$5$ Generative Models $[30$ points$]$

Let $q(x)$ be the true data distribution. Consider the following generative process:

$xq(x),z{q}_{}(zx),$tilde$(x)p_{}(tilde(x)|z).$

Design a GAN model to match the distribution of the generative data tilde$(x)$ and the true data

distribution $q.$

$[10$ points$]$ Draw the model graph and write out the objective function.

$[10$ points$]$ Can the objective achieve the same goal of VAE: to make tilde$(x)$ to be a recon$-$

struction of $x?$ Briefly explain your answer.

points$]$ Derive the gradient formula for $.($Hint: use the identity $p(x)gra{d}_{}logp(x)=$

$gra{d}_{}p(x),$ as in the policy gradient method$).$