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

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.