On rectangle-decomposable 2-parameter persistence modules - Inria - Institut national de recherche en sciences et technologies du numérique Access content directly
Conference Papers Leibniz International Proceedings in Informatics Year : 2020

On rectangle-decomposable 2-parameter persistence modules


This paper addresses two questions: (a) can we identify a sensible class of 2-parameter persistence modules on which the rank invariant is complete? (b) can we determine efficiently whether a given 2-parameter persistence module belongs to this class? We provide positive answers to both questions, and our class of interest is that of rectangle-decomposable modules. Our contributions include: on the one hand, a proof that the rank invariant is complete on rectangle-decomposable modules, together with an inclusion-exclusion formula for counting the multiplicities of the summands; on the other hand, algorithms to check whether a module induced in homology by a bifiltration is rectangle-decomposable, and to decompose it in the affirmative, with a better complexity than state-of-the-art decomposition methods for general 2-parameter persistence modules. Our algorithms are backed up by a new structure theorem, whereby a 2-parameter persistence module is rectangle-decomposable if, and only if, its restrictions to squares are. This local characterization is key to the efficiency of our algorithms, and it generalizes previous conditions derived for the smaller class of block-decomposable modules. It also admits an algebraic formulation that turns out to be a weaker version of the one for block-decomposability. By contrast, we show that general interval-decomposability does not admit such a local characterization, even when locality is understood in a broad sense. Our analysis focuses on the case of modules indexed over finite grids, the more general cases are left as future work.

Dates and versions

hal-02996045 , version 1 (09-11-2020)





Magnus Bakke Botnan, Vadim Lebovici, Steve Oudot. On rectangle-decomposable 2-parameter persistence modules. SoCG 2020 - 36th International Symposium on Computational Geometry, Sergio Cabello; Danny Z. Chen, Jun 2020, Zurich, Switzerland. pp.22:1-22:16, ⟨10.4230/LIPIcs.SoCG.2020.22⟩. ⟨hal-02996045⟩
27 View
0 Download



Gmail Facebook Twitter LinkedIn More