# 受限玻耳兹曼机

## 玻尔兹曼分布

玻尔兹曼分布是统计物理中的一种概率分布，描述系统处于某种状态的概率分布（能量越大概率越低）：

$$
p(x)=\frac{exp(-energy(x))}{z}
$$

可见单元和隐藏单元的值服从玻尔兹曼分布

$$
p(v,h)=\frac{1}{Z\_{\theta}}exp(-E\_{\theta}(v,h))=\frac{1}{Z\_{\theta}}exp(v^TWh+b^Tv+d^Th)
$$

其中能量定义为：

$$
E\_{\theta}(v,h)=-v^TWh-b^Tv-d^Th
$$

归一化因子为:

$$
Z\_{\theta}=\sum\_{(v,h)} exp(-E\_{\theta}(v,h))
$$

![](/files/-Lpsm_plyj_CJaxuEv1O)

上图概率分布表：

4个输入可见，3个隐藏，每个有0/1状态，部分表

![](/files/-Lpsm_pyUa6cPV2wq8U5)

## 用于特征提取

可见单元作为输入数据，隐藏单元作为特征向量，计算隐藏层神经元的激励能量

$$
a\_i=\sum\_{j}w\_{ij}v\_j+d\_i
$$

计算该隐藏单元的条件概率值，即状态为1的概率：

$$
p\_i=\sigma(a\_i)
$$

以p的概率将隐藏层神经元的状态值设置为1，以1-p的概率将其设置为0，也就是3神经元都有可能是0/1状态，1表示选择这个维度的特征

$$
\left\[
\begin{matrix}
0 & 1 \\
0 & 1 \\
0 & 1
\end{matrix}
\right]
$$

## 深度受限玻耳兹曼

拿多个玻耳兹曼串起来，和编码器一样


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