Radial Basis Function Approximation with Distributively Stored Data on Spheres

This paper proposes a distributed weighted regularized least squares algorithm (DWRLS) with radial basis functions to tackle spherical data that are stored across numerous local servers and cannot be shared with each other. Via developing a novel integral operator approach based on spherical quadrature rules, we succeed in deriving optimal approximation rates for DWRLS and theoretically demonstrate that DWRLS performs similarly as running a weighted regularized least squares algorithm on the whole data stored on a large enough machine. This interesting finding implies that distributed learning is capable of sufficiently exploiting potential values of distributively stored spherical data, even though local servers cannot access the whole data.