Geometry-Dependent Approximation and Stability Properties of Shallow ReLU$^k$ Networks on Bounded Euclidean Domains

원문 초록

On bounded Euclidean domains, we study how hidden-parameter geometry affects shallow ReLU$^k$ networks through results on approximation rates, saturation-type lower bounds, Gram and stiffness matrix spectra, Bernstein and Marcinkiewicz--Zygmund inequalities, discrete least-squares recovery, and inverse approximation. Our analysis uses smooth localization and dyadic Fourier estimates. We quantify parameter geometry by the fill distance $h$ and antipodal separation distance $\underline h$. For the prescribed network space $L_n^k$, representative approximation estimates take the form \begin{equation*} \begin{aligned} \inf_{g_n\in L_n^k}\|f-g_n\|_{\mathcal L^2(\Omega')}&\lesssim h^r\|f\|_{\mathcal H^r(\Omega)},\\ \inf_{g_n\in L_n^k}\|f-g_n\|_{\mathcal L^2(\Omega)}&\gtrsim \underline h^{\frac{d+2k+1}{2}}\|f\|_{\mathcal L^2(\Omega')}. \end{aligned} \end{equation*} The upper bound exploits target regularity up to the critical order $\frac{d+2k+1}{2}$, while the lower bound applies asymptotically to fixed targets smoother than this order. The upper estimate holds on an interior subdomain $\Omega'$, whereas the lower estimate concerns the larger domain $\Omega$. We also obtain geometry-dependent spectral estimates that quantify conditioning and coefficient stability. Further results establish norm discretization and regularized recovery from function values, and infer target regularity from approximation rates under coefficient or separation conditions. Altogether, these results describe how parameter geometry governs approximation accuracy, the stability of coefficient representations, discrete recovery, and the regularity information contained in convergence rates.