Singular persistent homology with geometrically parallelizable computation

TL;DR

This paper introduces singular persistent homology and a distributed computational workflow, enabling effective and feasible topological data analysis on large datasets by leveraging Mayer-Vietoris theorems.

Contribution

It develops a new theory of singular persistent homology and proves Mayer-Vietoris theorems for it, facilitating distributed computation for large data sets.

Findings

Mayer-Vietoris theorems are extended to persistent homology.

A distributed workflow for persistent homology is proposed.

Feasibility for large data sets is significantly improved.

Abstract

Persistent homology is a popular tool in Topological Data Analysis. It provides numerical characteristics of data sets which reflect global geometric properties. In order to be useful in practice, for example for feature generation in machine learning, it needs to be effectively computable. Classical homology is a computable topological invariant because of the Mayer-Vietoris exact and spectral sequences associated to coverings of a space. We state and prove versions of the Mayer-Vietoris theorem for persistent homology under mild and commonplace assumptions. This is done through the use of a new theory, the singular persistent homology, better suited for handling coverings of data sets. As an application, we create a distributed computational workflow where the advantage is not only or even primarily in speed improvement but in sheer feasibility for large data sets.