# Relocation of compact sets in ℝn by diffeomorphisms and linear separability of datasets in ℝn

arXiv: https://arxiv.org/abs/2604.21393 · HTML: https://mlsanity.com/paper/2604.21393/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 4 of its 10 references that are in the map.
- First posted on arXiv 23 Apr 2026.

## Builds on (most cited first, 4 of 4)

- [Universal Approximation with Deep Narrow Networks](https://mlsanity.com/paper/1905.08539/) (2019)
- [Approximating Continuous Functions by ReLU Nets of Minimal Width](https://mlsanity.com/paper/1710.11278/) (2017)
- [Coupling-based Invertible Neural Networks Are Universal Diffeomorphism Approximators](https://mlsanity.com/paper/2006.11469/) (2020)
- [Minimum Width of Deep Narrow Networks for Universal Approximation](https://mlsanity.com/paper/2511.06837/) (2025)

## Abstract (arXiv)

Relocation of compact sets in an n-dimensional manifold by self-diffeomorphism is of its own interest as well as significant potential applications to data classification in data science. This paper presents a theory for relocating a finite number of compact sets in ℝn to be relocated to arbitrary target domains in ℝn by diffeomorphisms of ℝn. Furthermore, we prove that for any such collection, there exists a differentiable embedding into ℝn+1 such that their images become linearly separable. As applications of the established theory, we show that a finite number of compact datasets in ℝn can be made linearly separable by width-n deep neural networks (DNNs) with Leaky-ReLU, ELU, or SELU activation functions, under a mild condition. In addition, we show that any finite number of mutually disjoint compact datasets in ℝn can be made linearly separable in ℝn+1 by a width-(n+1) DNN.