Function recovery on manifolds using scattered data

TL;DR

It is proved that the quality of the sample is given by the Lγ(M)-average of the geodesic distance to the point set and the value of γ ∈ (0, ∞) is determined, which extends the findings on bounded convex domains and yields that cubature formulas with random nodes are asymptotically as good as optimal cubatures if the weights are chosen correctly.

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 $γ\in (0,\infty]$. This extends our findings on bounded convex domains [IMA J. Numer. Anal., 44:1346--1371, 2024]. As a byproduct, we prove the optimal rate of convergence of the $n$-th minimal worst case error for $L_q(M)$-approximation for all $1\le q \le \infty$. 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 $γ<\infty$. 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].

Theorems & Definitions (19)

• Theorem 1
• Theorem 2: KS20
• Corollary 1
• Proposition 1
• Corollary 2
• Corollary 3
• Lemma 1
• Lemma 2
• proof
• Proposition 2