Variational autoencoders (VAE)

One-line definition

A variational autoencoder (Kingma & Welling, 2013) is a generative model with a latent variable $z$, learned encoder $q_{\varphi }(z\mid x)$, and decoder $p_{\theta }(x\mid z)$, trained to maximize the evidence lower bound (ELBO):

$$\log p_{\theta }(x)\ge {\mathbb{E}}_{q_{\varphi }(z|x)}[\log p_{\theta }(x\mid z)]-\mathrm{KL}(q_{\varphi }(z\mid x)\Vert p(z)).$$

Why it matters

VAEs introduced amortized variational inference to deep learning: a neural network learns to predict the posterior of a latent given the input, enabling end-to-end training of latent variable models with backprop. This idea now powers:

• Diffusion models (often built on top of a VAE in latent space. Stable Diffusion).
• Disentanglement research (β-VAE, factor-VAE).
• Generative pretraining for tabular and molecular data.
• Probabilistic recsys and time series.

The VAE is also the canonical example of the reparameterization trick, a tool used everywhere in modern probabilistic deep learning.

The model

• Prior: $p(z)=\mathcal{N}(0,I)$.
• Encoder: $q_{\varphi }(z\mid x)=\mathcal{N}({\mu }_{\varphi }(x),\mathrm{diag}({\sigma }_{\varphi }(x{)}^2))$. A neural net outputs mean and diagonal covariance.
• Decoder: $p_{\theta }(x\mid z)$. A neural net mapping $z$ back to a distribution over $x$ (Gaussian for continuous, Bernoulli for binary, categorical for discrete).

The ELBO

The ELBO has two terms:

$${\mathcal{L}}_{\text{ELBO}}=\underset{\text{reconstruction}}{\underbrace{{\mathbb{E}}_{q_{\varphi }(z|x)}[\log p_{\theta }(x|z)]}}-\underset{\text{regularizer}}{\underbrace{\mathrm{KL}(q_{\varphi }(z|x)\Vert p(z))}}.$$

• Reconstruction: rewards the decoder for assigning high probability to $x$ given samples from the encoder.
• KL: penalizes the encoder for diverging from the prior. Keeps the latent space compact and continuous.

Maximizing ELBO ≤ maximizing log-likelihood (true objective). The gap is $\mathrm{KL}(q_{\varphi }\Vert p_{\text{true posterior}})$.

The reparameterization trick

To backprop through sampling, write $z={\mu }_{\varphi }(x)+{\sigma }_{\varphi }(x)\odot \epsilon$ with $\epsilon \sim \mathcal{N}(0,I)$. The randomness is now external; the gradient flows through ${\mu }_{\varphi }$ and ${\sigma }_{\varphi }$ deterministically. Without this, the gradient of an expectation over a parameter-dependent distribution would require REINFORCE (high variance).

What VAEs are good and bad at

Good:

• Smooth, continuous latent space useful for interpolation and editing.
• Stable training (unlike GANs).
• Good likelihood estimation (after IWAE-style correction).
• Excellent as compressors / latent encoders for downstream models (Stable Diffusion’s first stage).

Bad:

• Image samples are blurry compared to GANs and diffusion. The Gaussian decoder + per-pixel MSE penalizes high-frequency detail.
• KL penalty causes posterior collapse in some configurations (decoder ignores $z$, output becomes nearly mean-only).
• Lower-quality samples than diffusion at the same parameter count.

Common pitfalls

• Posterior collapse. When the decoder is too powerful relative to the encoder, $q_{\varphi }(z|x)\to p(z)$ and the latent becomes useless. Mitigations: KL annealing, free bits, reduce decoder capacity, β-VAE with $\beta <1$ early in training.
• Forgetting the reparameterization trick. Sampling $z$ inside the network and trying to backprop through z = sample(N(mu, sigma)) doesn’t work; use z = mu + sigma * epsilon.
• Treating ELBO as the model’s likelihood. ELBO is a lower bound; for likelihood comparison use IWAE estimates.
• Using VAEs as competitive standalone image generators. They aren’t anymore; use them as latent compressors with diffusion / AR on top.

Variants

• β-VAE: scale the KL term by $\beta$. Higher $\beta$ encourages disentanglement; lower $\beta$ improves reconstruction.
• VQ-VAE: discrete (categorical) latent via vector quantization. Used in language-image models, audio.
• IWAE (importance-weighted): tighter ELBO via $K$-sample importance weighting.
• NVAE, Hierarchical VAEs: deep hierarchical latents for higher-fidelity generation.

Related

• Reparameterization trick. The gradient enabler.
• Autoregressive vs. diffusion. Alternative generative paradigms.