Pull-back Geometry of Persistent Homology Encodings

TL;DR

This paper introduces a novel geometric approach using pull-back metrics to analyze persistent homology encodings, providing intrinsic insights into their significance, sensitivity, and effectiveness for data analysis tasks.

Contribution

It proposes a new methodology based on pull-back geometry to understand persistent homology representations independently of downstream models.

Findings

Pull-back spectrum and eigenvectors reveal significant features captured by PH.

Pull-back norm indicates sensitivity and feature detection capability of PH.

Correlation between pull-back norm and downstream task performance guides PH encoding choice.

Abstract

Persistent homology (PH) is a method for generating topology-inspired representations of data. Empirical studies that investigate the properties of PH, such as its sensitivity to perturbations or ability to detect a feature of interest, commonly rely on training and testing an additional model on the basis of the PH representation. To gain more intrinsic insights about PH, independently of the choice of such a model, we propose a novel methodology based on the pull-back geometry that a PH encoding induces on the data manifold. The spectrum and eigenvectors of the induced metric help to identify the most and least significant information captured by PH. Furthermore, the pull-back norm of tangent vectors provides insights about the sensitivity of PH to a given perturbation, or its potential to detect a given feature of interest, and in turn its ability to solve a given classification or regression problem. Experimentally, the insights gained through our methodology align well with the existing knowledge about PH. Moreover, we show that the pull-back norm correlates with the performance on downstream tasks, and can therefore guide the choice of a suitable PH encoding.