Computing relative Betti diagrams of persistence modules over finite posets using Koszul complexes

In topological data analysis (TDA), functors from a poset category to vector spaces, called persistence modules, are a central object of investigation. Persistence modules can arise from data via several constructions, and often encode complex geometric information on the data. Finding informative and computable descriptors of persistence modules is crucial to extract this information and use it in data analysis.

A recent trend in TDA is to describe persistence modules over finite posets via relative homological invariants. Relative homological algebra extends constructions of standard homological algebra by redefining the notion of projective module, which depends on the choice of a family of “basic” modules. In this talk, we study relative resolutions, which are a way of approximating a given persistence module by a sequence of persistence modules that are projective relative to the chosen family. Under certain conditions, the multiplicities of the basic persistence modules in a relative resolution are unique, and define invariants called relative Betti diagrams. Using Koszul complexes, relative Betti diagrams can be computed in a simple and local way, avoiding the computation of the entire relative resolution.

The talk is based on a joint work with Wojciech Chachólski, Isaac Ren, Martina Scolamiero, and Francesca Tombari.