Hard thresholding hyperinterpolation over general regions

TL;DR

It is proved that hard thresholding hyperinterpolation is the unique solution to an ℓ0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek}-regularized weighted discrete least squares approximation problem.

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.

Figures (9)

• Figure 1: The geometric interpretation of hyperinterpolation and hard thresholding hyperinterpolation satisfying the Pythagorean theorem with respect to the discrete (semi) inner product \ref{['equ:semiinner']}.
• Figure 2: Approximate $f(\mathbf{x}) ={\frac{1}{3}} \sum_{i=1}^6 \Phi_2(\| {\mathbf{z}}_i - {\mathbf{x}}\|_2)$, perturbed by impulse noise ($a=0.02$) and Gaussian noise ($\sigma=0.02$), over the unit-sphere ${\mathbb{S}}^2$, via hyperinterpolation $\mathcal{L}_n f^{\epsilon}$, Lasso hyperinterpolation $\mathcal{L}_{n}^{\lambda}f^{\epsilon}$, and hard thresholding hyperinterpolation $\mathcal{H}_{n}^{\lambda} f^{\epsilon}$ at $n=15$.
• Figure 3: The choices of regularization parameter $\lambda$ for Lasso hyperinterpolation $\mathcal{L}_n^{\lambda} f^{\epsilon}$ and hard thresholding hyperinterpolation $\mathcal{H}_n^{\lambda} f^{\epsilon}$ at $n=15$ approximating $f(\mathbf{x}) ={\frac{1}{3}} \sum_{i=1}^6 \Phi_2(\| {\mathbf{z}}_i - {\mathbf{x}}\|_2)$, perturbed by impulse noise ($a=0.02$) and Gaussian noise ($\sigma=0.02$), over the unit-sphere ${\mathbb{S}}^2$.
• Figure 4: The domain $\mathcal{T}=\overset{\frown}{ABC}$ with vertices $A=[1,0,0]^{{\rm{T}}}$, $B=[0,1,0]^{{\rm{T}}},C=[0,0,1]^{{\rm{T}}}$ in which we perform our tests. We represent in black the $3N_{2n+m}=4620$ nodes of the cubature rule with $n=10$ and $m=34$.
• Figure 5: Quadrature nodes on a spherical triangle rotated with centroid at the north pole, completely contained in a hemisphere and lifted from the projected elliptical triangle, before compression.

Theorems & Definitions (18)

• Lemma 2.1: sloan1995hyperinterpolation
• Remark 2.1
• Definition 2.1: an2021lasso
• Definition 2.2: donoho1994ideal
• Definition 2.3
• Remark 2.2
• Theorem 2.1
• Theorem 3.1
• Remark 3.1
• Lemma 3.1