Essential rate for approximation by spherical neural networks

Authors:
Shaobo Lin;Feilong Cao;Zongben Xu
Affiliations:
Institute for Information and System Sciences, Xi'an Jiaotong University, Xi'an 710049, Shannxi Province, PR China;Institute of Metrology and Computational Science, China Jiliang University, Hangzhou 310018, Zhejiang Province, PR China;Institute for Information and System Sciences, Xi'an Jiaotong University, Xi'an 710049, Shannxi Province, PR China
Venue:
Neural Networks
Year:
2011

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Abstract

We consider the optimal rate of approximation by single hidden feed-forward neural networks on the unit sphere. It is proved that there exists a neural network with n neurons, and an analytic, strictly increasing, sigmoidal activation function such that the deviation of a Sobolev class W"2"r^2(S^d) from the class of neural networks @F"n^@f, behaves asymptotically as n^-^2^r^d^-^1. Namely, we prove that the essential rate of approximation by spherical neural networks is n^-^2^r^d^-^1.