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

What the math gives to ML

The paper provides a constructive alternative to learning all parameters of a shallow network: prescribe inner weights and biases using quasi-Chebyshev parameter sets, then fit only the output coefficients. Its key transferable asset is a deterministic fixed-feature space with a Sobolev approximation guarantee, avoiding the dimension-independent but often slow Monte Carlo behavior of generic random features. The most practical adaptation is a frozen analytic-activation feature layer whose parameters are generated from Chebyshev nodes and quasi-uniform ridge directions, followed by ridge regression or a trainable linear readout. This is especially attractive for repeated tasks on the same domain, where the feature matrix or its factorization can be reused.