Essential rate for approximation by spherical neural networks

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 _2r(Sd) from the class of neural networks Φn ^Φ, behaves asymptotically as n ^-2rd-1. Namely, we prove that the essential rate of approximation by spherical neural networks is n ^-2rd-1.