Curve Based Approximation of Measures on Manifolds by Discrepancy Minimization

TL;DR

This paper introduces a method for approximating probability measures on manifolds using Lipschitz curves by minimizing discrepancy, providing theoretical convergence rates and numerical algorithms with applications to various geometric spaces.

Contribution

It establishes optimal approximation rates for measures on manifolds via Lipschitz curves and develops a conjugate gradient algorithm utilizing Fourier techniques for practical computation.

Findings

Achieved optimal approximation rates in terms of Lipschitz constants.

Developed an efficient conjugate gradient algorithm on manifolds.

Demonstrated numerical effectiveness on complex geometric spaces.

Abstract

The approximation of probability measures on compact metric spaces and in particular on Riemannian manifoldsby atomic or empirical ones is a classical task in approximation and complexity theory with a wide range of applications. Instead of point measures we are concerned with the approximation by measures supported on Lipschitz curves. Special attention is paid to push-forward measures of Lebesgue measures on the interval by such curves. Using the discrepancy as distance between measures, we prove optimal approximation rates in terms of Lipschitz constants of curves. Having established the theoretical convergence rates, we are interested in the numerical minimization of the discrepancy between a given probability measure and the set of push-forward measures of Lebesgue measures on the interval by Lipschitz curves. We present numerical examples for measures on the 2- and 3-dimensional torus, the 2-sphere, the rotation group on $\mathbb R^3$ and the Grassmannian of all 2-dimensional linear subspaces of $\mathbb{R}^4$. Our algorithm of choice is a conjugate gradient method on these manifolds which incorporates second-oder information. For efficiently computing the gradients and the Hessians within the algorithm, we approximate the given measures by truncated Fourier series and use fast Fourier transform techniques on these manifolds.