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

TL;DR

This paper develops a theory for relocating finite compact sets in Euclidean space via diffeomorphisms and demonstrates how these sets can be made linearly separable using neural networks with specific activation functions.

Contribution

It introduces a method to reposition compact sets in $\

Findings

Finite datasets can be made linearly separable by width-$n$ DNNs with certain activations.

Any disjoint compact datasets can be separated in $\

Finite datasets can be embedded into $\

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.