Fundamental Limits of Two-layer Autoencoders, and Achieving Them with Gradient Methods

TL;DR

This paper investigates the fundamental limits of two-layer non-linear autoencoders in data compression, characterizes the solutions found by gradient methods, and demonstrates the universality of these results across datasets.

Contribution

It provides a theoretical analysis of the optimal autoencoder solutions and their training dynamics in the proportional regime, extending understanding beyond linear models.

Findings

Characterizes minimizers of population risk for non-linear autoencoders.

Shows gradient methods achieve these minimizers.

Establishes fundamental limits for Gaussian source compression with sign activation.

Abstract

Autoencoders are a popular model in many branches of machine learning and lossy data compression. However, their fundamental limits, the performance of gradient methods and the features learnt during optimization remain poorly understood, even in the two-layer setting. In fact, earlier work has considered either linear autoencoders or specific training regimes (leading to vanishing or diverging compression rates). Our paper addresses this gap by focusing on non-linear two-layer autoencoders trained in the challenging proportional regime in which the input dimension scales linearly with the size of the representation. Our results characterize the minimizers of the population risk, and show that such minimizers are achieved by gradient methods; their structure is also unveiled, thus leading to a concise description of the features obtained via training. For the special case of a sign activation function, our analysis establishes the fundamental limits for the lossy compression of Gaussian sources via (shallow) autoencoders. Finally, while the results are proved for Gaussian data, numerical simulations on standard datasets display the universality of the theoretical predictions.