Non-asymptotic Analysis of Matérn Regression: The Roles of Target and Kernel Lengthscales

What the math gives to ML

The paper provides a useful two-scale view of approximation error: the sampling design must resolve both the target signal scale and the model's kernel scale, while noise requires enough observations within the relevant scale volume. This suggests a concrete neural-network design rule for sampled spatial, temporal, or scientific data: select input token or patch spacing and receptive-field scales from estimated target-scale resolution and information, rather than treating patch size as a fixed hyperparameter. The most promising transfer is an adaptive multiscale front end that prevents aliasing of fine target structure while avoiding unnecessarily short learned kernels that cannot be supported by the sampling density.