Computing persistent homology with various coefficient fields in a single pass

Abstract

This article introduces an algorithm to compute the persistent homology of a filtered complex with various coefficient fields in a single matrix reduction. The algorithm is output-sensitive in the total number of distinct persistent homological features in the diagrams for the different coefficient fields. This computation allows us to infer the prime divisors of the torsion coefficients of the integral homology groups of the topological space at any scale, hence furnishing a more informative description of topology than persistence in a single coefficient field. We provide theoretical complexity analysis as well as detailed experimental results. The code is part of the Gudhi software library, and is available at Maria (in: GUDHI User and Reference Manual, GUDHI Editorial Board, 2015).

Arithmetic notations

• $\mathbb {Z}$ ring of integers,

• $\mathbb {Z}/n\mathbb {Z}$ ring of integers modulo $n \ge 2$,

• $\mathbb {Q}$ field of rationals,

• $q_1, \ldots , q_r$ the r first prime numbers, for $r \ge 1$,

• [r] the set $\{1, \ldots , r\}$,

• $Q := q_1 \times \cdots \times q_r$, product of first r prime numbers,

• $Q_S := \prod _{s \in S} q_s$, for a subset of indices $S \subset [r]$,

• Indices s, t, r, and set of indices S and T, are reserved to the indexing of prime numbers $\{q_1, \ldots , q_r\}$,

• Indices i, j, k and m refer to indices in the filtration of a complex, and hence indices for matrix columns and matrix reduction algorithms.