VAE

The Variational Auto-Encoder defines a latent-variable generative model $p_\theta(x, z) = p_\theta(x \mid z)\, p(z)$ with a standard-normal prior $p(z) = \mathcal{N}(0, \mathbf{I})$ and a neural decoder $p_\theta(x \mid z)$. Because the marginal likelihood $p_\theta(x) = \int p_\theta(x \mid z)\, p(z)\, dz$ is intractable, training maximises the evidence lower bound (ELBO) instead:

Here $q_\phi(z \mid x) = \mathcal{N}\!\big(z;\, \mu_\phi(x),\; \mathrm{diag}\,\sigma_\phi^2(x)\big)$ is the encoder (the variational posterior). Gradients flow through the stochastic sample via the reparameterization trick — $z = \mu_\phi(x) + \sigma_\phi(x) \odot \epsilon$, $\epsilon \sim \mathcal{N}(0, \mathbf{I})$ — which keeps the Monte-Carlo gradient low-variance and pathwise-differentiable. The KL term enjoys the closed form

The β-VAE (Higgins et al., 2017) variant rescales the KL term by a coefficient $\beta$ to trade reconstruction fidelity for posterior compression / latent disentanglement, recovering vanilla ELBO at $\beta = 1$. The hierarchical / Ladder VAE (Sønderby et al., 2016) extends the latent to a stack $z = (z_1, \dots, z_L)$, with one $z_\ell$ extracted per encoder stage and re-injected at the matching decoder resolution; the KL becomes a sum $\sum_\ell \mathrm{KL}(q_\phi(z_\ell \mid x) \,\|\, p(z_\ell))$, granting strictly tighter bounds and richer multi-scale priors.

The encoder/decoder backbones here are convolutional with stride-2 down-/up-sampling blocks; the reconstruction term is selectable between Gaussian (mse) and Bernoulli (bce) likelihoods to match the data domain. Despite being eclipsed by diffusion on raw sample quality, the VAE remains the workhorse of representation learning and a key component inside latent-diffusion pipelines (Stable Diffusion's first-stage autoencoder).