Fundamental Limits of Two-layer Autoencoders, and Achieving Them with Gradient Methods
This work addresses a gap in understanding autoencoders for machine learning and data compression, providing foundational insights into their optimization and limits, though it is incremental in extending analysis to non-linear cases and specific regimes.
The paper tackles the problem of understanding the fundamental limits, gradient method performance, and learned features of two-layer autoencoders in a proportional regime, showing that minimizers of population risk are achieved by gradient methods and establishing fundamental limits for lossy compression of Gaussian sources with sign activation.
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.