Foundation 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 local demo 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))

Reading map and next steps

Selected Foundation Object

Keep the equation fixed; move through the evidence.

Concept 7 of 100VAEsGenerative Models / Phase 4: Generative modeling families

Current question

Intuitive grasp of why ELBO works as both reconstruction + regularization

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

PredictionCommit before the demo.

Ask what should change when the equation is manipulated, then let the local visualization test that expectation.

EvidenceCompare local witness and source.

Use the local runnable panel, the key equation, and canonical papers as separate witnesses for the same object.

InvariantName what survives notation changes.

The useful learning product is the reusable mechanism you can carry into another model, paper, or engineering tradeoff.

Next moveContinue in the newer notebook.

This atlas page keeps a local demo; the domain notebook carries the fuller Intuition -> Math -> Code -> Demo sequence.

Open local demo Inspect equation Open newer notebook Next foundation

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 local 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: