Topological Data Analysis of Decision Boundaries with Application to Model Selection

TL;DR

This paper introduces novel topological methods to analyze decision boundaries in neural networks, enabling better understanding of model complexity and aiding dataset-model matching.

Contribution

It proposes new labeled and locally scaled Vietoris-Rips complexes for persistent homology inference of decision boundaries, with theoretical analysis and practical experiments.

Findings

Successfully recover decision boundary homology from samples.

Quantify neural network complexity effectively.

Match datasets to pre-trained models using topological features.

Abstract

We propose the labeled \v{C}ech complex, the plain labeled Vietoris-Rips complex, and the locally scaled labeled Vietoris-Rips complex to perform persistent homology inference of decision boundaries in classification tasks. We provide theoretical conditions and analysis for recovering the homology of a decision boundary from samples. Our main objective is quantification of deep neural network complexity to enable matching of datasets to pre-trained models; we report results for experiments using MNIST, FashionMNIST, and CIFAR10.