Interpolation and inverse problems in spectral Barron spaces
Shuai Lu, Peter Mathé
TL;DR
The paper develops a scale of spectral Barron spaces $B^{s}(\mathbb{R}^{d})$ built from the weakly sectorial operator $L=I-\Delta$, defining $B^{s}$ as the range of $L^{-s/2}$ and establishing a real interpolation inequality across the scale. By formulating inverse problems within this Barron framework through a link condition, it proves conditional stability and a modulus-of-continuity bound, and verifies the link for three linear settings: pseudo-differential operators, a Schrödinger inverse problem, and the Radon transform. It then shows that shallow two-layer networks can universally approximate Barron-regular functions with rates $\|u-u_n\|_{L^{2}(\Omega)} \lesssim \sqrt{|\Omega|/n}$ using RePU activations, via an integral-operator representation. Finally, a Barron-norm–penalized Tikhonov functional yields an error bound $\|u^{\dagger}-u_{n^{\delta}}\|_{L^{2}(\Omega)} \lesssim \delta^{p/(a+p)}$ with sublinear dependence on the noise level and a neuron count that does not suffer from the curse of dimensionality, highlighting a principled route to stable, dimension-insensitive inverse problem solutions using Barron-scale regularization.
Abstract
Spectral Barron spaces, which quantify the absolute value of weighted Fourier coefficients of a function, have gained considerable attention due to their capability for universal approximation across certain function classes. By establishing a connection between these spaces and a specific positive linear operator, we investigate the interpolation and scaling relationships among diverse spectral Barron spaces. Furthermore, we introduce a link condition by relating the spectral Barron space to inverse problems, illustrating this with three exemplary cases. We revisit the notion of universal approximation within the context of spectral Barron spaces and validate an error bound for Tikhonov regularization, penalized by the spectral Barron norm.