The bifiltration of a relation and extended Dowker duality

TL;DR

This paper extends Dowker's duality to a bifiltration framework that captures homotopical information of relations in data analysis, providing a functorial approach and a reconstruction method.

Contribution

It introduces a bifiltration of Dowker complexes and proves a functorial duality, advancing the understanding of relations in topological data analysis.

Findings

Extended Dowker duality to bifiltrations

Constructed a functorial version of the duality

Provided a reconstruction result for data matrices

Abstract

We explain how homotopical information of two composeable relations can be organized in two simplicial categories that augment the relations row and column complexes. We show that both of these categories realize to weakly equivalent spaces, thereby extending Dowker's duality theorem. We also prove a functorial version of this result. Specializing the above construction a bifiltration of Dowker complexes that coherently incorporates the total weights of a relation's row and column complex into one single object is introduced. This construction is motivated by challenges in data analysis that necessitate the simultaneous study of a data matrix rows and columns. To illustrate the applicability of our constructions for solving those challenges we give an appropriate reconstruction result.