Lasso hyperinterpolation over general regions

  title={Lasso hyperinterpolation over general regions},
  author={Congpei An and Hao-Ning Wu},
  journal={SIAM J. Sci. Comput.},
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… 

Figures and Tables from this paper

On the quadrature exactness in hyperinterpolation
This paper investigates the role of quadrature exactness in the approximation scheme of hyperinterpolation. Constructing a hyperinterpolant of degree n requires an m-point positive-weight quadrature


Polynomial interpolation and hyperinterpolation over general regions
  • I. Sloan
  • Mathematics, Computer Science
  • 1995
Abstract This paper studies a generalization of polynomial interpolation: given a continuous function over a rather general manifold, hyperinterpolation is a linear approximation that makes use of
Filtered hyperinterpolation: a constructive polynomial approximation on the sphere
This paper considers a fully discrete filtered polynomial approximation on the unit sphere $${\mathbb{S}^{d}.}$$ For $${f \in C(\mathbb{S}^{d}),V_{L,N}^{(a)} \, f}$$ is a polynomial approximation
The uniform norm of hyperinterpolation on the unit sphere in an arbitrary number of dimensions
In this paper we study the order of growth of the uniform norm of the hyperinterpolation operator on the unit sphere Sr−1 ⊂ Rr. The hyperinterpolation approximation Lnƒ, where ƒ ∈ C(Sr−1), is derived
Regularized Least Squares Approximations on the Sphere Using Spherical Designs
This work considers polynomial approximation on the unit sphere by a class of regularized discrete least squares methods with novel choices for the regularization operator and the point sets of the discretization.
Parameter Choice Strategies for Least-squares Approximation of Noisy Smooth Functions on the Sphere
A polynomial reconstruction of smooth functions from their noisy values at discrete nodes on the unit sphere by a variant of the regularized least-squares method, and a priori and a posteriori strategies for choosing these parameters are discussed.
Polynomial approximation on spheres - generalizing de la Vallée-Poussin
  • I. Sloan
  • Mathematics
    Comput. Methods Appl. Math.
  • 2011
An explicit generalization of the de la Vallée-Poussin construction to higher dimensional spheres S^d ≤ R^{d+1}.
Localized Linear Polynomial Operators and Quadrature Formulas on the Sphere
The purpose of this paper is to construct universal, auto-adaptive, localized, linear, polynomial (-valued) operators based on scattered data on the (hyper) sphere $q\ge2$ and to construct formulas exact for spherical polynomials of degree 178.
Local RBF-based penalized least-squares approximation on the sphere with noisy scattered data
How good can polynomial interpolation on the sphere be?
The quality of polynomial interpolation approximations over the sphere Sr−1⊂Rr in the uniform norm is explored, principally for r=3, and empirical evidence suggests that for points obtained by maximizing λmin , the growth in ‖Λn‖ is approximately n+1 for n<30.
Hyperinterpolation on the square