Effects of Data Geometry in Early Deep Learning

TL;DR

This paper investigates how the geometry of data manifolds influences the expressivity of randomly initialized deep neural networks, providing theoretical bounds and empirical validation on different data types.

Contribution

It extends theoretical understanding of neural network behavior on non-Euclidean data by deriving bounds on linear region boundaries and validating findings experimentally.

Findings

Number of linear regions varies across different data manifolds.

Complexity of linear regions differs between image manifolds and Euclidean space.

Results are consistent across various neural network architectures.

Abstract

Deep neural networks can approximate functions on different types of data, from images to graphs, with varied underlying structure. This underlying structure can be viewed as the geometry of the data manifold. By extending recent advances in the theoretical understanding of neural networks, we study how a randomly initialized neural network with piece-wise linear activation splits the data manifold into regions where the neural network behaves as a linear function. We derive bounds on the density of boundary of linear regions and the distance to these boundaries on the data manifold. This leads to insights into the expressivity of randomly initialized deep neural networks on non-Euclidean data sets. We empirically corroborate our theoretical results using a toy supervised learning problem. Our experiments demonstrate that number of linear regions varies across manifolds and the results hold with changing neural network architectures. We further demonstrate how the complexity of linear regions is different on the low dimensional manifold of images as compared to the Euclidean space, using the MetFaces dataset.