🤖 AI Summary
This work addresses the topological mechanisms underlying dimensionality reduction and width in deep neural networks, specifically focusing on preserving connectivity and ensuring class separability when embedding compact topological spaces into Euclidean space.
Method: We establish a rigorous mathematical framework grounded in topological degree theory, characterizing the intrinsic relationship between embedding connectivity and linear/nonlinear separability under dimension-reducing mappings. We quantitatively analyze how network width and input dimension compression affect classification separability and function approximation capacity.
Contribution/Results: We propose the first unified theoretical perspective integrating topological degree, embedding connectivity, and deep network architecture design—providing principled, mathematically grounded guidelines for width selection and dimension compression. Our analysis significantly enhances interpretability of deep models in classification and approximation tasks and rigorously delineates their topological performance limits.
📝 Abstract
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.