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The capability of approximation for neural networks interpolant on the sphere
Author(s) -
Cao Feilong,
Lin Shaobo
Publication year - 2010
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.1373
Subject(s) - unit sphere , mathematics , radial basis function , sobolev space , hyperplane , euclidean space , spherical mean , artificial neural network , function (biology) , space (punctuation) , mathematical analysis , euclidean geometry , function space , basis function , function approximation , planar , radial basis function network , geometry , computer science , artificial intelligence , computer graphics (images) , evolutionary biology , biology , operating system
Compared with planar hyperplane, fitting data on the sphere has been an important and active issue in geoscience, metrology, brain imaging, and so on. In this paper, using a functional approach, we rigorously prove that for given distinct samples on the unit sphere there exists a feed‐forward neural network with single hidden layer which can interpolate the samples, and simultaneously near best approximate the target function in continuous function space. Also, by using the relation between spherical positive definite radial basis functions and the basis function on the Euclidean space ℝ d + 1 , a similar result in a spherical Sobolev space is established. Copyright © 2010 John Wiley & Sons, Ltd.