Memorization in Overparameterized Autoencoders

TL;DR

This paper investigates how overparameterized autoencoders tend to memorize training data, revealing that they project data onto training examples and converge to them, which sheds light on their generalization behavior.

Contribution

It provides theoretical proofs showing memorization phenomena in both shallow and deep autoencoders, highlighting the role of depth in this process.

Findings

Single-layer autoencoders project data onto training examples.

Deep autoencoders are locally contractive at training points.

Depth is necessary and sufficient for memorization in convolutional autoencoders.

Abstract

The ability of deep neural networks to generalize well in the overparameterized regime has become a subject of significant research interest. We show that overparameterized autoencoders exhibit memorization, a form of inductive bias that constrains the functions learned through the optimization process to concentrate around the training examples, although the network could in principle represent a much larger function class. In particular, we prove that single-layer fully-connected autoencoders project data onto the (nonlinear) span of the training examples. In addition, we show that deep fully-connected autoencoders learn a map that is locally contractive at the training examples, and hence iterating the autoencoder results in convergence to the training examples. Finally, we prove that depth is necessary and provide empirical evidence that it is also sufficient for memorization in convolutional autoencoders. Understanding this inductive bias may shed light on the generalization properties of overparametrized deep neural networks that are currently unexplained by classical statistical theory.