VAE与CVAE

最编程 2024-07-24 10:15:39

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观测数据集

X=\left\{ \mathtt{x}^{(i)} \right\}^N_{i=1} i.i.d

（

X

本身可能是连续分布或者离散分布），对某个

x

的概率处理：

\begin{aligned}\log\; p_{\theta}(\mathtt{x}^{(i)} )&=\log\; p_{\theta }(\mathtt{x}^{(i)},\mathtt{z})- \log\; p_{\theta }(\mathtt{z}|\mathtt{x}^{(i)})\\ &=\log\; \frac{p_{\theta }(\mathtt{x}^{(i)},\mathtt{z})}{q_{\phi}(\mathtt{z}|\mathtt{x}^{(i)})}-\log\; \frac{p_{\theta }(\mathtt{z}|\mathtt{x}^{(i)})}{q_{\phi}(\mathtt{z}|\mathtt{x}^{(i)})}\; \; (q_{\phi}(\mathtt{z}|\mathtt{x}^{(i)})\neq 0).\end{aligned}\tag{1.1}

(2.2.2)式两边对

q_{\phi}(\mathtt{z}|\mathtt{x}^{(i)})

求期望得：

\log p_\theta(\mathtt{x}^{(i)})=D_{KL}{(q_{\phi}(\mathtt{z}|\mathtt{x}^{(i)})||p_{\theta }(\mathtt{z}|\mathtt{x}^{(i)}))}+\mathcal{L}(\theta,\phi;\mathtt{x}^{(i)})\tag{1.2}

其中：

\color{blue}\{ \begin{aligned} \mathcal{L}(\theta,\phi;\mathtt{x}^{(i)})&=\int_z q_{\phi}(\mathtt{z}|\mathtt{x}^{(i)}) \log\ \frac{p_{\theta }(\mathtt{z},\mathtt{x}^{(i)})}{q_{\phi}(\mathtt{z}|\mathtt{x}^{(i)})}dz\\ D_{KL}{(q_{\phi}(\mathtt{z}|\mathtt{x}^{(i)})||p_{\theta }(\mathtt{z}|\mathtt{x}^{(i)}))}&= - \int_z q_{\phi}(\mathtt{z}|\mathtt{x}^{(i)})\log\ \frac{p_{\theta }(\mathtt{z}|\mathtt{x}^{(i)})}{q_{\phi}(\mathtt{z}|\mathtt{x}^{(i)})}dz \end{aligned}

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