Radial Basis Function Approximation with Distributively Stored Data on Spheres

TL;DR

This paper introduces a distributed radial basis function approximation method for spherical data stored across multiple servers, achieving near-centralized performance through a novel integral operator approach.

Contribution

It develops a new distributed weighted regularized least squares algorithm using spherical basis functions and quadrature rules, with proven optimal approximation rates.

Findings

DWRLS performs similarly to centralized algorithms on large data sets.

The method effectively exploits distributed spherical data without data sharing.

Optimal approximation rates are theoretically established.

Abstract

This paper proposes a distributed weighted regularized least squares algorithm (DWRLS) based on spherical radial basis functions and spherical quadrature rules 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, we succeed in deriving optimal approximation rates for DWRLS and theoretically demonstrate that DWRLS performs similarly as running a weighted regularized least squares algorithm with the whole data 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 every local server cannot access all the data.