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 $s_1, s_2, \ldots, s_n > 0$. Specifically, we define a scaling function $S$ that maps each point $x = (x_1, x_2, \ldots, x_n) \in X$ to $$S(x) = (s_1 x_1, s_2 x_2, \ldots, s_n x_n).$$ 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: $$d_B(D, D_S) \leq (s_{\max} - s_{\min}) \cdot \operatorname{diam}(X),$$ where $s_{\min} = \min_{1 \leq i \leq n} s_i$, $s_{\max} = \max_{1 \leq i \leq n} s_i$, and $\operatorname{diam}(X)$ is the diameter of $X$. Based on this theoretical guarantee, we formulate an optimization problem to minimize the scaling variability $\Delta_s = s_{\max} - s_{\min}$ under the constraint $d_B(D, D_S) \leq \epsilon$, 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.