A Complete Symmetry Classification of Shallow ReLU Networks

TL;DR

This paper provides a complete classification of parameter space symmetries for shallow ReLU neural networks by leveraging the non-differentiability of ReLU, addressing a gap in existing symmetry analyses.

Contribution

It introduces a novel approach exploiting ReLU's non-differentiability to fully classify symmetries in shallow networks, unlike previous methods requiring analyticity.

Findings

Complete symmetry classification for shallow ReLU networks.

ReLU's non-differentiability is key to the classification.

Impacts understanding of optimization and parameter identifiability.

Abstract

Parameter space is not function space for neural network architectures. This fact, investigated as early as the 1990s under terms such as ``reverse engineering," or ``parameter identifiability", has led to the natural question of parameter space symmetries\textemdash the study of distinct parameters in neural architectures which realize the same function. Indeed, the quotient space obtained by identifying parameters giving rise to the same function, called the \textit{neuromanifold}, has been shown in some cases to have rich geometric properties, impacting optimization dynamics. Thus far, techniques towards complete classifications have required the analyticity of the activation function, notably excising the important case of ReLU. Here, in contrast, we exploit the non-differentiability of the ReLU activation to provide a complete classification of the symmetries in the shallow case.