A directed persistent homology theory for dissimilarity functions

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.