The Stability of Persistence Diagrams Under Non-Uniform Scaling

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 $s_1,s_2,\dots ,s_n>0$ 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 $$d_B(D,D_S)\le \frac{1}{2}(s_{\max }-s_{\min })\cdot \mathrm{diam}(X),$$ where $s_{\min }$ and $s_{\max }$ are the smallest and largest scaling factors, respectively, and $\mathrm{diam}(X)$ 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.