Minimum-Norm Interpolation in Weighted Variation Spaces: Direct and Inverse Problems

Abstract

We study minimum-norm interpolation in weighted variation spaces generated by ReLU$^s$ ridge functions on the unit ball, where $s$ is a positive integer. For a target $f$ in the weighted variation space and a node set with sufficiently small fill distance $h$, every minimum-norm interpolant satisfies the direct bound $\|f-\mathscr{I}_{X_h}f\|_{L_2(\mathbb{B}^d)}\lesssim_{d,s}h^{(2s+1)/4}\|f\|_{\mathcal{V}_s}$. Under a boundary separation condition on the nodes, we construct a matching lower bound for the worst-case interpolation error. For the inverse problem, we introduce dual certificates and derive an estimate analogous to a Bernstein inequality. Under suitable upper bounds on dictionary size and spectral lower bounds for the atomic Gram matrices, sufficiently fast $L_2$ convergence of minimum-norm interpolants along geometrically decreasing fill distances implies that the target belongs to $\mathcal{V}_s$. As a byproduct, the best $L_2$ approximation error $E_n(f)_{L_2(\mathbb{B}^d)}$ for shallow ReLU$^s$ networks of width at most $n$ satisfies $E_n(f)_{L_2(\mathbb{B}^d)}\lesssim_{d,s}n^{-1/2-(2s+1)/(2d)}\|f\|_{\mathcal{V}_s}$, with the weighted atomic cost of an approximating network bounded by a constant multiple of $\|f\|_{\mathcal{V}_s}$, independently of $n$.