2024/11/25 by Vu-Anh Le, Le, Vu-Anh, Mehmet Dık +1
Computer Science · Physics and Astronomy · #Algebraic Topology (math.AT) #Complex Network Analysis Techniques #Data Management and Algorithms #FOS: Mathematics #Geometric Topology (math.GT) #Metric Geometry (math.MG) #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2411.16126
openalex publication_date 2024/11/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We investigate the stability of persistence diagrams \( D \) under non-uniform scaling transformations \( S \) in \( ℝn \). Given a finite metric space \( X ⊂ ℝn \) with Euclidean distance \( dX \), and scaling factors \( s1, s2, …, sn > 0 \) applied to each coordinate, we derive explicit bounds on the bottleneck distance \( dB(D, DS) \) between the persistence diagrams of \( X \) and its scaled version \( S(X) \). Specifically, we show that dB(D, DS) ≤ (1)/(2) (smax - smin) ⋅ diam(X), where \( smin \) and \( smax \) are the smallest and largest scaling factors, respectively, and \( diam(X) \) is the diameter of \( X \). We extend this analysis to higher-dimensional homological features, alternative metrics such as the Wasserstein distance, and iterative or probabilistic scaling scenarios. Our results provide a framework for quantifying the effects of non-uniform scaling on persistence diagrams.