[Paper Review] Function recovery on manifolds using scattered data

David Krieg, Mathias Sonnleitner|arXiv (Cornell University)|Sep 9, 2021

Numerical methods in inverse problems4 citations

TL;DR

This paper establishes a sharp characterization of the quality of scattered sampling points on compact Riemannian manifolds for Sobolev function recovery, showing that the worst-case recovery error is governed by the $L_\gamma(M)$-average of geodesic distances to the point set, with $\gamma = d/s$ for $s > d/p$. It proves that random sampling is asymptotically as effective as optimal sampling when $\gamma < \infty$, closing a logarithmic gap in cubature formula efficiency.

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 $γ&lt;\infty$. 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].

Motivation & Objective

• To characterize the worst-case error in recovering Sobolev functions on compact Riemannian manifolds from scattered data.
• To determine the dependence of recovery error on the distribution of sampling points, especially their spatial spread.
• To establish when random sampling is asymptotically as effective as optimally chosen sampling points.
• To close a logarithmic gap in the efficiency of random cubature formulas compared to optimal ones.

Proposed method

• Uses Bessel potential (Sobolev) spaces $H^s_p(M)$ on compact Riemannian manifolds to model function regularity.
• Defines the worst-case recovery error in $L_q(M)$-norm for both general and linear reconstruction maps.
• Introduces a geometric quality measure based on the $L_\gamma(M)$-average of geodesic distances from points on $M$ to the sampling set $P$.
• Proves that the recovery error is equivalent (up to constants) to this $L_\gamma$-average of distances, with $\gamma = d/s$.
• Analyzes moments of the average distance for i.i.d. uniform sampling points using probabilistic and geometric arguments.
• Applies the dominated convergence theorem and Lebesgue differentiation to show convergence of moment generating functions, proving asymptotic equivalence of random and optimal sampling.

Experimental results

Research questions

• RQ1 What geometric property of a scattered point set determines the worst-case error in Sobolev function recovery on a compact Riemannian manifold?
• RQ2 For which values of $\gamma$ is random sampling asymptotically as effective as optimal sampling in function recovery?
• RQ3 Can the error of cubature formulas with random nodes match that of optimal formulas when weights are chosen appropriately?
• RQ4 How does the $L_\gamma$-average of geodesic distances relate to the approximation power of scattered data?

Key findings

• The worst-case recovery error for functions in $H^s_p(M)$ is equivalent to the $L_\gamma(M)$-average of geodesic distances to the sampling set, with $\gamma = d/s$.
• When $\gamma < \infty$, random sampling is asymptotically as good as optimal sampling for function recovery.
• Cubature formulas with random nodes achieve the same asymptotic error rate as optimal cubature formulas when weights are chosen correctly.
• The limit theorem for moments of average distance confirms that random point sets achieve optimal asymptotic performance in the $\gamma < \infty$ regime.
• The result extends prior findings on convex domains to general compact Riemannian manifolds, including spheres and other curved spaces.
• The equivalence $e(P, H^s_p(M), L_q(M)) \asymp \| \operatorname{dist}_M(\cdot, P) \|_{L_\gamma(M)}$ holds uniformly across all finite point sets $P$.