On Space Folds of ReLU Neural Networks

TL;DR

This paper provides a quantitative analysis of how ReLU neural networks geometrically fold input space, revealing patterns of self-similarity and non-convexity, with empirical validation on benchmarks like MNIST.

Contribution

It introduces a novel measure for space folding in ReLU networks and proves the equivalence of convexity notions between input and activation spaces.

Findings

ReLU networks cause space folding leading to non-convex activation regions.

The introduced measure effectively quantifies the folding phenomenon.

Empirical analysis confirms the theoretical insights on benchmarks like MNIST.

Abstract

Recent findings suggest that the consecutive layers of ReLU neural networks can be understood geometrically as space folding transformations of the input space, revealing patterns of self-similarity. In this paper, we present the first quantitative analysis of this space folding phenomenon in ReLU neural networks. Our approach focuses on examining how straight paths in the Euclidean input space are mapped to their counterparts in the Hamming activation space. In this process, the convexity of straight lines is generally lost, giving rise to non-convex folding behavior. To quantify this effect, we introduce a novel measure based on range metrics, similar to those used in the study of random walks, and provide the proof for the equivalence of convexity notions between the input and activation spaces. Furthermore, we provide empirical analysis on a geometrical analysis benchmark (CantorNet) as well as an image classification benchmark (MNIST). Our work advances the understanding of the activation space in ReLU neural networks by leveraging the phenomena of geometric folding, providing valuable insights on how these models process input information.