Beyond Diagonal Covariance: Flexible Posterior VAEs via Free-Form Injective Flows

TL;DR

This paper introduces a regularized Free-form Injective Flow (FIF) approach that enhances Variational Autoencoders by enabling flexible, full covariance posteriors without increasing computational costs, leading to better likelihoods on image data.

Contribution

It proposes a novel regularized FIF method that allows VAEs to use full covariance posteriors efficiently, overcoming limitations of diagonal Gaussian assumptions.

Findings

Full covariance posteriors improve model likelihood.

Regularized FIF achieves flexibility comparable to full Gaussian posteriors.

Method maintains computational efficiency similar to diagonal VAEs.

Abstract

Variational Autoencoders (VAEs) are powerful generative models widely used for learning interpretable latent spaces, quantifying uncertainty, and compressing data for downstream generative tasks. VAEs typically rely on diagonal Gaussian posteriors due to computational constraints. Using arguments grounded in differential geometry, we demonstrate inherent limitations in the representational capacity of diagonal covariance VAEs, as illustrated by explicit low-dimensional examples. In response, we show that a regularized variant of the recently introduced Free-form Injective Flow (FIF) can be interpreted as a VAE featuring a highly flexible, implicitly defined posterior. Crucially, this regularization yields a posterior equivalent to a full Gaussian covariance distribution, yet maintains computational costs comparable to standard diagonal covariance VAEs. Experiments on image datasets validate our approach, demonstrating that incorporating full covariance substantially improves model likelihood.