Optimising the topological information of the $A_\infty$-persistence groups

Persistent homology typically studies the evolution of homology groups $Hp(X)$ (with coefficients in a field) along a filtration of topological spaces. $A\infty$-persistence extends this theory by analysing the evolution of subspaces such as $V := \text{Ker}\, {\Deltan}{| Hp(X)} \subseteq Hp(X)$, where ${\Deltam}{m\geq1}$ denotes a structure of $A\infty$-coalgebra on $H*(X)$. In this paper we illustrate how $A\infty$-persistence can be useful beyond persistent homology by discussing the topological meaning of $V$, which is the most basic form of $A\infty$-persistence group. In addition, we explore how to choose $A\infty$-coalgebras along a filtration to make the $A\infty$-persistence groups carry more faithful information.