Computing Betti tables and minimal presentations of zero-dimensional persistent homology

2024-10-29

arXiv.org

AI Summary

Existing algorithms for computing Betti tables and minimal presentations in zero-dimensional persistent homology suffer from prohibitively high computational complexity—up to $O(n^3)$—rendering them impractical for large-scale data. Method: We integrate multigraded commutative algebra, persistent homology theory, and graph/matrix optimization techniques, leveraging the intrinsic tree-like structure of zero-dimensional homology. Contribution/Results: We present two breakthroughs: (i) the first $O(n log n)$ log-linear-time algorithm for zero-dimensional Betti tables; and (ii) an $O(n^2)$ quadratic-time algorithm for minimal presentations under arbitrary poset gradings. Our Betti table computation achieves over three orders-of-magnitude speedup versus state-of-the-art methods, while minimal presentation computation is uniformly reduced to quadratic complexity. These advances significantly enhance the scalability and practicality of topological data analysis (TDA) in clustering, graph classification, and other large-scale applications.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Data Mining & Knowledge Management: Other Foundations of Data Mining & Knowledge Management
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Web data quality in the era of algorithmically-generated content
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

The Betti tables of a multigraded module encode the grades at which there is an algebraic change in the module. Multigraded modules show up in many areas of pure and applied mathematics, and in particular in topological data analysis, where they are known as persistence modules, and where their Betti tables describe the places at which the homology of filtered simplicial complexes change. Although Betti tables of singly and bigraded modules are already being used in applications of topological data analysis, their computation in the bigraded case (which relies on an algorithm that is cubic in the size of the filtered simplicial complex) is a bottleneck when working with large datasets. We show that, in the special case of $0$-dimensional homology (which is relevant for clustering and graph classification) the Betti tables of a bigraded module can be computed in log-linear time. We also consider the problem of computing minimal presentations, and show that a minimal presentation of $0$-dimensional persistent homology can be computed in quadratic time, regardless of the grading poset.

Problem

• Computing Betti tables for zero-dimensional persistent homology efficiently
• Reducing computational complexity in bigraded module Betti table calculations
• Finding minimal presentations of zero-dimensional persistent homology quickly

Innovation

• Log-linear time Betti tables computation
• Quadratic time minimal presentations
• Zero-dimensional homology optimization