Abstract
arXiv:2605.18468v4 Announce Type: replace-cross Abstract: This paper studies approximation by shallow ReLU$^s$ networks, $\sigma_s(t)=\max\{0,t\}^s$, together with their generalization behavior under $\ell_1$ path-norm control. For the $L^p$-type integral spaces $\widetilde{\mathcal{F}}_{p,\tau_d,s}$, $1\le p\le2$, spherical harmonic analysis yields approximation bounds for shallow networks. In particular, when $\tau_d$ is the uniform measure and $1\le p<2$, where $p^*=\frac{2d+2}{d+3}$. Approximation bounds for Sobolev spaces $W^{\alpha,p}$, $1\le p<2$, are obtained through embeddings into spectral Barron spaces. For nonparametric regression with sub-Gaussian noise, path-norm-regularized shallow ReLU$^s$ networks achieve minimax-optimal rates $O\!\left(n^{-\frac{d+2s+1}{2d+2s+1}}\log n\right)$ over $\mathscr{B}_s$ and $O\!\left(n^{-\frac{2\alpha}{2\alpha+d}}\log n\right)$ over $W^{\alpha,\infty}$, with matching lower bounds up to logarithmic factors.