Topological consistency via kernel estimation

TL;DR

This paper presents a kernel-based estimator for the homology of level sets of density and regression functions, providing a consistent method for topological inference in noisy data and demonstrating its effectiveness through simulations.

Contribution

It introduces a novel, consistent kernel estimation approach for homology and persistent homology, applicable to noisy data and validated with simulations.

Findings

Proves consistency of the estimator for homology and persistent homology

Successfully applies the method to simulated binary regression data

Demonstrates potential for topological inference in noisy observational data

Abstract

We introduce a consistent estimator for the homology (an algebraic structure representing connected components and cycles) of level sets of both density and regression functions. Our method is based on kernel estimation. We apply this procedure to two problems: (1) inferring the homology structure of manifolds from noisy observations, (2) inferring the persistent homology (a multi-scale extension of homology) of either density or regression functions. We prove consistency for both of these problems. In addition to the theoretical results, we demonstrate these methods on simulated data for binary regression and clustering applications.