Variational Autoencoders: Understanding the Reparameterization Trick

Variational autoencoders (VAEs) are a powerful class of generative models that combine ideas from latent variable models and neural networks. This article provides an in-depth look at VAEs, with a particular focus on the reparameterization trick that enables gradient computation.

Latent Variable Models

VAEs belong to the broader category of latent variable models. In a latent variable model, the distribution over observed data x is defined as a marginal over a joint distribution with some unobserved (latent) variables z:

P(x) = ∫ P(x,z) dz

The goal is to learn both the model parameters θ and an approximate posterior distribution over the latent variables q(z|x).

The VAE Objective

VAEs aim to maximize a lower bound on the log-likelihood, known as the evidence lower bound (ELBO):

ELBO = E[log P(x|z)] - KL(q(z|x) || P(z))

Where:

• E[log P(x|z)] is the expected reconstruction error
• KL(q(z|x) || P(z)) is the KL divergence between the approximate posterior and the prior

The Reparameterization Trick

A key challenge in training VAEs is computing gradients of the ELBO with respect to the parameters of q(z|x). The reparameterization trick provides an elegant solution:

1. Express z as a deterministic function of x, some parameters φ, and a noise variable ε: z = g(x, φ, ε)

2. Sample ε from a simple distribution (e.g. standard normal)

3. Compute z using the sampled ε

This allows gradients to flow through the sampling process.

Example: Gaussian Case

For a Gaussian q(z|x), we can reparameterize as:

z = μ(x) + σ(x) * ε

Where μ(x) and σ(x) are outputs of the encoder network, and ε ~ N(0, I).

Training Process

The VAE training process involves:

1. Encoding: Pass x through the encoder to get μ(x) and σ(x)
2. Sampling: Sample ε and compute z
3. Decoding: Pass z through the decoder to reconstruct x
4. Loss computation: Compute reconstruction error and KL divergence
5. Backpropagation: Compute gradients and update parameters

Inference and Generation

After training:

• For inference (encoding): Pass x through the encoder
• For generation: Sample z from the prior and pass through the decoder

Conclusion

The reparameterization trick is a crucial component of VAEs, enabling efficient training of powerful generative models. By understanding this technique, we gain insight into how VAEs bridge the gap between traditional latent variable models and modern deep learning approaches.

In future articles, we'll explore extensions to the basic VAE framework, including more expressive posterior approximations and applications to specific domains like image and text generation.

Article created from: https://youtu.be/c475SLygCK4?feature=shared