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Decomposing Multiparameter Persistence Modules

Tamal K. Dey, Jan Jendrysiak, Michael Kerber

TL;DR

The paper advances the decomposition of multiparameter persistence modules by extending the Generalized Persistence Algorithm to all finitely presented modules, removing the uniquely graded restriction. It develops a suite of matrix-reduction techniques (notably BlockReduce and its generalizations), introduces parameter-restriction concepts (including $U$-concentrated sequences and $\mathcal{P}_U$-presentations), and proves fixed-parameter tractability with respect to the maximum number of relations sharing a degree. The resulting Automorphism-invariant Iterative Decomposition Algorithm (AIDA) delivers practical performance on large-scale inputs, achieving $O(n^3)$ time for interval-decomposable modules and enabling interval-decomposability testing; an implementation is provided as the AIDA library. Extensive benchmarks show speed-ups over prior methods on real and synthetic data, and the approach opens avenues for improved invariants and downstream tasks in multiparameter persistence. Overall, the work significantly extends the computational toolkit for MPMs, enabling decomposition-based analysis at scale and broadening applicability across posets and higher-parameter settings.

Abstract

Dey and Xin (J.Appl.Comput.Top., 2022, arXiv:1904.03766) describe an algorithm to decompose finitely presented multiparameter persistence modules using a matrix reduction algorithm. Their algorithm only works for modules whose generators and relations are distinctly graded. We extend their approach to work on \emph{all} finitely presented modules and introduce several improvements that lead to significant speed-ups in practice. Our algorithm is fixed parameter tractable with respect to the maximal number of relations with the same degree and with further optimisation we obtain an $O(n^3)$ algorithm for interval-decomposable modules. In particular, we can decide interval-decomposability in this time. As a by-product to the proofs of correctness we develop a theory of parameter restriction for persistence modules. Our algorithm is implemented as a software library \textsc{aida} which is the first to enable the decomposition of large inputs. We show its capabilities via extensive experimental evaluation.

Decomposing Multiparameter Persistence Modules

TL;DR

The paper advances the decomposition of multiparameter persistence modules by extending the Generalized Persistence Algorithm to all finitely presented modules, removing the uniquely graded restriction. It develops a suite of matrix-reduction techniques (notably BlockReduce and its generalizations), introduces parameter-restriction concepts (including -concentrated sequences and -presentations), and proves fixed-parameter tractability with respect to the maximum number of relations sharing a degree. The resulting Automorphism-invariant Iterative Decomposition Algorithm (AIDA) delivers practical performance on large-scale inputs, achieving time for interval-decomposable modules and enabling interval-decomposability testing; an implementation is provided as the AIDA library. Extensive benchmarks show speed-ups over prior methods on real and synthetic data, and the approach opens avenues for improved invariants and downstream tasks in multiparameter persistence. Overall, the work significantly extends the computational toolkit for MPMs, enabling decomposition-based analysis at scale and broadening applicability across posets and higher-parameter settings.

Abstract

Dey and Xin (J.Appl.Comput.Top., 2022, arXiv:1904.03766) describe an algorithm to decompose finitely presented multiparameter persistence modules using a matrix reduction algorithm. Their algorithm only works for modules whose generators and relations are distinctly graded. We extend their approach to work on \emph{all} finitely presented modules and introduce several improvements that lead to significant speed-ups in practice. Our algorithm is fixed parameter tractable with respect to the maximal number of relations with the same degree and with further optimisation we obtain an algorithm for interval-decomposable modules. In particular, we can decide interval-decomposability in this time. As a by-product to the proofs of correctness we develop a theory of parameter restriction for persistence modules. Our algorithm is implemented as a software library \textsc{aida} which is the first to enable the decomposition of large inputs. We show its capabilities via extensive experimental evaluation.

Paper Structure

This paper contains 24 sections, 14 theorems, 25 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Classical Techniques.
  3. Previous Results for Multiparameter Persistence Modules.
  4. Previous Results for Interval-Decopmposability.
  5. On Decompositions.
  6. Persistence Modules and their Presentations
  7. Introduction
  8. Notation
  9. Simplicial filtrations.
  10. Presentations:
  11. Presentations
  12. Minimal Presentations
  13. Decompositions
  14. Decomposing Uniquely Graded Presentations
  15. The Generalized Persistence Algorithm
  16. ...and 9 more sections

Key Result

Proposition 1

CZMP Let $A\text{-GrMod}$ be the category of $\mathbb{Z}^d$-graded $A$-modules and linear maps. The direct sum of all vector spaces in a persistence module and the functor which sends a graded $A$-module to its homogeneous summands define an equivalence of categories

Figures (3)

  • Figure 1: Philippe Chaperon's set design for Giuseppe Verdi's opera Aida, 1880
  • Figure 2: The matrix in the middle presents the module on the left. The picture on the right indicates the dimensions of the vector spaces and locations of generators (blue points and brackets) and relations (red point and bracket) over a continuous plane.
  • Figure 3: A (4,3)-decomposed presentation. There are two blocks in $M_\mathscr{B}$ and they present the interval modules (\ref{['def:interval']}) on the right. Some admissible operations have been indicated by arrows both in the matrix and between the corresponding generators and relations.

Theorems & Definitions (44)

  • Definition 1
  • Proposition 1
  • Definition 2: Interval Module
  • Definition 3
  • Definition 4: DeyXin
  • Remark 1
  • Definition 5
  • Definition 6
  • Proposition 2
  • Corollary 1
  • ...and 34 more