Shallow ReLU$^s$ Networks in $L^p$-Type and Sobolev Spaces: Approximation and Path-Norm Controlled Generalization

ArXi:2605.18468v1 Announce Type: cross We study approximation by shallow ReLU$^s$ networks, $\sigma_s(t)=\max{0,t}^s$, and the generalization behavior of such networks under $\ell_1$ path-norm control. For the $L^p$-type integral spaces $\widetilde{\mathcal{F}}_{p,\tau_d,s}$, $1\le p\le2$, we establish approximation bounds for shallow networks using spherical harmonic analysis. In particular, when the parameter measure is the uniform measure $\tau_d$ and $p <2$ by embedding them into spectral Barron spaces.