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 ℝn ) with Euclidean distance ( dX ), we consider scaling transformations defined by scaling factors ( s1, s2, ldots, sn > 0 ). Specifically, we define a scaling function ( S ) that maps each point ( x = (x1, x2, ldots, xn) ∈ X ) to [ S(x) = (s1 x1, s2 x2, ldots, sn xn). ] Our main result establishes that the bottleneck distance ( dB(D, DS) ) between the persistence diagrams ( D ) of ( X ) and ( DS ) of ( S(X) ) satisfies: [ dB(D, DS) ≤ (smax - smin) · operatornamediam(X), ] where ( smin = min1 ≤ i ≤ n si ), ( smax = max1 ≤ i ≤ n si ), and ( operatornamediam(X) ) is the diameter of ( X ). Based on this theoretical guarantee, we formulate an optimization problem to minimize the scaling variability ( Deltas = smax - smin ) under the constraint ( dB(D, DS) ≤ ε ), where ( ε > 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.