Legacy Concept Lab

Variational Autoencoders & Variational Inference

Stable Diffusion is a latent diffusion model: an autoencoder maps images ↔ compressed latent space where diffusion operates

Concept 7 of 100Generative ModelsPhase 4

Open New Notebook All Foundations

#7VAEsGenerative Models

key equation

\log p_\theta(x) \ge \mathbb E_{q_\phi}[\log p_\theta(x\mid z)] - \mathrm{KL}(q_\phi\,\|\,p(z))

Phase 4: Generative modeling familiesConcept 7 of 100

Why It Matters for Modern Models

Stable Diffusion is a latent diffusion model: an autoencoder maps images ↔ compressed latent space where diffusion operates
VAEs underpin many multimodal encoders (audio, video latents) used as building blocks

What Tutorials Skip

What is still poorly explained in textbooks and papers:

Intuitive grasp of why ELBO works as both reconstruction + regularization
Visualizations of how the prior p(z) and posterior families affect sample quality/diversity

Interactive Visualization

Core Math (Optional Deep Dive)

If you want intuition first, start with the key equation and the visualization. Come back here for the full walkthrough.

Key Equation

\log p_\theta(x) \ge \mathbb E_{q_\phi}[\log p_\theta(x\mid z)] - \mathrm{KL}(q_\phi\,\|\,p(z))

Latent variable model $p_\theta(x,z) = p(z)p_\theta(x\mid z)$ with intractable posterior. Introduce variational encoder $q_\phi(z\mid x)$ and maximize ELBO: