The Stability of Persistence Diagrams Under Non-Uniform Scaling

TL;DR

This paper establishes explicit bounds on how non-uniform coordinate scaling in Euclidean spaces affects the stability of persistence diagrams, providing a quantitative framework for understanding their robustness under such transformations.

Contribution

It derives explicit bounds on the bottleneck distance between persistence diagrams before and after non-uniform scaling in Euclidean spaces, extending stability analysis to higher dimensions and alternative metrics.

Findings

Bottleneck distance bounded by half the product of scaling range and data diameter.

Results extend to higher-dimensional homological features and Wasserstein distances.

Provides a quantitative measure of persistence diagram stability 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.