Side Effects of Learning from Low-dimensional Data Embedded in a Euclidean Space

TL;DR

This paper investigates how neural networks behave when trained on low-dimensional data embedded in high-dimensional Euclidean space, focusing on their variation outside the training subspace and effects of noise and depth.

Contribution

It provides theoretical estimates on neural network variation outside the training subspace and explores regularization effects related to network depth and noise.

Findings

Variation of neural networks outside the data subspace can be bounded.

Deeper networks and noise influence regularization and side effects.

Noise introduces additional side effects during training.

Abstract

The low-dimensional manifold hypothesis posits that the data found in many applications, such as those involving natural images, lie (approximately) on low-dimensional manifolds embedded in a high-dimensional Euclidean space. In this setting, a typical neural network defines a function that takes a finite number of vectors in the embedding space as input. However, one often needs to consider evaluating the optimized network at points outside the training distribution. This paper considers the case in which the training data is distributed in a linear subspace of $\mathbb R^d$. We derive estimates on the variation of the learning function, defined by a neural network, in the direction transversal to the subspace. We study the potential regularization effects associated with the network's depth and noise in the codimension of the data manifold. We also present additional side effects in training due to the presence of noise.