sum\_\{i=1\}^\{m\} \\sum\_\{k=1\}^\{K\} 1\\left\\\{y^\{(i)\} = k\\right\\\} \\log \\frac\{\\exp(\\theta^\{(k)\\top\} x^\{(i)\})\}\{\\sum\_\{j=1\}^K \\exp(\\theta^\{(j)\\top\} x^\{(i)\})\}\\right\] \\\\ &= - \\left\[ \\sum\_\{i=1\}^m (1-y^\{(i)\}) \\log (1-h\_\\theta(x^\{(i)\})) + y^\{(i)\} \\log h\_\\theta(x^\{(i)\}) \\right\] \\\\ &= - \\left\[ \\sum\_\{i=1\}^\{m\} \\sum\_\{k=0\}^\{1\} 1\\left\\\{y^\{(i)\} = k\\right\\\} \\log P(y^\{(i)\} = k | x^\{(i)\} ; \\theta) \\right\] \\end\{aligned\} J(θ)=−\[i=1∑mk=1∑K1\{ y(i)=k\}log∑j=1Kexp(θ(j)⊤x(i))exp(θ(k)⊤x(i))\]=−\[i=1∑m(1−y(i))log(1−hθ(x(i)))\+y(i)loghθ(x(i))\]=−\[i=1∑mk=0∑11\{ y(i)=k\}logP(y(i)=k∣x(i);θ)\]  
说明比较重要的几点：

*  `A = exp([theta' * X; zeros(1,m)]);`是计算  exp ⁡ ( θ ⊤ X ) \\exp(\\theta^\\top X) exp(θ⊤X)，其中`zeros(1,m)`就是上面提到的偏置项。
    
    `A`的维度是  \[ ( K − 1 ) × n \] × \[ n × m \] + 1 × m = \[ ( K − 1 ) × m \] + \[ 1 × m \] = K × m \[(K-1)×n\]×\[n×m\] + 1×m = \[(K-1)×m\] + \[1×m\] = K×m \[(K−1)×n\]×\[n×m\]\+1×m=\[(K−1)×m\]\+\[1×m\]=K×m，具体在此处为10×60000。
 *  `I = sub2ind(size(C),y,1:size(C,2));` 和 `f = (-1) * sum(C(I));` 完成的实际上是  1 \{ y ( i ) = k \} 1\\left\\\{y^\{(i)\} = k\\right\\\} 1\{ y(i)=k\} 这个函数。即筛选出预测类别中正确类别的概率，来计算目标函数。
    
    其中前一句是得到预测类别中正确类别的概率所在的**索引**（`sub2ind`产生索引为线性索引），而 `y` 指明了获取的索引为**预测类别中正确类别的概率**。
    
    后一句通过访问索引，得到  log ⁡ P ( y ( i ) = k ∣ x ( i ) ; θ ) \\log P(y^\{(i)\} = k | x^\{(i)\} ; \\theta) logP(y(i)=k∣x(i);θ) ，计算目标函数。

