Computing persistent homology

We show that the persistent homology of a filtered ddimensional simplicial complex is simply the standard homology of a particular graded module over a polynomial ring. Our analysis establishes the existence of a simple description of persistent homology groups over arbitrary fields. It also enables us to derive a natural algorithm for computing persistent homology of spaces in arbitrary dimension over any field. This result generalizes and extends the previously known algorithm that was restricted to subcomplexes of S3 and Z2 coefficients. Finally, our study implies the lack of a simple classification over non-fields. Instead, we give an algorithm for computing individual persistent homology groups over an arbitrary principal ideal domains in any dimension.

• Type: Conference
• Year: 2004
• Where: COMPGEOM
• Authors: Afra Zomorodian, Gunnar Carlsson
• Keywords: COMPGEOM 2004, Individual Persistent Homology, Persistent Homology, Persistent Homology Groups