A. Zomorodian, G. Carlsson, Computing Persistent Homology, Discrete and Computational Geometry 33 (2005), 249-274.

Abstract:

We show that the persistent homology of a filtered d-dimensional 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 results generalizes and extends the previously known algorithm that was restricted to subcomplexes of S^3 and Z_2 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 PIDs in any dimension.

Bibtex: 

@article{zc-cph-05,
  author = "Zomorodian, A. and Carlsson, G.",
  title = "Computing Persistent Homology",
  journal = "Discrete Comput. Geom.",
  year = 2005,
  volume = 33,
  number = 2,
  pages = "249--274",
 }