Addressing caveats of neural persistence with deep graph persistence

TL;DR

This paper critically examines neural persistence, revealing its limitations related to weight variance and spatial structure, and introduces deep graph persistence as a more robust measure for neural network complexity.

Contribution

It identifies key limitations of neural persistence and proposes deep graph persistence, a new method that captures network complexity more effectively across layers.

Findings

Neural persistence is mainly influenced by weight variance and spatial concentration.

In later layers, neural persistence lacks meaningful spatial structure.

Deep graph persistence improves robustness by considering the entire network structure.

Abstract

Neural Persistence is a prominent measure for quantifying neural network complexity, proposed in the emerging field of topological data analysis in deep learning. In this work, however, we find both theoretically and empirically that the variance of network weights and spatial concentration of large weights are the main factors that impact neural persistence. Whilst this captures useful information for linear classifiers, we find that no relevant spatial structure is present in later layers of deep neural networks, making neural persistence roughly equivalent to the variance of weights. Additionally, the proposed averaging procedure across layers for deep neural networks does not consider interaction between layers. Based on our analysis, we propose an extension of the filtration underlying neural persistence to the whole neural network instead of single layers, which is equivalent to calculating neural persistence on one particular matrix. This yields our deep graph persistence measure, which implicitly incorporates persistent paths through the network and alleviates variance-related issues through standardisation. Code is available at https://github.com/ExplainableML/Deep-Graph-Persistence .