How shallow neural networks approximate smooth functions

This work analyzes how shallow neural networks with ReLU^s activations (raising ReLU outputs to power s) approximate functions in smooth function spaces. Using spherical harmonic analysis, the authors derive approximation rates that beat random-feature baselines for L^p and Sobolev spaces. They also prove that regularizing the network by path norm—a measure of parameter size—achieves minimax-optimal generalization bounds for regression, meaning no algorithm can do substantially better without stronger assumptions.