Computing Minimal Presentations and Bigraded Betti Numbers of 2-Parameter Persistent Homology

Abstract

Motivated by applications to topological data analysis, we give an efficient algorithm for computing a (minimal) presentation of a bigraded K[x, y]-module M , where K is a field. The algorithm takes as input a short chain complex of free modulesmemory, where |F i | denotes the size of a basis of F i . We observe that, given the presentation computed by our algorithm, the bigraded Betti numbers of M are readily computed. We also introduce a different but related algorithm, based on Koszul homology, which computes the bigraded Betti numbers without computing a presentation, with these same complexity bounds. These algorithms have been implemented in RIVET, a software tool for the visualization and analysis of two-parameter persistent homology. In experiments on topological data analysis problems, our approach outperforms the standard computational commutative algebra packages Singular and Macaulay2 by a wide margin.

Rectangular Approximation and Stability of $2$-parameter Persistence Modules

“…1. Compute the minimal presentations of M and N by the algorithm given in [Lesnick and Wright, 2019]. 2.…”

“…In practice, the data is often given as a simplicial filtration which induces the persistence module. It is shown in [Lesnick and Wright, 2019] that for a 2-parameter persistence module, a finite presentation can be computed in cubic time.…”

Bigraded Betti numbers and Generalized Persistence Diagrams

“…Asashiba et al provided a criterion for determining whether or not a given multiparameter persistence module is interval decomposable without having to explicitly compute decompositions [1]. Efficient algorithms for computing minimal presentations and the bigraded Betti numbers of 2-parameter persistence modules have been studied in [33,39]. In the special case of 2-parameter persistence modules that arise from the zeroth homology of Rips bifiltrations, an efficient combinatorial method to compute the bigraded Betti numbers has been proposed in [10].…”

Morse inequalities for the Koszul complex of multi-persistence

“…Proof. Proposition 2.3 of [LW19] states the following relation between the point-wise dimension of a (finitely presented) n-parameter persistence module V and its Betti tables ξ j , which is an easy consequence of Hilbert's Syzygy theorem:…”

“…For details on persistence modules not satisfying this property, see, e.g., [CSV17]. The corresponding pth Betti table ξ i p : Z n → N is obtained as the pth homology of the Koszul complex associated with the persistence module, a strategy already used in [Knu08] and later in [LW19]. As for the number c i (u) of the critical points of index i at grade u ∈ Z n of the filtration, we count them as the dimension of the homology at grade u relative to the previous grades: c i (u) := dim H i (X u , ∪ j X u−ej ).…”