A directed persistent homology theory for dissimilarity functions

TL;DR

This paper introduces a novel directed persistent homology framework for analyzing directed simplicial complexes, enabling detection of persistent directed cycles and establishing connections with classical homology.

Contribution

It develops a new directed persistent homology theory based on semiring coefficients, addressing computational challenges and stability, and extending classical homology concepts.

Findings

Relates directed persistent homology to classical persistent homology.

Proves stability results for the new theory.

Discusses computational challenges and solutions.

Abstract

We develop a theory of persistent homology for directed simplicial complexes which detects persistent directed cycles in odd dimensions. We relate directed persistent homology to classical persistent homology, prove some stability results, and discuss the computational challenges of our approach. Our directed persistent homology theory is motivated by homology with semiring coefficients: by explicitly removing additive inverses, we are able to detect directed cycles algebraically.