Generative Adversarial Network

Objective:

$$\begin{array}{rlr}V(D,G)& =\stackrel{\text{ Good at recog. real img? }}{\overbrace{{\mathbb{E}}_{x\sim p_{\text{real}}(x)}[\ln D_{{\theta }_d}(x)]A}}& +\stackrel{\text{ Good at recog. fake img? }}{\overbrace{{\mathbb{E}}_{z\sim N(0,1)}[\ln 1-D_{{\theta }_d}(G_{{\theta }_g}(z))]}}\\ & ={\int }_x{p}_{\text{real}}(x)\ln D_{{\theta }_d}(x)dx& +\underset{\text{ change of variables }}{\underbrace{{\int }_z\varphi (z)\ln 1-D_{{\theta }_d}(G_{{\theta }_g}(z))dz}}\\ & =\dots & +{\int }_x{\hat{p}}_G\ln 1-D_{{\theta }_d}(x)dx\\ & ={\int }_x{p}_{\text{real}}(x)\ln D_{{\theta }_d}(x)& +{\hat{p}}_G(x)\ln 1-D_{{\theta }_d}(x)dx\end{array}$$

where

• Discrimminator $D$ and parameters ${\theta }_d$
• Generator $G$ and parameters ${\theta }_g$
• real distribution is $x\sim p_{\text{real}}(x)$
• latent is simply $\mathcal{N}(0,1)$ but $G$ transforms it into implicit distribution $x\sim {\hat{p}}_G(x)$