A higher homotopic extension of persistent (co)homology

TL;DR

This paper introduces a higher homotopic extension of persistent (co)homology, constructing an A_infinity-algebra structure on persistent cohomology and defining a refined metric for topological data analysis.

Contribution

It develops a novel A_infinity-algebra framework for persistent cohomology and proposes a new metric to distinguish complex topological features.

Findings

Constructed a (pseudo)metric on generalized barcodes

Refined the bottleneck metric with A_infinity-structure

Demonstrated the metric's ability to detect linking patterns like Borromean rings

Abstract

Our objective in this article is to show a possibly interesting structure of homotopic nature appearing in persistent (co)homology. Assuming that the filtration of the (say) simplicial set embedded in a finite dimensional vector space induces a multiplicative filtration (which would not be a so harsh hypothesis in our setting) on the dg algebra given by the complex of simplicial cochains, we may use a result by T. Kadeishvili to get a unique (up to noncanonical equivalence) A_infinity-algebra structure on the complete persistent cohomology of the filtered simplicial (or topological) set. We then provide a construction of a (pseudo)metric on the set of all (generalized) barcodes (that is, of all cohomological degrees) enriched with the A_infinity-algebra structure stated before, refining the usual bottleneck metric, and which is also independent of the particular A_infinity-algebra structure chosen (among those equivalent to each other). We think that this distance might deserve some attention for topological data analysis, for it in particular can recognize different linking or foldings patterns, as in the Borromean rings. As an aside, we give a simple proof of a result relating the barcode structure between persistent homology and cohomology. This result was observed in a recent article by V. de Silva, D. Morozov and M. Vejdemo-Johansson under some restricted assumptions, which we do not suppose.