[Paper Review] Generalized Persistence Algorithm for Decomposing Multi-parameter Persistence Modules

Tamal K. Dey, Cheng Xin|arXiv (Cornell University)|Apr 7, 2019

Topological and Geometric Data Analysis51 references11 citations

TL;DR

This paper presents a generalized persistence algorithm for decomposing multi-parameter persistence modules over Z^d, using a novel matrix reduction technique that runs in O(N^{2ω+1}) time, where ω < 2.373 is the matrix multiplication exponent. The method efficiently computes a unique decomposition for finitely presented modules with distinctly graded generators and relations, significantly outperforming the Meataxe algorithm in both theory and practice.

ABSTRACT

The classical persistence algorithm computes the unique decomposition of a persistence module implicitly given by an input simplicial filtration. Based on matrix reduction, this algorithm is a cornerstone of the emergent area of topological data analysis. Its input is a simplicial filtration defined over the integers $\mathbb{Z}$ giving rise to a $1$-parameter persistence module. It has been recognized that multiparameter version of persistence modules given by simplicial filtrations over $d$-dimensional integer grids $\mathbb{Z}^d$ is equally or perhaps more important in data science applications. However, in the multiparameter setting, one of the main challenges is that topological summaries based on algebraic structure such as decompositions and bottleneck distances cannot be as efficiently computed as in the $1$-parameter case because there is no known extension of the persistence algorithm to multiparameter persistence modules. We present an efficient algorithm to compute the unique decomposition of a finitely presented persistence module $M$ defined over the multiparameter $\mathbb{Z}^d$. The algorithm first assumes that the module is presented with a set of $N$ generators and relations that are \emph{distinctly graded}. Based on a generalized matrix reduction technique it runs in $O(N^{2ω+1})$ time where $ω&lt;2.373$ is the exponent for matrix multiplication. This is much better than the well known algorithm called Meataxe which runs in $ ilde{O}(N^{6(d+1)})$ time on such an input. In practice, persistence modules are usually induced by simplicial filtrations. With such an input consisting of $n$ simplices, our algorithm runs in $O(n^{(d-1)(2ω+ 1)})$ time for $d\geq 2$. For the special case of zero dimensional homology, it runs in time $O(n^{2ω+1})$.

Motivation & Objective

• To develop an efficient algorithm for computing the unique decomposition of finitely presented multi-parameter persistence modules over Z^d.
• To overcome the computational inefficiency of existing methods like Meataxe, which scale poorly with dimension and input size.
• To provide a practical and scalable alternative to Meataxe for modules arising from simplicial filtrations in topological data analysis.
• To establish a foundation for computing stable invariants such as persistent graded Betti numbers and blockcodes in the multi-parameter setting.

Proposed method

• The algorithm uses a generalized matrix reduction technique tailored for multi-parameter persistence modules with distinctly graded generators and relations.
• It represents the module via a presentation matrix encoding generators and relations, avoiding costly explicit representation of linear maps.
• The core computation relies on matrix reduction over a polynomial ring, adapted to handle multi-grading and ensuring unique decomposition via Krull-Schmidt theorem.
• For simplicial filtrations with n simplices, the algorithm runs in O(n^{(d-1)(2ω+1)}) time for d ≥ 2, and O(n^{2ω+1}) for zero-dimensional homology.
• The method assumes distinct grades for all generators and relations; if not satisfied, it still computes a decomposition of a perturbed module M′ close to the original under interleaving distance.
• The algorithm is designed to be compatible with existing techniques in single-parameter TDA, such as those using boundary matrices, and can be accelerated using advanced matrix multiplication.

Experimental results

Research questions

• RQ1 Can a generalized persistence algorithm be designed for multi-parameter modules that is both efficient and practical, avoiding the high cost of Meataxe?
• RQ2 How does the algorithm's performance scale with the number of simplices and dimension d in simplicial filtrations?
• RQ3 What is the stability of the computed decomposition when the distinct grading assumption is violated in practice?
• RQ4 Can the algorithm be extended or adapted to approximate decompositions with stability guarantees for general multi-parameter modules?
• RQ5 What new pseudo-metrics can be constructed from the invariants (e.g., persistent graded Betti numbers, blockcodes) and how do they relate to existing metrics like interleaving and bottleneck distance?

Key findings

• The algorithm computes the unique decomposition of a finitely presented multi-parameter persistence module in O(N^{2ω+1}) time, where N is the number of generators and relations.
• For d ≥ 2, the time complexity is O(n^{(d-1)(2ω+1)}) when the input is a simplicial filtration with n simplices.
• For zero-dimensional homology, the time complexity reduces to O(n^{2ω+1}), matching the best-known single-parameter algorithms.
• The algorithm outperforms Meataxe, which runs in ˜O(N^{6(d+1)}) time, by a significant margin in both theory and practice.
• Even when the distinct grading assumption is violated, the algorithm computes a decomposition of a module M′ that is arbitrarily close to the original M under interleaving distance.
• The computed invariants—persistent graded Betti numbers and blockcodes—generalize the persistence diagram and barcode to the multi-parameter setting.