Distributed Learning via Filtered Hyperinterpolation on Manifolds

Abstract

Learning mappings of data on manifolds is an important topic in contemporary machine learning, with applications in astrophysics, geophysics, statistical physics, medical diagnosis, biochemistry, and 3D object analysis. This paper studies the problem of learning real-valued functions on manifolds through filtered hyperinterpolation of input–output data pairs where the inputs may be sampled deterministically or at random and the outputs may be clean or noisy. Motivated by the problem of handling large data sets, it presents a parallel data processing approach which distributes the data-fitting task among multiple servers and synthesizes the fitted sub-models into a global estimator. We prove quantitative relations between the approximation quality of the learned function over the entire manifold, the type of target function, the number of servers, and the number and type of available samples. We obtain the approximation rates of convergence for distributed and non-distributed approaches. For the non-distributed case, the approximation order is optimal.

Is hyperinterpolation efficient in the approximation of singular and oscillatory functions?

“…We refer the reader to [17,19,21,29,32,36,37,42] for some follow-up works on the general analysis of hyperinterpolation and [2,22,28,38] for some variants of classical hyperinterpolation.…”

“…The fact that L n ∞ is not uniformly bounded has spurred the development of filtered hyperinterpolation on the sphere and then on general regions [22,28,38]. The filtered hyperinterpolation operator, as an operator from C(Ω) → C(Ω), has a uniformly bounded norm.…”