Optimal Approximation of Zonoids and Uniform Approximation by Shallow Neural Networks

TL;DR

This paper advances the understanding of geometric approximation of zonoids in all dimensions and improves uniform approximation rates for shallow ReLU$^k$ neural networks, including their derivatives.

Contribution

It completes the solution to zonoid approximation in all dimensions and enhances approximation rates for shallow neural networks with ReLU$^k$ activations.

Findings

Closed the gap in zonoid approximation for dimensions 2 and 3.

Improved approximation rates for shallow ReLU$^k$ neural networks.

Achieved uniform approximation of functions and derivatives.

Abstract

We study the following two related problems. The first is to determine to what error an arbitrary zonoid in $\mathbb{R}^{d+1}$ can be approximated in the Hausdorff distance by a sum of $n$ line segments. The second is to determine optimal approximation rates in the uniform norm for shallow ReLU$^k$ neural networks on their variation spaces. The first of these problems has been solved for $d\neq 2,3$, but when $d=2,3$ a logarithmic gap between the best upper and lower bounds remains. We close this gap, which completes the solution in all dimensions. For the second problem, our techniques significantly improve upon existing approximation rates when $k\geq 1$, and enable uniform approximation of both the target function and its derivatives.