Optimal Approximation of Zonoids and Uniform Approximation by Shallow Neural Networks
This work addresses foundational mathematical problems in approximation theory, with implications for neural network design, but is incremental in refining existing bounds.
The paper closes a logarithmic gap in approximating zonoids by sums of line segments for dimensions d=2,3, completing the solution in all dimensions, and significantly improves uniform approximation rates for shallow ReLU^k neural networks when k≥1, enabling approximation of both functions and their derivatives.
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.