# Topology-Preserving Scaling in Data Augmentation

arXiv: https://arxiv.org/abs/2411.19512 · HTML: https://mlsanity.com/paper/2411.19512/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 14 references that are in the map.
- First posted on arXiv 29 Nov 2024.

## Builds on (most cited first, 5 of 5)

- [mixup: Beyond Empirical Risk Minimization](https://mlsanity.com/paper/1710.09412/) (2017)
- [Group Normalization](https://mlsanity.com/paper/1803.08494/) (2018)
- [A roadmap for the computation of persistent homology](https://mlsanity.com/paper/1506.08903/) (2015)
- [RegBN: Batch Normalization of Multimodal Data with Regularization](https://mlsanity.com/paper/2310.00641/) (2023)
- [Feature transforms for image data augmentation](https://mlsanity.com/paper/2201.09700/) (2022)

## Abstract (arXiv)

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.