用拓扑度理论揭示深度网络降维与宽度的内在关系

Dimensionality reduction and width of deep neural networks based on topological degree theory

• 基于拓扑度理论建立嵌入映射与降维可分性的数学关联
• 证明了特定降维下嵌入保持分离性的条件
• 为深层网络的宽度设计提供理论依据，适合理论研究者

本文构建了一个将紧致拓扑空间嵌入欧氏空间的嵌入映射与特定类降维映射下嵌入可分性相联系的数学框架。基于该理论，我们对深度学习中分类与逼近问题提供了新视角，特别是在深度神经网络设置下，揭示了拓扑结构在降维过程中的保持机制及其对网络宽度的影响。

原文摘要 · Abstract (English)

In this paper we present a mathematical framework on linking of embeddings of compact topological spaces into Euclidean spaces and separability of linked embeddings under a specific class of dimension reduction maps. As applications of the established theory, we provide some fascinating insights into classification and approximation problems in deep learning theory in the setting of deep neural networks.

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