Loss Landscapes of Regularized Linear Autoencoders

TL;DR

This paper analyzes the loss landscapes of regularized linear autoencoders, revealing their symmetry properties and how they learn principal directions, with implications for PCA, neuroscience, and topology.

Contribution

It proves that $L_2$-regularized LAEs are symmetric at critical points and learn principal directions as left singular vectors, connecting to probabilistic PCA.

Findings

Regularized LAEs are symmetric at all critical points.

LAEs learn principal directions as left singular vectors.

Empirical illustrations support theoretical results.

Abstract

Autoencoders are a deep learning model for representation learning. When trained to minimize the distance between the data and its reconstruction, linear autoencoders (LAEs) learn the subspace spanned by the top principal directions but cannot learn the principal directions themselves. In this paper, we prove that $L_2$-regularized LAEs are symmetric at all critical points and learn the principal directions as the left singular vectors of the decoder. We smoothly parameterize the critical manifold and relate the minima to the MAP estimate of probabilistic PCA. We illustrate these results empirically and consider implications for PCA algorithms, computational neuroscience, and the algebraic topology of learning.