Approximation Rates for Shallow ReLU$^k$ Neural Networks on Sobolev Spaces via the Radon Transform

• URL: http://arxiv.org/abs/2408.10996v1
• Date: Tue, 20 Aug 2024 16:43:45 GMT
• Title: Approximation Rates for Shallow ReLU$^k$ Neural Networks on Sobolev Spaces via the Radon Transform
• Authors: Tong Mao, Jonathan W. Siegel, Jinchao Xu,
• Abstract summary: We consider the problem of how efficiently shallow neural networks with the ReLU$k$ activation function can approximate functions from Sobolev spaces. We provide a simple proof of nearly optimal approximation rates in a variety of cases, including when $qleq p$, $pgeq 2$, and $s leq k + (d+1)/2$.
• Score: 4.096453902709292
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Let $\Omega\subset \mathbb{R}^d$ be a bounded domain. We consider the problem of how efficiently shallow neural networks with the ReLU$^k$ activation function can approximate functions from Sobolev spaces $W^s(L_p(\Omega))$ with error measured in the $L_q(\Omega)$-norm. Utilizing the Radon transform and recent results from discrepancy theory, we provide a simple proof of nearly optimal approximation rates in a variety of cases, including when $q\leq p$, $p\geq 2$, and $s \leq k + (d+1)/2$. The rates we derive are optimal up to logarithmic factors, and significantly generalize existing results. An interesting consequence is that the adaptivity of shallow ReLU$^k$ neural networks enables them to obtain optimal approximation rates for smoothness up to order $s = k + (d+1)/2$, even though they represent piecewise polynomials of fixed degree $k$.

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