Persistent (Co)Homology in Matrix Multiplication Time

作     者：Morozov, Dmitriy Skraba, Primoz 

作者机构：Lawrence Berkeley National Laboratory United States School of Mathematical Sciences Queen Mary University of London United Kingdom 

出 版 物：《arXiv》 (arXiv)

年 卷 期：2024年

核心收录：

主　　题：Consensus algorithm 

摘      要：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 for the more general case of zigzag persistent homology [1], 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 [1], but returns a different set of representatives than the standard algorithm [2]. We then give a fast version of a variant called the row algorithm [3], which returns the same representatives as the standard algorithm. © 2024, CC BY.