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 $s_1, s_2, \ldots, s_n > 0$ 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.