Estimating the persistent homology of $\mathbb{R}^n$-valued functions using function-geometric multifiltrations

TL;DR

This paper extends the approximation of persistent homology from real-valued functions to vector-valued functions using function-geometric multifiltrations, demonstrating robustness, convergence, and practical algorithms through theoretical analysis and experiments.

Contribution

It introduces a novel approach for approximating persistent homology of vector-valued functions using multifiltrations, expanding prior work limited to scalar functions.

Findings

Robustness of approximation to input noise

Good statistical convergence properties

Effective algorithm validated on synthetic and biological data

Abstract

Given an unknown $\mathbb{R}^n$-valued function $f$ on a metric space $X$, can we approximate the persistent homology of $f$ from a finite sampling of $X$ with known pairwise distances and function values? This question has been answered in the case $n=1$, assuming $f$ is Lipschitz continuous and $X$ is a sufficiently regular geodesic metric space, and using filtered geometric complexes with fixed scale parameter for the approximation. In this paper we answer the question for arbitrary $n$, under similar assumptions and using function-geometric multifiltrations. Our analysis offers a different view on these multifiltrations by focusing on their approximation properties rather than on their stability properties. We also leverage the multiparameter setting to provide insight into the influence of the scale parameter, whose choice is central to this type of approach. From a practical standpoint, we show that our approximation results are robust to input noise, and that function-geometric multifiltrations have good statistical convergence properties. We also provide an algorithm to compute our estimators, and we use its implementation to conduct extensive experiments, on both synthetic and real biological data, in order to validate our theoretical results and to assess the practicality of our approach.