Function recovery on manifolds using scattered data

Abstract

We consider the task of recovering a Sobolev function on a connected compact Riemannian manifold M when given a sample on a finite point set. We prove that the quality of the sample is given by the L γ (M )-average of the geodesic distance to the point set and determine the value of γ ∈ (0, ∞]. This extends our findings on bounded convex domains [arXiv:2009[arXiv: .11275, 2020. Further, a limit theorem for moments of the average distance to a set consisting of i.i.d. uniform points is proven. This yields that a random sample is asymptotically as good as an optimal sample in precisely those cases with γ < ∞. In particular, we obtain that cubature formulas with random nodes are asymptotically as good as optimal cubature formulas if the weights are chosen correctly. This closes a logarithmic gap left open by Ehler, Gräf and Oates [Stat. Comput., 29:1203-1214, 2019.Disclaimer. This paper relies on our previous work [15] which is still in the revision process. The reader may therefore not yet find a complete proof of all the results that are presented in this paper. However, this only concerns the lower bound of Theorem 2 for smoothness s ∈ N and the lower bound of Corollary 1 in the case that the dimension d is even. See Remark 2 below.We are interested in the problem of recovering a real-valued function on a connected compact Riemannian manifold which is only known through a finite number of function values, obtained on a finite set of sampling points. We study this problem

On the power of iid information for linear approximation

“…In this section we take a closer look at L q -approximation in isotropic Sobolev spaces for which we have a characterization of the quality of (random) samples due to [70,71] which implies asymptotic optimality of n or n log n iid measurements depending on the parameters involved. There are also generalizations to similarly structured isotropic function spaces such as Holder, Triebel-Lizorkin or Besov spaces.…”

“…For simplicity, in the remainder of this section, the domain D will be a bounded convex domain (and in particular Lipschitz) and we refer to [71] for (almost) analogous results on manifolds. We will suppress D in the notation.…”