Chapter 6: Variational Autoencoders (VAEs) for Structured Latent Spaces

Previous chapters introduced autoencoders primarily as tools for dimensionality reduction and feature learning, where the network learns to reconstruct its input. This chapter presents Variational Autoencoders (VAEs), a distinct class of autoencoders that incorporate a probabilistic perspective. VAEs are designed not just for reconstruction but also to learn a continuous and structured latent space, making them particularly effective for generative modeling – the creation of new data samples.

By working through this chapter, you will gain an understanding of:

The basic ideas behind generative modeling and how VAEs fit into this area.
The core principles of Variational Autoencoders, including their probabilistic encoder and decoder.
How the VAE encoder is structured to output parameters (typically mean $\mu$ and log-variance $\log(\sigma^2)$ ) that define a probability distribution in the latent space for each input.
The reparameterization trick, a method that allows gradients to be backpropagated through the sampling process, which is essential for training VAEs.
How the VAE decoder, acting as a generator, uses samples drawn from the latent distribution to produce new data.
The VAE loss function, which typically consists of two main parts: a reconstruction loss (e.g., Mean Squared Error or Binary Cross-Entropy) and a Kullback-Leibler (KL) divergence term. The KL divergence acts as a regularizer, encouraging the learned latent distributions to be close to a prior distribution (often a standard normal distribution $\mathcal{N}(0, I)$ ). The combined loss can be expressed as $L_{VAE} = \text{ReconstructionLoss} + D_{KL}(q(z|x) || p(z))$
The characteristics of the VAE latent space, such as its smoothness and how it facilitates interpolation between data points.
Methods for using the mean vectors (or samples) from the VAE's latent space as features for other machine learning tasks.

A hands-on section will demonstrate how to construct a VAE and visually inspect its latent space to observe its learned structure.

Sections

6.1 Introduction to Generative Modeling with Autoencoders
6.2 Principles of Variational Autoencoders
6.3 The VAE Encoder: Outputting Distribution Parameters
6.4 The Reparameterization Trick Explained
6.5 The VAE Decoder: Generating Data from Latent Samples
6.6 The VAE Loss Function: Balancing Reconstruction and Regularization
6.7 Characteristics of the VAE Latent Space
6.8 Using VAE Latent Representations as Features
6.9 Hands-on: Building a VAE and Inspecting Its Latent Space