Quasi Monte Carlo integration and kernel-based function approximation on Grassmannians

TL;DR

This paper develops and tests numerical methods for integration and function approximation on Grassmannian manifolds, extending techniques traditionally used on spheres, and confirms their effectiveness through experiments.

Contribution

It derives feasible expressions for approximation schemes on Grassmannians and validates them with numerical experiments matching theoretical expectations.

Findings

Numerical schemes for Grassmannians are feasible and effective.

Experimental results align with theoretical predictions.

Extension of sphere-based methods to Grassmannian manifolds.

Abstract

Numerical integration and function approximation on compact Riemannian manifolds based on eigenfunctions of the Laplace-Beltrami operator have been widely studied in the recent literature. The standard example in numerical experiments is the Euclidean sphere. Here, we derive numerically feasible expressions for the approximation schemes on the Grassmannian manifold, and we present the associated numerical experiments on the Grassmannian. Indeed, our experiments illustrate and match the corresponding theoretical results in the literature.