Minimax Rates for Homology Inference

TL;DR

This paper investigates the statistical challenge of estimating the homology groups of a manifold from noisy high-dimensional data, providing bounds on the minimax risk under various noise models.

Contribution

It introduces new upper and lower bounds on the minimax risk for homology inference from noisy samples, advancing understanding of the problem's fundamental limits.

Findings

Derived upper bounds using union of balls estimators

Established lower bounds via Le Cam's lemma

Quantified the minimax risk for different noise models

Abstract

Often, high dimensional data lie close to a low-dimensional submanifold and it is of interest to understand the geometry of these submanifolds. The homology groups of a manifold are important topological invariants that provide an algebraic summary of the manifold. These groups contain rich topological information, for instance, about the connected components, holes, tunnels and sometimes the dimension of the manifold. In this paper, we consider the statistical problem of estimating the homology of a manifold from noisy samples under several different noise models. We derive upper and lower bounds on the minimax risk for this problem. Our upper bounds are based on estimators which are constructed from a union of balls of appropriate radius around carefully selected points. In each case we establish complementary lower bounds using Le Cam's lemma.