$A_\infty$ persistent homology estimates the topology from pointcloud datasets

Abstract

Let $X$ be a closed subspace of a metric space $M$. Under mild hypotheses, one can estimate the Betti numbers of $X$ from a finite set $P \subset M$ of points approximating $X$. In this paper, we show that one can also use $P$ to estimate much more detailed topological properties of $X$. These properties are computed via $A_\infty$-structures, and are therefore related to the cup and Massey products of $X$, its loop space $\Omega X$, its formality, linking numbers, etc. Additionally, we study the following setting: given a continuous function $f \colon Y \longrightarrow \mathbb R$ on a topological space $Y$, $A_\infty$ persistent homology builds a family of barcodes presenting a highly detailed description of some geometric and topological properties of $Y$. We prove here that under mild assumptions, these barcodes are stable: small perturbations in the function $f$ imply at most small perturbations in the barcodes.