Sharp Sobolev Approximation on General Domains by Linearized Shallow Networks with Analytic Activations

Abstract: We study Sobolev approximation on bounded domains by linearized shallow neural networks whose inner parameters are prescribed independently of the target function. Our main step is a one-dimensional construction for analytic activations. We prove that quasi-Chebyshev parameter sets with univariate resolution $m$ generate fixed feature spaces attaining the sharp $H<sup>r$-to-$H<sup>s$ approximation order $m<sup>{-(r-s)}$ for a class of analytic activations satisfying a quantitative non-cancellation condition on their Taylor coefficients. Combining this result with the ridge-function lifting theorem in [SIAM J. Math. Anal. 30 (1998), pp. 155-189] and its extension to arbitrary quasi-uniform direction sets established in this work, we construct tensor-product-type parameter sets that attain the sharp rate $$|f-f_n|<em>{L^{2(Ω)}\lesssim} n^{-\frac rd}|f|</em>{H^r(Ω)},\quad f\in H<sup>r(Ω)$$ for all $r&gt;0$. In contrast to the finite-difference construction in [Neural Comput. 8 (1996), pp. 164-177], whose explicit admissibility condition may require an extremely small parameter scale, the proposed parameter sets remain distributed over fixed intervals and are therefore more amenable to practical computation.