Learning Variational Autoencoders Foundations

VAE architecture: encoder, latent space, decoder

In a variational autoencoder (VAE), the architecture is typically described as three connected parts: an encoder, a latent space, and a decoder. The encoder takes an input data point x (e.g., an image) and maps it into a probabilistic representation in the latent space, producing parameters of a variational distribution qϕ(z|x) rather than a single latent vector. This latent distribution is usually modeled as a multivariate Gaussian, so the encoder outputs values such as the mean Eϕ(x) and (optionally) a variance σϕ(x), defining how z is distributed for each input. The latent space is the probabilistic bottleneck where sampling occurs. During training, a latent variable z is sampled from qϕ(z|x) and then passed to the decoder. The decoder Dθ maps from the latent space back to the input space, producing a reconstruction x′ (or parameters of a likelihood distribution for x given z). In practice, the decoder is often implemented so that x|z is Gaussian with mean Dθ(z), and the reconstruction quality is measured with a reconstruction error (e.g., mean squared error or cross-entropy). Training the encoder and decoder jointly is done by maximizing a lower bound on the data likelihood, the ELBO. The ELBO combines (1) a reconstruction term that encourages x′ to match x and (2) a regularization term that makes the approximate posterior qϕ(z|x) align with the prior pθ(z), commonly via the Kullback–Leibler divergence. To enable gradient-based learning through the sampling step, VAEs use the reparameterization trick, rewriting the random sampling as a deterministic function of encoder outputs plus external noise.

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Probabilistic view: prior, likelihood, posterior

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