Hybrid hyperinterpolation over general regions

TL;DR

Hybrid hyperinterpolation addresses noisy data by uniting filtered hyperinterpolation and Lasso hyperinterpolation into an $\ell^2_2+\ell^1$-regularized discrete least-squares framework. It constructs the estimator $\mathscr{H}_{L}^{\lambda} f$ through a soft-thresholded, filtered coefficient update, bridging coefficient shrinkage and basis selection. The paper provides $L^2$ error analysis, including stability and convergence results, and demonstrates via numerical experiments on the union of disks and the unit sphere that the method achieves favorable $L^2$ and $L_{\infty}$ errors with robust denoising, at a reasonable sparsity level. The results suggest practical applicability to nonstandard regions where explicit orthonormal bases are unavailable and cubature rules are used. The work highlights potential extensions to alternative penalties, filter designs, and broader region classes.

Abstract

We present an $\ell^2_2+\ell_1$-regularized discrete least squares approximation over general regions under assumptions of hyperinterpolation, named hybrid hyperinterpolation. Hybrid hyperinterpolation, using a soft thresholding operator and a filter function to shrink the Fourier coefficients approximated by a high-order quadrature rule of a given continuous function with respect to some orthonormal basis, is a combination of Lasso and filtered hyperinterpolations. Hybrid hyperinterpolation inherits features of them to deal with noisy data once the regularization parameter and the filter function are chosen well. We derive $L_2$ errors in theoretical analysis for hybrid hyperinterpolation to approximate continuous functions with noise data on sampling points. Numerical examples illustrate the theoretical results and show that well chosen regularization parameters can enhance the approximation quality over the unit-sphere and the union of disks.

Figures (5)

• Figure 1: The domain $\Omega=\Omega^{(r_1)} \cup \Omega^{(r_2)}$ in which we perform our tests. We represent in red the $N=496$ nodes of the cubature rule for algebraic degree of exactness=$30$.
• Figure 2: Approximate $f(x,y)=(1-(x^2+y^2)) \exp ( x \, \cos(y))$, perturbed by Gaussian noise ($\sigma=0.05$), over the union of disks, via hyperinterpolation $\mathscr{L}_L f^{\epsilon}$, filtered hyperinterpolation $\mathscr{F}_{L,N}f^{\epsilon}$, and hybrid hyperinterpolation $\mathscr{H}_{L}^{\lambda} f^{\epsilon}$ with $L=15$.
• Figure 3: The choices of regularization parameter $\lambda$ for hybrid hyperinterpolation $\mathscr{H}_L^{\lambda} f^{\epsilon}$ with $L=15$ approximating $f(x,y)=(1-(x^2+y^2)) \exp ( x \, \cos(y))$, perturbed by Gaussian noise ($\sigma=0.05$), over the union of disks.
• Figure 4: Approximate $f(x) ={\frac{1}{3}} \sum_{i=1}^6 \Phi_2(\| {\mathbf{z}}_i - {\mathbf{x}}\|_2)$, perturbed by impulse noise ($a=0.02$) and Gaussian noise ($\sigma=0.02$), over the unit-sphere ${\mathbb{S}}^2$, via hyperinterpolation $\mathscr{L}_L f^{\epsilon}$, filtered hyperinterpolation $\mathscr{F}_{L,N}f^{\epsilon}$, and hybrid hyperinterpolation $\mathscr{H}_{L}^{\lambda} f^{\epsilon}$ with $L=15$.
• Figure 5: The choices of regularization parameter $\lambda$ for hybrid hyperinterpolation $\mathscr{H}_L^{\lambda} f^{\epsilon}$ with $L=15$ approximating $f(x) ={\frac{1}{3}} \sum_{i=1}^6 \Phi_2(\| {\mathbf{z}}_i - {\mathbf{x}}\|_2)$, perturbed by impulse noise ($a=0.02$) and Gaussian noise ($\sigma=0.02$), over the unit-sphere ${\mathbb{S}}^2$.

Theorems & Definitions (12)

• Lemma 2.1: Lemma 5 in sloan1995polynomial
• Definition 2.1: sloan2012filtered
• Definition 2.2: donoho1994ideal
• Theorem 2.1: Theorem 3.4 in an2021lasso
• Definition 3.1: Hybrid hyperinterpolation
• Theorem 3.1
• Remark 3.1
• Lemma 4.1: Theorem 1 in sloan1995polynomial
• Lemma 4.2
• Remark 4.1