Radial Basis Function Approximation with Distributively Stored Data on Spheres

Noname manuscript No. (will be inserted by the editor) Abstract This paper proposes a distributed weighted regularized lea st squares algorithm (DWRLS) with radial basis functions to tackle spherical data that are stored acr oss numerous local servers and cannot be shared with each other. Via developing a novel integral operator approa ch based on spherical quadrature rules, we succeed in deriving optimal approximation rates for DWRLS and theor etically demonstrate that DWRLS performs similarly as running a weighted regularized least squares algo rithm 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 thoug h local servers cannot access the whole data. Keywords Distributed learning Scattered data approximation Sphere Integral operator 1 Introduction In geophysics, solar system, climate prediction, environm ent governance and meteorology, and image rendering, samples formed as input-output pairs are collected over spheres [13, 15, 49], such as the surface of the earth and the direction of radiance. Due to the storage bo ttleneck and data privacy, these spherical data are often distributively stored across numerous computati onal servers.