Variational Autoencoder — Step-by-Step Visualization

Algorithm Pattern

Encode → Sample → Decode

Key Idea

Unlike a standard autoencoder, VAE encodes to a DISTRIBUTION (μ, σ²) not a point. The reparameterization trick (z = μ + σ·ε) keeps gradients flowing while allowing stochastic sampling.

Step-by-Step Approach

1. Encoder maps x → μ and log(σ²) — the parameters of a Gaussian.
2. Reparameterize: z = μ + σ·ε where ε ~ N(0,1) — enables backprop through sampling.
3. Decoder maps z → x̂ (reconstruction).
4. Reconstruction loss: MSE(x̂, x) — how well we rebuilt the input.
5. KL loss: −0.5·Σ(1 + log(σ²) − μ² − σ²) — regularizes latent space toward N(0,1).

Common Gotchas

• Without KL loss, the encoder collapses to a point — equivalent to a standard autoencoder.
• The reparameterization trick is what makes VAEs trainable end-to-end.
• VAE latent spaces are smooth — interpolating between two z vectors produces valid outputs.