变量的内容：

*  `A` ：每一列代表一个样本，每一行代表对应类别的评分。
    
    <table> 
     <thead> 
      <tr> 
       <th align="center">1.059044</th> 
       <th align="center">1.016396</th> 
       <th align="center">1.044481</th> 
       <th align="center">1.094523</th> 
       <th align="center">1.013291</th> 
       <th align="center">……</th> 
      </tr> 
     </thead> 
     <tbody> 
      <tr> 
       <td align="center">1.042245</td> 
       <td align="center">1.009933</td> 
       <td align="center">1.044206</td> 
       <td align="center">1.097755</td> 
       <td align="center">1.011094</td> 
       <td align="center">……</td> 
      </tr> 
      <tr> 
       <td align="center">1.055637</td> 
       <td align="center">1.012596</td> 
       <td align="center">1.034192</td> 
       <td align="center">1.084184</td> 
       <td align="center">1.002852</td> 
       <td align="center">……</td> 
      </tr> 
      <tr> 
       <td align="center">1.065303</td> 
       <td align="center">1.014536</td> 
       <td align="center">1.034789</td> 
       <td align="center">1.09325</td> 
       <td align="center">1.009064</td> 
       <td align="center">……</td> 
      </tr> 
      <tr> 
       <td align="center">1.063292</td> 
       <td align="center">1.014994</td> 
       <td align="center">1.026204</td> 
       <td align="center">1.085303</td> 
       <td align="center">1.017293</td> 
       <td align="center">……</td> 
      </tr> 
      <tr> 
       <td align="center">1.05112</td> 
       <td align="center">1.016776</td> 
       <td align="center">1.039049</td> 
       <td align="center">1.088784</td> 
       <td align="center">1.014322</td> 
       <td align="center">……</td> 
      </tr> 
      <tr> 
       <td align="center">1.060112</td> 
       <td align="center">1.019359</td> 
       <td align="center">1.047857</td> 
       <td align="center">1.10238</td> 
       <td align="center">1.010763</td> 
       <td align="center">……</td> 
      </tr> 
      <tr> 
       <td align="center">1.064268</td> 
       <td align="center">1.006946</td> 
       <td align="center">1.039117</td> 
       <td align="center">1.097793</td> 
       <td align="center">1.007203</td> 
       <td align="center">……</td> 
      </tr> 
      <tr> 
       <td align="center">1.055498</td> 
       <td align="center">1.020393</td> 
       <td align="center">1.027037</td> 
       <td align="center">1.076283</td> 
       <td align="center">1.006618</td> 
       <td align="center">……</td> 
      </tr> 
      <tr> 
       <td align="center">1</td> 
       <td align="center">1</td> 
       <td align="center">1</td> 
       <td align="center">1</td> 
       <td align="center">1</td> 
       <td align="center">……</td> 
      </tr> 
     </tbody> 
    </table>
 *  `B`：`A`的归一化
    
    <table> 
     <thead> 
      <tr> 
       <th align="center">0.100703</th> 
       <th align="center">0.100316</th> 
       <th align="center">0.101044</th> 
       <th align="center">0.101155</th> 
       <th align="center">0.1004</th> 
       <th align="center">……</th> 
      </tr> 
     </thead> 
     <tbody> 
      <tr> 
       <td align="center">0.099106</td> 
       <td align="center">0.099678</td> 
       <td align="center">0.101017</td> 
       <td align="center">0.101454</td> 
       <td align="center">0.100183</td> 
       <td align="center">……</td> 
      </tr> 
      <tr> 
       <td align="center">0.100379</td> 
       <td align="center">0.099941</td> 
       <td align="center">0.100048</td> 
       <td align="center">0.100199</td> 
       <td align="center">0.099366</td> 
       <td align="center">……</td> 
      </tr> 
      <tr> 
       <td align="center">0.101298</td> 
       <td align="center">0.100133</td> 
       <td align="center">0.100106</td> 
       <td align="center">0.101037</td> 
       <td align="center">0.099982</td> 
       <td align="center">……</td> 
      </tr> 
      <tr> 
       <td align="center">0.101107</td> 
       <td align="center">0.100178</td> 
       <td align="center">0.099276</td> 
       <td align="center">0.100303</td> 
       <td align="center">0.100797</td> 
       <td align="center">……</td> 
      </tr> 
      <tr> 
       <td align="center">0.099949</td> 
       <td align="center">0.100354</td> 
       <td align="center">0.100518</td> 
       <td align="center">0.100625</td> 
       <td align="center">0.100503</td> 
       <td align="center">……</td> 
      </tr> 
      <tr> 
       <td align="center">0.100804</td> 
       <td align="center">0.100609</td> 
       <td align="center">0.10137</td> 
       <td align="center">0.101881</td> 
       <td align="center">0.10015</td> 
       <td align="center">……</td> 
      </tr> 
      <tr> 
       <td align="center">0.1012</td> 
       <td align="center">0.099383</td> 
       <td align="center">0.100525</td> 
       <td align="center">0.101457</td> 
       <td align="center">0.099797</td> 
       <td align="center">……</td> 
      </tr> 
      <tr> 
       <td align="center">0.100366</td> 
       <td align="center">0.100711</td> 
       <td align="center">0.099356</td> 
       <td align="center">0.099469</td> 
       <td align="center">0.099739</td> 
       <td align="center">……</td> 
      </tr> 
      <tr> 
       <td align="center">0.095088</td> 
       <td align="center">0.098698</td> 
       <td align="center">0.096741</td> 
       <td align="center">0.092419</td> 
       <td align="center">0.099083</td> 
       <td align="center">……</td> 
      </tr> 
     </tbody> 
    </table>
 *  `C`：对`B`取对数
    
    <table> 
     <thead> 
      <tr> 
       <th align="center">-2.29558</th> 
       <th align="center">-2.29943</th> 
       <th align="center">-2.2922</th> 
       <th align="center">-2.2911</th> 
       <th align="center">-2.29859</th> 
       <th align="center">……</th> 
      </tr> 
     </thead> 
     <tbody> 
      <tr> 
       <td align="center">-2.31157</td> 
       <td align="center">-2.30581</td> 
       <td align="center">-2.29247</td> 
       <td align="center">-2.28815</td> 
       <td align="center">-2.30076</td> 
       <td align="center">……</td> 
      </tr> 
      <tr> 
       <td align="center">-2.2988</td> 
       <td align="center">-2.30317</td> 
       <td align="center">-2.3021</td> 
       <td align="center">-2.30059</td> 
       <td align="center">-2.30894</td> 
       <td align="center">……</td> 
      </tr> 
      <tr> 
       <td align="center">-2.28969</td> 
       <td align="center">-2.30126</td> 
       <td align="center">-2.30153</td> 
       <td align="center">-2.29227</td> 
       <td align="center">-2.30277</td> 
       <td align="center">……</td> 
      </tr> 
      <tr> 
       <td align="center">-2.29158</td> 
       <td align="center">-2.30081</td> 
       <td align="center">-2.30986</td> 
       <td align="center">-2.29956</td> 
       <td align="center">-2.29465</td> 
       <td align="center">……</td> 
      </tr> 
      <tr> 
       <td align="center">-2.30309</td> 
       <td align="center">-2.29905</td> 
       <td align="center">-2.29742</td> 
       <td align="center">-2.29636</td> 
       <td align="center">-2.29757</td> 
       <td align="center">……</td> 
      </tr> 
      <tr> 
       <td align="center">-2.29457</td> 
       <td align="center">-2.29652</td> 
       <td align="center">-2.28898</td> 
       <td align="center">-2.28395</td> 
       <td align="center">-2.30109</td> 
       <td align="center">……</td> 
      </tr> 
      <tr> 
       <td align="center">-2.29066</td> 
       <td align="center">-2.30877</td> 
       <td align="center">-2.29735</td> 
       <td align="center">-2.28812</td> 
       <td align="center">-2.30462</td> 
       <td align="center">……</td> 
      </tr> 
      <tr> 
       <td align="center">-2.29893</td> 
       <td align="center">-2.2955</td> 
       <td align="center">-2.30905</td> 
       <td align="center">-2.30791</td> 
       <td align="center">-2.3052</td> 
       <td align="center">……</td> 
      </tr> 
      <tr> 
       <td align="center">-2.35295</td> 
       <td align="center">-2.31569</td> 
       <td align="center">-2.33572</td> 
       <td align="center">-2.38142</td> 
       <td align="center">-2.31179</td> 
       <td align="center">……</td> 
      </tr> 
     </tbody> 
    </table>
 *  `I`：正确概率的线性索引
    
    <table> 
     <thead> 
      <tr> 
       <th align="center">6</th> 
       <th align="center">14</th> 
       <th align="center">21</th> 
       <th align="center">37</th> 
       <th align="center">44</th> 
       <th align="center">……</th> 
      </tr> 
     </thead> 
     <tbody></tbody> 
    </table>

#### 计算梯度 ####

% calculate g 
    Y = repmat(y',1,num_classes);
    for i=1:num_classes
        Y(Y(:,i)~=i,i) = 0;       % 第i列中，仅标签为i的样本对应的位置不为0。
    end
    Y(Y~=0)=1;                % 第i列非零元所在的位置，表示标签为i的元素的索引
    g = (-1) * X * (Y(:,1:(size(Y,2)-1))-B(1:(size(B,1)-1),:)');

