Computing Persistent Homology

We study the homology of a filtered d-dimensional simplicial complex K as a single algebraic entity and establish a correspondence that provides a simple description over fields. Our analysis enables us to derive a natural algorithm for computing persistent homology over an arbitrary field in any dimension. Our study also 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.

[1]  A. O. Houcine.  On hyperbolic groups , 2006 .

[2]  Leonidas J. Guibas,et al.  Persistence barcodes for shapes , 2004, SGP '04.

[3]  B.R. Donald,et al.  On the complexity of computing the homology type of a triangulation , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[4]  Leonidas J. Guibas,et al.  A Barcode Shape Descriptor for Curve Point Cloud Data , 2022 .

[5]  Gunnar E. Carlsson,et al.  Topological estimation using witness complexes , 2004, PBG.

[6]  Herbert Edelsbrunner,et al.  Computing and Comprehending Topology: Persistence and Hierarchical Morse Complexes , 2001 .

[7]  Steven J. Leon.  Transform linear algebra , 2002 .

[8]  Erik Carlsson,et al.  c ○ World Scientific Publishing Company AN ALGEBRAIC TOPOLOGICAL METHOD FOR FEATURE IDENTIFICATION , 2022 .

[9]  Herbert Edelsbrunner,et al.  Topological Persistence and Simplification , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.

[10]  H. O. Foulkes.  Abstract Algebra , 1967, Nature.

[11]  David A. Cox,et al.  Ideals, Varieties, and Algorithms , 1997 .

[12]  Saugata Basu,et al.  On Bounding the Betti Numbers and Computing the Euler Characteristic of Semi-Algebraic Sets , 1996, STOC '96.

[13]  Herbert Edelsbrunner,et al.  Topological persistence and simplification , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.

[14]  Tommy Färnqvist.  Number Theory Meets Cache Locality – Efficient Implementation of a Small Prime FFT for the GNU Multiple Precision Arithmetic Library , 2005 .

[15]  Jean-Guillaume Dumas,et al.  Computing Simplicial Homology Based on Efficient Smith Normal Form Algorithms , 2003, Algebra, Geometry, and Software Systems.

[16]  J. Friedman,et al.  Computing Betti Numbers via Combinatorial Laplacians , 1996, STOC '96.

[17]  James R. Munkres,et al.  Elements of algebraic topology , 1984 .