Regular Article: Computing Fourier Transforms and Convolutions on the 2-Sphere
Advances in Applied Mathematics
Fast algorithms for discrete polynomial transforms
Mathematics of Computation
Approximation by ridge functions and neural networks
SIAM Journal on Mathematical Analysis
Spherical Marcinkiewicz-Zygmund inequalities and positive quadrature
Mathematics of Computation
Kolmogorov width of classes of smooth functions on the sphere Sd-1
Journal of Complexity
On the tractability of multivariate integration and approximation by neural networks
Journal of Complexity
Polynomial operators and local smoothness classes on the unit interval
Journal of Approximation Theory
Optimal lower bounds for cubature error on the sphere S2
Journal of Complexity
Bounds on rates of variable-basis and neural-network approximation
IEEE Transactions on Information Theory
Quadrature in Besov spaces on the Euclidean sphere
Journal of Complexity
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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.