通过微分同胚变换，可将任意有限紧凑数据集变为线性可分。
Relocation of compact sets in $\mathbb{R}^n$ by diffeomorphisms and linear separability of datasets in $\mathbb{R}^n$
- 利用微分同胚实现数据集在高维空间的任意重定位。
- 证明了任意有限个紧致数据集可在n+1维空间中线性可分。
- 适用于深度神经网络的理论支持，尤其适合复杂分类任务。
通过ℝⁿ上的自微分同胚对有限个紧致集进行重定位，不仅具有理论意义，也对数据科学中的数据分类有潜在应用价值。本文建立了一套理论，证明可将ℝⁿ中任意有限个紧致集通过ℝⁿ的微分同胚映射到任意目标区域。此外，我们证明对于任意此类集合族，存在一个ℝ^{n+1}中的可微嵌入，使其像集成为线性可分。作为理论的应用，我们表明在满足弱条件下，任意有限个紧致数据集可通过宽度为n的深度神经网络（使用Leaky-ReLU、ELU或SELU激活函数）实现线性可分。进一步地，任意有限个互不相交的紧致数据集在ℝ^{n+1}中可通过宽度为n+1的深度神经网络实现线性可分。
原文摘要 · Abstract (English)
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.
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