对应公式：  
 ∇ θ ( k ) J ( θ ) = − ∑ i = 1 m \[ x ( i ) ( 1 \{ y ( i ) = k \} − P ( y ( i ) = k ∣ x ( i ) ; θ ) ) \] \\begin\{aligned\} \\nabla\_\{\\theta^\{(k)\}\} J(\\theta) = - \\sum\_\{i=1\}^\{m\}\{ \\left\[ x^\{(i)\} \\left( 1\\\{ y^\{(i)\} = k\\\} - P(y^\{(i)\} = k | x^\{(i)\}; \\theta) \\right) \\right\] \} \\end\{aligned\} ∇θ(k)J(θ)=−i=1∑m\[x(i)(1\{ y(i)=k\}−P(y(i)=k∣x(i);θ))\]  
写成向量格式：  
 ∇ θ J ( θ ) = − \[ X ( 1 \{ Y = K \} − P ( Y = K ∣ X ; θ ) ) \] \\begin\{aligned\} \\nabla\_\{\\theta\} J(\\theta) = - \{ \\left\[ X \\left( 1\\\{ Y = K\\\} - P(Y = K | X; \\theta) \\right) \\right\] \} \\end\{aligned\} ∇θJ(θ)=−\[X(1\{ Y=K\}−P(Y=K∣X;θ))\]  
尤其注意一下几个变量的维度和内容：

*  `X`： n × m n×m n×m = 758×60000。
 *  `Y`：为  1 \{ Y = K \} 1\\\{ Y = K\\\} 1\{ Y=K\}，维度  n × ( K − 1 ) n×(K-1) n×(K−1)。对应代码中`Y(:,1:(size(Y,2)-1)`要**去掉非参数的偏置项**。
    
    <table> 
     <thead> 
      <tr> 
       <th align="center">0</th> 
       <th align="center">0</th> 
       <th align="center">0</th> 
       <th align="center">……</th> 
      </tr> 
     </thead> 
     <tbody> 
      <tr> 
       <td align="center">0</td> 
       <td align="center">0</td> 
       <td align="center">0</td> 
       <td align="center">……</td> 
      </tr> 
      <tr> 
       <td align="center">1</td> 
       <td align="center">0</td> 
       <td align="center">0</td> 
       <td align="center">……</td> 
      </tr> 
      <tr> 
       <td align="center">0</td> 
       <td align="center">0</td> 
       <td align="center">0</td> 
       <td align="center">……</td> 
      </tr> 
      <tr> 
       <td align="center">0</td> 
       <td align="center">0</td> 
       <td align="center">0</td> 
       <td align="center">……</td> 
      </tr> 
      <tr> 
       <td align="center">0</td> 
       <td align="center">0</td> 
       <td align="center">0</td> 
       <td align="center">……</td> 
      </tr> 
      <tr> 
       <td align="center">0</td> 
       <td align="center">0</td> 
       <td align="center">0</td> 
       <td align="center">……</td> 
      </tr> 
      <tr> 
       <td align="center">0</td> 
       <td align="center">0</td> 
       <td align="center">0</td> 
       <td align="center">……</td> 
      </tr> 
      <tr> 
       <td align="center">0</td> 
       <td align="center">0</td> 
       <td align="center">0</td> 
       <td align="center">……</td> 
      </tr> 
      <tr> 
       <td align="center">0</td> 
       <td align="center">1</td> 
       <td align="center">0</td> 
       <td align="center">……</td> 
      </tr> 
      <tr> 
       <td align="center">0</td> 
       <td align="center">0</td> 
       <td align="center">0</td> 
       <td align="center">……</td> 
      </tr> 
      <tr> 
       <td align="center">0</td> 
       <td align="center">0</td> 
       <td align="center">0</td> 
       <td align="center">……</td> 
      </tr> 
      <tr> 
       <td align="center">0</td> 
       <td align="center">0</td> 
       <td align="center">1</td> 
       <td align="center">……</td> 
      </tr> 
      <tr> 
       <td align="center">……</td> 
       <td align="center">……</td> 
       <td align="center">……</td> 
       <td align="center">……</td> 
      </tr> 
     </tbody> 
    </table>
 *  `B`：样本被分类为各个类别的概率， K × n K×n K×n。对应代码中`B(1:(size(B,1)-1),:)'`要**去掉非参数的偏置项，并转置**。最后维度为 n × ( K − 1 ) n×(K-1) n×(K−1)。

[UFLDL-Tutorial-notes-code]: https://github.com/laisimiao/UFLDL-Tutorial-notes-code