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This paper introduces a method for relocating compact sets in $\mathbb{R}^n$ to arbitrary target domains using diffeomorphisms. It further proves the existence of differentiable embeddings into $\mathbb{R}^{n+1}$ that render these sets linearly separable. As a practical application, the authors demonstrate that finite compact datasets in $\mathbb{R}^n$ can be made linearly separable using deep neural networks with specific activation functions.
Compact datasets in n-dimensional space can be transformed into linearly separable sets using diffeomorphisms and shallow, wide neural networks, challenging the need for complex architectures in certain classification tasks.
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.
