2024/11/15 by Jiaxing He, He, Jiaxing, Bingzhe Hou +5
Computer Science · Mathematics · #47B15 #68T09 #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Primary 55N31 #Secondary 15A18 #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2411.09960
openalex publication_date 2024/11/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Classification in the sense of similarity is an important issue. In this paper, we study similarity classification in Topological Data Analysis. We define a pseudometric dS(p) to measure the distance between barcodes generated by persistent homology groups of topological spaces, and we provide that our pseudometric dS(2) is a similarity invariant. Thereby, we establish a connection between Operator Theory and Topological Data Analysis. We give the calculation formula of the pseudometric dS(2) (dS(1)) by arranging all eigenvalues of matrices determined by barcodes in descending order to get the infimum over all matchings. Since conformal linear transformation is one representative type of similarity transformations, we construct comparative experiments on both synthetic datasets and waves from an online platform to demonstrate that our pseudometric dS(2) (dS(1)) is stable under conformal linear transformations, whereas the bottleneck and Wasserstein distances are not. In particular, our pseudometric on waves is only related to the waveform but is independent on the frequency and amplitude. Furthermore, the computation time for dS(2) (dS(1)) is significantly less than the computation time for bottleneck distance and is comparable to the computation time for accelerated Wasserstein distance between barcodes.