[Paper Review] A New Matching Algorithm for Multidimensional Persistence

Madjid Állili, Tomasz Kaczyński+2 more|arXiv (Cornell University)|Nov 17, 2015

Topological and Geometric Data Analysis17 references3 citations

TL;DR

This paper presents a novel matching algorithm for multidimensional persistence that constructs an acyclic partial matching on simplicial complexes using vector-valued functions extended via least common upper bounds. By processing lower stars of simplices of all dimensions—not just vertices—it achieves substantial cell reduction while preserving multidimensional persistent homology, significantly outperforming prior vertex-based methods in reducing spurious critical cells.

ABSTRACT

An algorithm is presented that constructs an acyclic partial matching on the cells of a given simplicial complex from a vector-valued function defined on the vertices and extended to each simplex by taking the least common upper bound of the values on its vertices. The resulting acyclic partial matching may be used to construct a reduced filtered complex with the same multidimensional persistent homology as the original simplicial complex filtered by the sublevel sets of the function. Numerical tests show that in practical cases the rate of reduction in the number of cells achieved by the algorithm is substantial. This promises to be useful for the computation of multidimensional persistent homology of simplicial complexes filtered by sublevel sets of vector-valued functions.

Motivation & Objective

• To address the limitation of existing discrete Morse matching algorithms in multidimensional persistence, which produce excessive spurious critical cells.
• To improve matching optimality in multidimensional filtrations by extending the scope beyond vertex-based processing to include all simplex dimensions.
• To develop a method that preserves multidimensional persistent homology while achieving high reduction rates in cell count for both synthetic and real-world data.
• To provide a practical and scalable algorithm for computing persistent homology in multidimensional settings, especially for complex data like 3D meshes and topological surfaces.

Proposed method

• The algorithm extends a vector-valued function defined on vertices to all simplices by computing the least common upper bound of vertex values.
• It processes the lower stars of all simplices (not just vertices), enabling better handling of non-comparable values in multidimensional filtrations.
• An acyclic partial matching is constructed on the complex’s cells, partitioning them into matched pairs and critical cells.
• The resulting matching enables the construction of a reduced Lefschetz complex with identical multidimensional persistent homology as the original.
• The method uses topological sorting and iterative matching to ensure acyclicity and correctness.
• The algorithm is implemented and tested on triangulated surfaces (torus, Klein bottle) and real 3D meshes with component-wise injective perturbations of the function.

Experimental results

Research questions

• RQ1 Can a matching algorithm that processes lower stars of all simplices, not just vertices, reduce the number of spurious critical cells in multidimensional persistence?
• RQ2 How does the performance of this new matching algorithm compare to prior vertex-based approaches in terms of cell reduction and preservation of persistent homology?
• RQ3 To what extent can the algorithm scale and achieve high reduction rates on large and complex 3D simplicial complexes?
• RQ4 Does the use of least common upper bounds for extending vector-valued functions to higher-dimensional simplices lead to better topological fidelity in the reduced complex?

Key findings

• On a 7200-simplex torus triangulation, the algorithm reduced the number of critical simplices to 156, representing only 2.17% of the original complex.
• For a Klein bottle approximation with 1881 simplices, the algorithm achieved 257 critical simplices, or 13.67% of the original size.
• On real 3D meshes such as the space shuttle, the algorithm reduced the total number of critical cells to just 3.82% of the original, with only 225 critical vertices out of 2376.
• The method achieved a 96.18% reduction in critical cells for the space shuttle mesh, demonstrating strong scalability and efficiency on real-world data.
• The algorithm significantly outperformed the vertex-based recursive method from prior work, especially in reducing critical faces and edges.
• Experiments confirmed that the algorithm scales well with increasing complex size and maintains high reduction rates across diverse topological structures.