[Paper Review] Matroid Filtrations and Computational Persistent Homology

Gregory Henselman, Robert Ghrist|arXiv (Cornell University)|Jun 1, 2016

Topological and Geometric Data Analysis21 references47 citations

TL;DR

This paper introduces a novel algorithm for efficient computation of persistent homology using matroid theory, discrete Morse theory, and matrix factorization. By leveraging acyclic matchings to avoid constructing large boundary matrices, the method drastically reduces memory usage—achieving up to several orders of magnitude compression—while enabling computation of representative homology generators. The approach is implemented in the EIRENE library and demonstrates significant performance gains on high-dimensional point clouds, including a 10,000-point complex with over 160 billion simplices.

ABSTRACT

This technical report introduces a novel approach to efficient computation in homological algebra over fields, with particular emphasis on computing the persistent homology of a filtered topological cell complex. The algorithms here presented rely on a novel relationship between discrete Morse theory, matroid theory, and classical matrix factorizations. We provide background, detail the algorithms, and benchmark the software implementation in the Eirene package.

Motivation & Objective

• To develop a more memory-efficient and computationally scalable method for computing persistent homology in topological data analysis.
• To address the high memory cost of storing full boundary matrices in large simplicial complexes.
• To enable computation of representative homology generators in the original complex, not just barcode summaries.
• To integrate matroid theory and discrete Morse theory into a unified framework for computational homology.
• To implement and benchmark the algorithm in the EIRENE software package for real-world datasets.

Proposed method

• The method uses matroid theory to reframe persistent homology concepts like rank, bases, and circuits in a combinatorial framework that supports efficient computation.
• Discrete Morse theory is applied via acyclic matchings to simplify the complex and avoid explicit construction of large boundary operators.
• The algorithm leverages matrix factorization theorems to compute persistent homology efficiently and recover representative cycles in the original complex.
• A dynamic construction subroutine in EIRENE generates only the essential cells of the Morse complex, avoiding storage of irrelevant simplices.
• The approach uses a greedy algorithm on matroid bases to identify minimal sets of cells that preserve homology, reducing computational load.
• The method is optimized for top-dimensional boundary operators, which are typically the largest and most memory-intensive components.

Experimental results

Research questions

• RQ1 Can matroid-theoretic formulations improve the efficiency and clarity of persistent homology computation?
• RQ2 How can discrete Morse theory be used to avoid constructing and storing full boundary matrices in persistent homology?
• RQ3 To what extent can acyclic matchings be used to compress the cellular complex while preserving homology and enabling generator recovery?
• RQ4 Can the combination of matroid theory, Morse theory, and matrix factorization yield a practical, scalable algorithm for large-scale persistent homology?
• RQ5 What is the memory and time performance gain of this method on real-world high-dimensional datasets?

Key findings

• The algorithm reduced memory usage by up to several orders of magnitude by avoiding construction of large boundary matrices, particularly for top-dimensional operators.
• For a 10,000-point random geometric complex in R20, the method computed persistent H1 with 35.7 GB of heap memory and 3,120 seconds, handling a 2-skeleton with over 160 billion simplices.
• The compression ratio (size of Morse complex / size of 4-skeleton) dropped to as low as 0.02 for n=240 vertices, indicating extreme sparsification.
• The method successfully computed persistent homology with representative generators for a 1-dimensional class in a noisy torus sample, visualized via Plotly.
• On challenging unfiltered complexes like the chessboard and matching complexes, performance remained high but was limited by the intrinsic complexity of the spaces.
• The EIRENE implementation achieved significant speedups and memory savings, especially on high-dimensional point clouds, outperforming prior benchmarks on large-scale datasets.