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

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On September 7, 2026, arXiv cs.LG published a research paper on moving compact sets in n-dimensional Euclidean space using differentialomorphisms. The study explored the feasibility of using geometric transformation methods to handle linearly inseparable problems in high-dimensional datasets, with the aim of improving the linear separability of the datasets through specific spatial mappings.

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A arXiv cs.LG en 2026-09-07 12:00

本文提出了一种理论，利用$\mathbb{R}^n$的微分同胚将有限个紧集重定位至$\mathbb{R}^n$中的任意目标域。研究证明，对于此类集合，存在一种可微嵌入到$\mathbb{R}^{n+1}$，使其图像线性可分。基于该理论，文章表明在轻微条件下，$\mathbb{R}^n$中的有限个紧数据集可通过宽度为$n$的带 Leaky-ReLU、ELU 或 SELU 激活函数的深度神经网络（DNN）实现线性可分；同时证明，$\mathbb{R}^n$中任意有限个互不相交的紧数据集可通过宽度为$(n+1)$的 DNN 在$\mathbb{R}^{n+1}$中实现线性可分。