🤖 AI Summary
This paper addresses whether the persistent homology of an unknown $mathbb{R}^n$-valued function $f$ on a metric space $X$ can be reliably approximated from a finite sample—containing pairwise distances and function values—extending prior theoretical guarantees, which exist only for $n=1$. Method: We generalize the function–geometry bifiltration to arbitrary $n$, assuming $f$ is Lipschitz continuous and $X$ is a regular geodesic metric space. Our approach establishes provable approximation error bounds for multiparameter persistent homology, explicitly characterizing how scale parameters affect estimation accuracy—going beyond mere stability analysis. Contribution/Results: We provide the first theoretically grounded, general framework for high-dimensional topological data analysis, overcoming the limitations of single-parameter persistence. The results enable robust modeling of multiscale and multidirectional topological features, with quantifiable approximation guarantees for multiparameter persistent homology under realistic sampling assumptions.
📝 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.