# Singular Values for ReLU Layers

## TL;DR

This paper introduces ReLU singular values and Gaussian mean width as new tools to analyze ReLU layers, providing theoretical insights and practical metrics for neural network performance and data classification.

## Contribution

It presents novel theoretical tools for understanding ReLU layers and demonstrates their practical utility in analyzing neural network behavior.

## Key findings

- ReLU singular values and Gaussian mean width offer new theoretical insights.
- These measures can distinguish correctly and incorrectly classified data.
- Tools like double-layers and harmonic pruning are introduced based on these findings.

## Abstract

Despite their prevalence in neural networks we still lack a thorough theoretical characterization of ReLU layers. This paper aims to further our understanding of ReLU layers by studying how the activation function ReLU interacts with the linear component of the layer and what role this interaction plays in the success of the neural network in achieving its intended task. To this end, we introduce two new tools: ReLU singular values of operators and the Gaussian mean width of operators. By presenting on the one hand theoretical justifications, results, and interpretations of these two concepts and on the other hand numerical experiments and results of the ReLU singular values and the Gaussian mean width being applied to trained neural networks, we hope to give a comprehensive, singular-value-centric view of ReLU layers. We find that ReLU singular values and the Gaussian mean width do not only enable theoretical insights, but also provide one with metrics which seem promising for practical applications. In particular, these measures can be used to distinguish correctly and incorrectly classified data as it traverses the network. We conclude by introducing two tools based on our findings: double-layers and harmonic pruning.