Decomposing filtered chain complexes: geometry behind barcoding algorithms

TL;DR

This paper introduces an algorithm to decompose filtered chain complexes into interval spheres, offering geometric insights into persistence algorithms and enabling broader applications in topological data analysis.

Contribution

It provides a novel decomposition algorithm for filtered chain complexes into interval spheres, enhancing understanding and application of persistence in TDA.

Findings

Algorithm successfully decomposes filtered chain complexes into interval spheres

Provides geometric insights into persistence algorithms and optimizations

Applicable to general filtered chain complexes, including filtered kernels

Abstract

In Topological Data Analysis, filtered chain complexes enter the persistence pipeline between the initial filtering of data and the final persistence invariants extraction. It is known that they admit a tame class of indecomposables, called interval spheres. In this paper, we provide an algorithm to decompose filtered chain complexes into such interval spheres. This algorithm provides geometric insights into various aspects of the standard persistence algorithm and two of its run-time optimizations. Moreover, since it works for any filtered chain complexes, our algorithm can be applied in more general cases. As an application, we show how to decompose filtered kernels with it.