Variational Autoencoder
Amortized inference and generation through a learned latent space.
Motivation
The variational autoencoder pairs a probabilistic decoder with an amortized inference network, learning a smooth latent representation of the data. It is the canonical example of turning intractable posterior inference into optimization of a variational bound.
Method
Training maximizes the evidence lower bound (ELBO): a reconstruction term plus a KL regularizer that pulls the approximate posterior toward the prior. The reparameterization trick makes the whole objective differentiable end-to-end.
Mathematical core
The ELBO is the same variational principle that reappears across inference and diffusion. Studying the VAE clarifies the trade-off between reconstruction fidelity and latent regularity, and the geometry the KL term imposes on the latent manifold.