Approximation Error and Complexity Bounds for ReLU Networks on Low-Regular Function Spaces
Abstract
In this work, we consider the approximation of a large class of bounded functions, with minimal regularity assumptions, by ReLU neural networks. We show that the approximation error can be bounded from above by a quantity proportional to the uniform norm of the target function and inversely proportional to the product of network width and depth. We inherit this approximation error bound from Fourier features residual networks, a type of neural network that uses complex exponential activation functions. Our proof is constructive and proceeds by conducting a careful complexity analysis associated with the approximation of a Fourier features residual network by a ReLU network.
Cite
@article{arxiv.2405.06727,
title = {Approximation Error and Complexity Bounds for ReLU Networks on Low-Regular Function Spaces},
author = {Owen Davis and Gianluca Geraci and Mohammad Motamed},
journal= {arXiv preprint arXiv:2405.06727},
year = {2026}
}
Comments
Theorem 1 and 2 proofs are incorrect; the proof strategy is flawed and not easily fixed. The Fourier feature residual network weights in [12] must be controlled to avoid infinite constants C. Any fix would require a near-rewrite, producing a manuscript of different scope and not suitable as a replacement. A corrected, modified Theorem 1 appears in arXiv:2207.09511v2