Relocation of compact sets in R^n by diffeomorphisms and linear separability of datasets in R^n

arXiv:2604.21393v1 Announce Type: new 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 R^n to be relocated to arbitrary target domains in R^n by diffeomorphisms of R^n. Furthermore, we prove that for any such collection, there exists a differentiable embedding into 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 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 R^n can be made linearly separable in R^{n+1} by a width-(n+1) DNN.

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Source: arXiv cs.LG | 2026-04-24