Sobolev Algorithm for Local Smoothness Analysis (SALSA) via Sharp Direct and Inverse Statements

TL;DR

This paper introduces SALSA, a new algorithm for analyzing local smoothness of data using sharp approximation statements for kernel methods, extending existing theory to broader schemes.

Contribution

It develops a Sobolev-based algorithm for local smoothness analysis grounded in sharp approximation statements, applicable beyond interpolation.

Findings

SALSA effectively detects local smoothness properties in data.

Numerical experiments demonstrate the algorithm's robustness and accuracy.

Theoretical extension to broader approximation schemes enhances existing kernel methods.

Abstract

We extend sharp direct and inverse approximation statements for kernel-based methods for finitely smooth kernels, i.e. those whose native spaces are norm-equivalent to Sobolev spaces. In particular, our inverse results are now formulated for a broad class of approximation schemes beyond interpolation, extending existing theory. Building on these results, we propose a novel Sobolev Algorithm for Local Smoothness Analysis (SALSA) for detecting local smoothness properties of target data, including their degree of smoothness and non-smoothness. The method is rigorously grounded based on the sharp direct and inverse statements. Numerical experiments in various settings highlight the effectiveness of the proposed algorithm.