Shallow neural network approximation in mixed Sobolev spaces

arXiv:2609.05263v1 Announce Type: cross Abstract: We investigate the best L_2 approximation of mixed Sobolev spaces by shallow neural networks with n neurons and general activation functions. We first establish an activation-independent Fourier-block principle: if an activation has univariate approximation order rho in the sense of the Fourier-block property, then the global approximation rate has algebraic order min{alpha,rho} for target functions of mixed smoothness alpha, up to explicit logarithmic factors. To verify this property for concrete activations, we introduce a structured univariate approximation condition that implies the Fourier-block property with explicit parameters. For ReLU^k, a matching algebraic lower bound identifies min{alpha,k+1} as the optimal algebraic approximation exponent in any dimension, up to logarithmic factors in the upper bound. The framework also yields the exponent min{alpha,k+1} for cardinal B-splines and soft-ReLU^k, and the full mixed-smoothness exponent alpha for ELU and cosine activations, again up to logarithmic~factors.

Source: arXiv cs.LG | 2026-09-07