[Paper Review] Variational Information Bottleneck on Vector Quantized Autoencoders

Hanwei Wu, Markus Flierl|arXiv (Cornell University)|Aug 2, 2018

Generative Adversarial Networks and Image Synthesis5 references4 citations

TL;DR

This paper establishes a theoretical connection between Vector Quantized Variational Autoencoders (VQ-VAEs) and information bottleneck principles. It shows that the original VQ-VAE loss is derived from the Variational Deterministic Information Bottleneck (VDIB), while the EM-trained VQ-VAE approximates the Variational Information Bottleneck (VIB), with improved latent perplexity due to increased conditional entropy.

ABSTRACT

In this paper, we provide an information-theoretic interpretation of the Vector Quantized-Variational Autoencoder (VQ-VAE). We show that the loss function of the original VQ-VAE can be derived from the variational deterministic information bottleneck (VDIB) principle. On the other hand, the VQ-VAE trained by the Expectation Maximization (EM) algorithm can be viewed as an approximation to the variational information bottleneck(VIB) principle.

Motivation & Objective

• To provide an information-theoretic interpretation of the VQ-VAE training objective.
• To clarify the theoretical relationship between VQ-VAE and the Variational Information Bottleneck (VIB) and Variational Deterministic Information Bottleneck (VDIB) principles.
• To explain why EM-based training of VQ-VAE improves latent representation quality via increased conditional entropy.
• To formalize the connection between the VQ-VAE loss function and information-theoretic compression objectives.

Proposed method

• Derives the VDIB loss function by upper-bounding the entropy H(Z) using a variational distribution r(z), leading to a cross-entropy regularization term.
• Derives the VIB loss function by upper-bounding the KL divergence term I(I,Z) using variational approximations, resulting in a KL regularization on the posterior p(Z|I).
• Maps the VQ-VAE's reconstruction and commitment loss to the VDIB objective, where the reconstruction term corresponds to the distortion and the commitment loss to the cross-entropy regularization.
• Uses the stop-gradient operator to decouple the encoder and quantizer, enabling backpropagation only through the decoder, which aligns with the VDIB training objective.
• Applies soft clustering via the EM algorithm to the VQ-VAE bottleneck, increasing the conditional entropy H(p(Z|I)) and approximating the VIB objective.
• Shows that the original VQ-VAE corresponds to VDIB with uniform r(z), while EM-trained VQ-VAE corresponds to VIB with non-uniform p(Z|I) due to soft assignments.

Experimental results

Research questions

• RQ1 Can the original VQ-VAE loss be derived from a known information-theoretic principle?
• RQ2 How does EM-based training of VQ-VAE relate to the Variational Information Bottleneck (VIB) principle?
• RQ3 Why does EM-based VQ-VAE achieve higher latent perplexity than standard VQ-VAE?
• RQ4 What is the role of the conditional entropy H(p(Z|I)) in improving representation quality?
• RQ5 How do the VDIB and VIB principles compare in their application to discrete autoencoders?

Key findings

• The original VQ-VAE loss function is equivalent to the Variational Deterministic Information Bottleneck (VDIB) objective when the prior r(z) is uniform.
• The EM-trained VQ-VAE approximates the Variational Information Bottleneck (VIB) principle by explicitly modeling soft assignments, increasing the conditional entropy H(p(Z|I)).
• The soft clustering in EM-trained VQ-VAE increases the latent representation's perplexity, as confirmed by empirical results in prior work.
• The commitment loss in VQ-VAE enforces the encoder to commit to the codebook, which corresponds to minimizing the cross-entropy term in VDIB.
• The reconstruction term in VQ-VAE corresponds to the distortion term in both VIB and VDIB, minimizing the KL divergence between true and reconstructed data distributions.
• The use of stop-gradient in VQ-VAE ensures that only the decoder is trained via backpropagation through the quantized code, aligning with the VDIB training procedure.