Lasso hyperinterpolation over general regions

TL;DR

This paper introduces Lasso hyperinterpolation, a discrete polynomial approximation method that employs $\,l_1$-regularization and soft thresholding to effectively handle noisy data over general regions, improving robustness compared to traditional hyperinterpolation.

Contribution

The paper develops a novel Lasso hyperinterpolation method that combines hyperinterpolation with $\,l_1$-regularization and provides theoretical analysis and numerical examples over various regions.

Findings

Operator norm is bounded independently of polynomial degree.

$L_2$ error bound is smaller than hyperinterpolation under high noise.

Method is effective over intervals, discs, spheres, and cubes.

Abstract

This paper develops a fully discrete soft thresholding polynomial approximation over a general region, named Lasso hyperinterpolation. This approximation is an $\ell_1$-regularized discrete least squares approximation under the same conditions of hyperinterpolation. Lasso hyperinterpolation also uses a high-order quadrature rule to approximate the Fourier coefficients of a given continuous function with respect to some orthonormal basis, and then it obtains its coefficients by acting a soft threshold operator on all approximated Fourier coefficients. Lasso hyperinterpolation is not a discrete orthogonal projection, but it is an efficient tool to deal with noisy data. We theoretically analyze Lasso hyperinterpolation for continuous and smooth functions. The principal results are twofold: the norm of the Lasso hyperinterpolation operator is bounded independently of the polynomial degree, which is inherited from hyperinterpolation; and the $L_2$ error bound of Lasso hyperinterpolation is less than that of hyperinterpolation when the level of noise becomes large, which improves the robustness of hyperinterpolation. Explicit constructions and corresponding numerical examples of Lasso hyperinterpolation over intervals, discs, spheres, and cubes are given.