Statistical topology via Morse theory, persistence and nonparametric estimation

TL;DR

This paper explores the use of topological data analysis, specifically persistent homology, for nonparametric estimation in multivariate statistics, providing theoretical guarantees on estimator convergence.

Contribution

It introduces a novel estimator of persistent homology from data and establishes minimax bounds for its accuracy under the sup-norm loss in nonparametric regression.

Findings

Establishes asymptotic minimax bounds for persistent homology estimation.

Provides convergence guarantees for the estimator in terms of expected bottleneck distance.

Connects topological methods with classical statistical risk bounds.

Abstract

In this paper we examine the use of topological methods for multivariate statistics. Using persistent homology from computational algebraic topology, a random sample is used to construct estimators of persistent homology. This estimation procedure can then be evaluated using the bottleneck distance between the estimated persistent homology and the true persistent homology. The connection to statistics comes from the fact that when viewed as a nonparametric regression problem, the bottleneck distance is bounded by the sup-norm loss. Consequently, a sharp asymptotic minimax bound is determined under the sup-norm risk over Holder classes of functions for the nonparametric regression problem on manifolds. This provides good convergence properties for the persistent homology estimator in terms of the expected bottleneck distance.