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arXiv · 2606.07120

Beyond Linear and Overcomplete Regimes: A Mean-Field Analysis of Bottleneck Autoencoders

Santanu Das·Ramyak Bilas·Pascal Esser·Satyaki Mukherjee
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Abstract

Autoencoders (AEs) learn low-dimensional representations by mapping data into a latent space while minimizing reconstruction error. Despite their empirical success, theoretical understanding remains limited and largely restricted to linear models or settings without a bottleneck. In this work, we study nonlinear AEs with a fixed finite-dimensional bottleneck in the mean-field (MF) regime. We derive explicit MF learning dynamics for both encoder and decoder, providing a tractable characterization of training in the nonlinear setting. We show that, over finite time horizons, the empirical risk of finite-width networks trained with stochastic gradient descent closely tracks the MF risk trajectory with high probability. At optimality, we further establish that the finite-width risk converges to the MF optimum, demonstrating that finite networks are sufficiently expressive to approximate the infinite-width solution.

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Santanu Das, Ramyak Bilas, Pascal Esser, Satyaki Mukherjee. 2026-06-05. Beyond Linear and Overcomplete Regimes: A Mean-Field Analysis of Bottleneck Autoencoders. https://arxiv.org/abs/2606.07120

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