Distributed Learning via Filtered Hyperinterpolation on Manifolds

TL;DR

This paper introduces a distributed filtered hyperinterpolation method for learning functions on manifolds, providing theoretical guarantees on approximation quality and convergence rates, suitable for large-scale data processing in various scientific fields.

Contribution

It proposes a parallel data processing framework for manifold learning with filtered hyperinterpolation, including theoretical analysis of approximation rates and optimality in non-distributed settings.

Findings

Distributed approach effectively handles large datasets.

Theoretical bounds relate approximation quality to data and model parameters.

Non-distributed method achieves optimal approximation order.

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, 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.