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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 $P \subset M$ ofpoints approximating $X$. In this paper, we show that one can also use $P$ toestimate much more detailed topological properties of $X$. These properties arecomputed via $A_\infty$-structures, and are therefore related to the cup andMassey products of $X$, its loop space $\Omega X$, its formality, linkingnumbers, 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 detaileddescription of some geometric and topological properties of $Y$. We prove herethat under mild assumptions, these barcodes are stable: small perturbations inthe function $f$ imply at most small perturbations in the barcodes.

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