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

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.