Hard thresholding hyperinterpolation over general regions

TL;DR

This paper introduces hard thresholding hyperinterpolation, a new approximation method that enhances function approximation and denoising over general regions using a novel coefficient filtering technique.

Contribution

The paper develops and analyzes a new hyperinterpolation variant employing hard thresholding, with proven uniqueness, algebraic properties, and practical denoising capabilities.

Findings

Proves the uniqueness of hard thresholding hyperinterpolation solution.

Demonstrates denoising and basis selection abilities similar to Lasso hyperinterpolation.

Shows numerical effectiveness on sphere, spherical triangle, and cube.

Abstract

This paper proposes a novel variant of hyperinterpolation, called hard thresholding hyperinterpolation. This approximation scheme of degree $n$ leverages a hard thresholding operator to filter all hyperinterpolation coefficients, which approximate the Fourier coefficients of a continuous function by a quadrature rule with algebraic exactness $2n$. We prove that hard thresholding hyperinterpolation is the unique solution to an $\ell_0$-regularized weighted discrete least squares approximation problem. Hard thresholding hyperinterpolation is not only idempotent and commutative with hyperinterpolation, but also adheres to the Pythagorean theorem in terms of the discrete (semi) inner product. By the estimate of the reciprocal of Christoffel function, we present the upper bound of the uniform norm of hard thresholding hyperinterpolation operator. Additionally, hard thresholding hyperinterpolation possesses denoising and basis selection abilities akin to Lasso hyperinterpolation. To judge the $L_2$ errors of both hard thresholding and Lasso hyperinterpolations, we propose a criterion that integrates the regularization parameter with the product of noise coefficients and the signs of hyperinterpolation coefficients. Numerical examples on the sphere, spherical triangle and the cube demonstrate the denoising ability of hard thresholding hyperinterpolation.