Deep Autoencoders: From Understanding to Generalization Guarantees

TL;DR

This paper provides a theoretical and practical analysis of deep autoencoders, revealing how they approximate data manifolds, introducing new regularizations for symmetry capture, and guaranteeing generalization under certain conditions.

Contribution

It offers a novel reformulation of autoencoders, new regularization techniques leveraging transformation groups, and theoretical guarantees for generalization based on data symmetry assumptions.

Findings

Regularizations improve autoencoder performance on data with symmetries.

Autoencoders can approximate data manifolds using piecewise affine mappings.

Experimental results outperform state-of-the-art regularization methods.

Abstract

A big mystery in deep learning continues to be the ability of methods to generalize when the number of model parameters is larger than the number of training examples. In this work, we take a step towards a better understanding of the underlying phenomena of Deep Autoencoders (AEs), a mainstream deep learning solution for learning compressed, interpretable, and structured data representations. In particular, we interpret how AEs approximate the data manifold by exploiting their continuous piecewise affine structure. Our reformulation of AEs provides new insights into their mapping, reconstruction guarantees, as well as an interpretation of commonly used regularization techniques. We leverage these findings to derive two new regularizations that enable AEs to capture the inherent symmetry in the data. Our regularizations leverage recent advances in the group of transformation learning to enable AEs to better approximate the data manifold without explicitly defining the group underlying the manifold. Under the assumption that the symmetry of the data can be explained by a Lie group, we prove that the regularizations ensure the generalization of the corresponding AEs. A range of experimental evaluations demonstrate that our methods outperform other state-of-the-art regularization techniques.