Relocation of compact sets in $\mathbb{R}^n$ by diffeomorphisms and linear separability of datasets in $\mathbb{R}^n$

2026-04-23

AI Summary

This work investigates how smooth invertible mappings—specifically, diffeomorphisms—can reposition any finite collection of pairwise disjoint compact datasets in $\mathbb{R}^n$ to achieve linear separability. By integrating tools from differential topology and embedding theory, the authors construct a diffeomorphism that maps the data into $\mathbb{R}^{n+1}$ and demonstrate that a deep neural network of width merely $n$ or $n+1$, equipped with Leaky-ReLU, ELU, or SELU activations, suffices to realize this separation. This study establishes, for the first time, a rigorous connection between diffeomorphic repositioning theory and the expressive power of neural networks, providing theoretical guarantees for linear separability of compact sets under mild conditions and revealing the remarkable classification capability of shallow yet wide networks in high-dimensional representations.

Technology Category

• Machine Learning: Learning with Manifolds
• Computer Vision: Representation Learning for Vision
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

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 $\mathbb{R}^n$ to be relocated to arbitrary target domains in $\mathbb{R}^n$ by diffeomorphisms of $\mathbb{R}^n$. Furthermore, we prove that for any such collection, there exists a differentiable embedding into $\mathbb{R}^{n+1}$ such that their images become linearly separable. As applications of the established theory, we show that a finite number of compact datasets in $\mathbb{R}^n$ 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 $\mathbb{R}^n$ can be made linearly separable in $\mathbb{R}^{n+1}$ by a width-$(n+1)$ DNN.

Problem

• compact sets
• diffeomorphisms
• linear separability
• data classification
• deep neural networks

Innovation

• diffeomorphism
• linear separability
• compact sets
• deep neural networks
• differentiable embedding