Topological optimization with birth and death cochains

TL;DR

This paper introduces birth and death cochains as unique, generalized tools for persistent cohomology, enabling new loss functions for topological optimization across diverse data types and a novel application to arctic ice imagery.

Contribution

It proposes birth and death cochains as a novel, unique alternative to simplices in persistent cohomology, facilitating advanced topological optimization methods.

Findings

Birth and death cochains are always unique for a given persistent cohomology class.

Birth and death content can be effectively used as loss functions in topological optimization.

Application to arctic ice images demonstrates practical utility.

Abstract

We introduce the notion of birth and death cochains as generalized versions of birth and death simplices in persistent cohomology. We show that birth and death cochains (unlike birth and death simplices) are always unique for a given persistent cohomology class. We use birth and death cochains to define birth and death content as generalizations of birth and death times. We then demonstrate the advantages of using that birth and death content as loss functions on a variety of topological optimization tasks with point clouds, time series and scalar fields. We close with a novel application of topological optimization to a dataset of arctic ice images.