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

TL;DR

This paper introduces a mathematical framework connecting topological embeddings and dimension reduction maps, offering new insights into deep learning classification and approximation capabilities.

Contribution

It develops a novel topological approach to analyze neural network embeddings and their separability after dimension reduction, advancing theoretical understanding.

Findings

Provides a topological perspective on neural network embeddings

Analyzes separability of embeddings under dimension reduction

Offers insights into classification and approximation in deep learning

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.