Topology-Preserving Scaling in Data Augmentation

We propose an algorithmic framework for dataset normalization in data augmentation pipelines that preserves topological stability under non-uniform scaling transformations. Given a finite metric space $X\subset {\mathbb{R}}^n$ with Euclidean distance $d_X$, we consider scaling transformations defined by scaling factors $s_1,s_2,\dots ,s_n>0$. Specifically, we define a scaling function $S$ that maps each point $x=(x_1,x_2,\dots ,x_n)\in X$ to $$S(x)=(s_1{x}_1,s_2{x}_2,\dots ,s_n{x}_n)\mathrm{.}$$ Our main result establishes that the bottleneck distance $d_B(D,D_S)$ between the persistence diagrams $D$ of $X$ and $D_S$ of $S(X)$ satisfies: $$d_B(D,D_S)\le (s_{\max }-s_{\min })\cdot \mathrm{diam}(X),$$ where $s_{\min }={\min }_{1\le i\le n}{s}_i$, $s_{\max }={\max }_{1\le i\le n}{s}_i$, and $\mathrm{diam}(X)$ is the diameter of $X$. Based on this theoretical guarantee, we formulate an optimization problem to minimize the scaling variability ${\mathrm{\Delta }}_s=s_{\max }-s_{\min }$ under the constraint $d_B(D,D_S)\le \epsilon$, where $\epsilon >0$ is a user-defined tolerance. We develop an algorithmic solution to this problem, ensuring that data augmentation via scaling transformations preserves essential topological features. We further extend our analysis to higher-dimensional homological features, alternative metrics such as the Wasserstein distance, and iterative or probabilistic scaling scenarios. Our contributions provide a rigorous mathematical framework for dataset normalization in data augmentation pipelines, ensuring that essential topological characteristics are maintained despite scaling transformations.