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

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.

Machine Learning (cs.LG)

arXiv:2606.07120 [cs.LG]

(or arXiv:2606.07120v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2606.07120 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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