Improved Universal Approximation with Neural Networks Studied via Affine-Invariant Subspaces of \(L_2(\mathbb{R}^n)\)
Year: 2025
Type: article
Abstract: We show that there are no non-trivial closed subspaces of \(L_2(\mathbb{R}^n)\) that are invariant under invertible affine transformations. We apply this result to neural networks showing that any nonzero \(L_2(\mathbb{R})\) function is an adequate activation function in a one hidden layer neural network in order to approximate every function in \(L_2(\mathbb{R})\) with any desired accuracy. This generalizes the universal approximation properties of neural networks in \(L_2(\mathbb{R})\) related to Wiener's Tauberian Theorems. Our results extend to the spaces \(L_p(\mathbb{R})\) with \(p>1\).
Authors: Cornelia Schneider, Cornelia Schneider
Institution: Friedrich-Alexander-Universität Erlangen-Nürnberg
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W4409503657