Weighted Sobolev Approximation Rates for Neural Networks on Unbounded Domains
This work addresses a theoretical limitation in neural network approximation for unbounded domains, which is incremental but relevant for mathematical analysis and applications in machine learning.
The paper extends approximation theory for shallow neural networks to weighted Sobolev spaces on unbounded domains, establishing asymptotic approximation rates without the curse of dimensionality.
In this work, we consider the approximation capabilities of shallow neural networks in weighted Sobolev spaces for functions in the spectral Barron space. The existing literature already covers several cases, in which the spectral Barron space can be approximated well, i.e., without curse of dimensionality, by shallow networks and several different classes of activation function. The limitations of the existing results are mostly on the error measures that were considered, in which the results are restricted to Sobolev spaces over a bounded domain. We will here treat two cases that extend upon the existing results. Namely, we treat the case with bounded domain and Muckenhoupt weights and the case, where the domain is allowed to be unbounded and the weights are required to decay. We first present embedding results for the more general weighted Fourier-Lebesgue spaces in the weighted Sobolev spaces and then we establish asymptotic approximation rates for shallow neural networks that come without curse of dimensionality.