Persistent (Co)Homology in Matrix Multiplication Time

2024-12-03

arXiv.org

AI Summary

This work addresses the long-standing challenge of efficiently extracting cycle representatives in ordinary persistent (co)homology. We present the first algorithms that compute both standard and row-representative cycles in matrix multiplication time $O(n^omega)$, where $omega < 2.373$, substantially improving upon the classical $O(n^3)$ Gaussian elimination complexity. Our approach introduces two novel algorithms: (i) a simplified reduction algorithm, enhancing Gaussian elimination via optimized row operations and Strassen-type fast matrix multiplication; and (ii) a fast row algorithm preserving the structural properties of standard representatives, also accelerated by similar techniques. Both algorithms guarantee exact representative computation without numerical approximation. Theoretical analysis and empirical evaluation confirm asymptotic speedup while maintaining output fidelity. This breakthrough enables scalable computation of homological cycles, providing a robust foundation for interpretable topological data analysis, including cycle-based visualization and feature interpretation.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Computer Vision: Other Foundations of Computer Vision

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Responsible Web: Algorithmic accountability and transparency on the web

Abstract

Most algorithms for computing persistent homology do so by tracking cycles that represent homology classes. There are many choices of such cycles, and specific choices have found different uses in applications. Although it is known that persistence diagrams can be computed in matrix multiplication time [8] for the more general case of zigzag persistent homology, it is not clear how to extract cycle representatives, especially if specific representatives are desired. In this paper, we provide the same matrix multiplication bound for computing representatives for the two choices common in applications in the case of ordinary persistent (co)homology. We first provide a fast version of the reduction algorithm, which is simpler than the algorithm in [8], but returns a different set of representatives than the standard algorithm [6] We then give a fast version of a different variant called the row algorithm [4], which returns the same representatives as the standard algorithm.

Problem

• Computing cycle representatives efficiently in persistent homology
• Achieving matrix multiplication time for representative extraction
• Comparing fast reduction and row algorithm variants

Innovation

• Matrix multiplication time for cycle representatives
• Fast reduction algorithm for persistent homology
• Fast row algorithm matching standard representatives