Paper Info
Title
Weighted quadrature formulas and approximation by zonal function networks on the sphere
Abstract
Let q ≥ 1 be an integer, Sq be the unit sphere embedded in the Euclidean space Rq+1. A zonal function (ZF) network with an activation function φ : [-1, 1] → R and n neurons is a function on Sq of the form x ↦ Σk=1n akφ(x.ξk), where ak's are real numbers, ξk's are points on Sq. We consider the activation functions φ for which the coefficients {φ(l)} in the appropriate ultraspherical polynomial expansion decay as a power of (l + 1)-1. We construct ZF networks to approximate functions in the Sobolev classes on the unit sphere embedded in a Euclidean space, yielding an optimal order of decay for the degree of approximation in terms of n, compared with the nonlinear n-widths of these classes. Our networks do not require training in the traditional sense. Instead, the network approximating a function is given explicitly as the value of a linear operator at that function. In the case of uniform approximation, our construction utilizes values of the target function at scattered sites. The approximation bounds are used to obtain error bounds on a very general class of quadrature formulas that are exact for the integration of high degree polynomials with respect to a weighted integral. The bounds are better than those expected from a straightforward application of the Sobolev embeddings.
Year
DOI
Venue
2006
10.1016/j.jco.2005.10.003
J. Complexity
Keywords
DocType
Volume
Quadrature formulas,Approximation on the sphere,Neural networks,Spherical basis functions,approximation bound,approximate function,zonal function network,Euclidean space,Sobolev class,target function,activation function,zonal function,Weighted quadrature formula,Euclidean space R,uniform approximation,Sobolev embeddings,Radial basis functions,unit sphere
Journal
22
Issue
ISSN
Citations 
3
Journal of Complexity
15
PageRank 
References 
Authors
1.22
13
1
Name
Order
Citations
PageRank
Hrushikesh Narhar Mhaskar125761.07