$1$ Diffusion Models: Questions $(40$ points$)$

$1.$ Detail the forward and reverse processes of Diffusion Model $($DDPM$)[1],$ with equations and a simple diagram$/$Figure$?$

$2.$ What are the key differences between DMs $[1,4,7]$ and Hierarchical Variational Autoencoders $[8,9]?$

$3.$ Discuss the advantages of the Latent Diffusion Model $[10]$ in detail and elaborate why it can achieve these advantages.

$4.$ What is the difference between Classifier $[11]$ and Classifier$-$Free $[12]$ Guidance mechanisms to condition DMs$?$ Discuss advantages and disadvantages.

$5.$ Read the Cascaded Diffusion Models paper $[13]$ and discuss its main idea. Then, discuss the main challenge they face while connecting the different stages together and discuss the proposed solution.

$6.$ How and why SDE $[7]$ is connected with SBGMs $[4]?$