The Stability of Persistence Diagrams Under Non-Uniform Scaling

Abstract: We investigate the stability of persistence diagrams ( D ) under non-uniform scaling transformations ( S ) in ( \mathbb{R}^n ). Given a finite metric space ( X \subset \mathbb{R}^n ) with Euclidean distance ( d_X ), and scaling factors ( s_1, s_2, \ldots, s_n > 0 ) applied to each coordinate, we derive explicit bounds on the bottleneck distance ( d_B(D, D_S) ) between the persistence diagrams of ( X ) and its scaled version ( S(X) ). Specifically, we show that [ d_B(D, D_S) \leq \frac{1}{2} (s_{\max} - s_{\min}) \cdot \operatorname{diam}(X), ] where ( s_{\min} ) and ( s_{\max} ) are the smallest and largest scaling factors, respectively, and ( \operatorname{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.