Optimal Sobolev Approximation by Deterministic and Random Shallow Sigmoidal Networks

What the math gives to ML

The paper gives an explicit Sobolev approximation law for shallow ridge-feature dictionaries: with M features, the error in H^m for an H^k target decays as M^{-(k-m)/d}, matching the Kolmogorov-width-optimal exponent. The transferable asset is that hidden parameters can be fixed in advance while only output coefficients are trained, and i.i.d. feature sampling preserves the exponent with only logarithmic oversampling. This suggests frozen random-feature or deterministic-feature front ends for smooth low-dimensional regression, scientific surrogate models, and PINNs, where derivatives of the network output are part of the loss.