  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.
