The Research On Spheircal Learning Algorithms And Their Rates

During recent decades, it has become more and more popular to study func-tion approximation and data mining on spheres. And it has attracted much attention ofmany researchers in the world for its remarkable and practical applications. Actually,the main reasons for this are as follows. On one hand, the sphere is a compact Rieman-nian manifold, which owns some good properties such as symmetry and homogeneousnature. On the other hand, the studied results for the sphere can be used in Meteorol-ogy, Oceanography, Geoscience and Geoengineering in general. Hence the researchfor the sphere owns the obvious practical values. In this paper, we mainly discuss thefollowing three questions.First of all, we study the approximating and learning by spherical Lipschitz kernelon the sphere. Comparing with linear approximation on the sphere, we deduce a betterconvergence rate for approximation by shifts of Lipschitz kernel on the sphere by meansof a method of nonlinear approximation on the sphere, which is faster than O(n1/2),where n is the number of parameters needed in the approximation. By using the ap-proximation, we also deduce a learning rate of regularized least square algorithm withthe Lipschitz kernel on the sphere. The obtained results showed that spherical Lipschitzkernel possesses better properties of approximation and faster rate of learning.Secondly, we give a theoretical analysis of the performance of the regularizationleast-square algorithm on a reproducing kernel Hilbert space associated with sphericalharmonics kernels on the unit sphere. A spherical operator, called the best approxi-mation operator, is used to estimate the upper bound of generalization error, and thelower bound of the error is estimated by using the entropy number of the set includingthe regression function. The obtained estimations are indeed optimal over a suitableclass of priors defined by the considered kernel. We can obtain the best learning rateby spherical regularized least-square algorithm under some assumptions.Finally, by the probability inequalities and somewhat standard in learning theory,we consider convergence rate by moving lest square on a reproducing Hilbert space onthe sphere when regression function belonging to the Lipschitz class.