High-dimensional Asymptotics of VAEs: Threshold of Posterior Collapse and Dataset-Size Dependence of Rate-Distortion Curve

Yuma Ichikawa, Koji Hukushima

Keywords: statistical physics, replica method, variational autoencoder, exact asymptotics

TL;DR: We provide sharp asymptotics for the generalization error of linear variational autoencoders in high dimension

Abstract: In variational autoencoders (VAEs), the variational posterior often aligns closely with the prior, known as posterior collapse, which leads to poor representation learning quality. An adjustable hyperparameter beta has been introduced in VAE to address this issue. This study sharply evaluates the conditions under which the posterior collapse occurs with respect to beta and dataset size by analyzing a minimal VAE in a high-dimensional limit. Additionally, this setting enables the evaluation of the rate-distortion curve in the VAE. This result shows that, unlike typical regularization parameters, VAEs face "inevitable posterior collapse" beyond a certain beta threshold, regardless of dataset size. The dataset-size dependence of the derived rate-distortion curve also suggests that relatively large datasets are required to achieve a rate-distortion curve with high rates. These results robustly explain generalization behavior across various real datasets with highly non-linear VAEs.

Primary Area: unsupervised, self-supervised, semi-supervised, and supervised representation learning

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Submission Number: 5872