Sufficient Conditions for the Shrinking Wellness Lemma

TL;DR

This paper examines the Shrinking Wellness Lemma related to well groups, providing counterexamples to its general validity and establishing conditions under which the lemma holds, thereby clarifying its applicability in topological data analysis.

Contribution

The paper identifies limitations of the original lemma and introduces conditions ensuring its correctness, expanding understanding of well group robustness measures.

Findings

Counterexample showing the lemma does not always hold

Conditions under which the lemma is valid

Most practical applications of well groups are covered by these conditions

Abstract

The well groups were introduced by Edelsbrunner, Morozov, and Patel to measure the robustness of geometric features of a function with respect to perturbations. Roughly speaking, the $r$-th well group measures the number of features that cannot be removed by perturbing the function by at most $r$. The Shrinking Wellness Lemma states that the rank of these groups decreases as $r$ increases. In the generality originally stated, it is wrong. We present a counterexample and give conditions under which the result holds. These conditions are general enough to cover most cases in which the well groups have been applied.