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Main Title: Curve based approximation of measures on manifolds by discrepancy minimization
Author(s): Ehler, Martin
Gräf, Manuel
Neumayer, Sebastian
Steidl, Gabriele
Type: Article
Is Part Of: 10.14279/depositonce-11298
Language Code: en
Abstract: The approximation of probability measures on compact metric spaces and in particular on Riemannian manifolds by 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 unit interval by such curves. Using the discrepancy as distance between measures, we prove optimal approximation rates in terms of the curve’s length and Lipschitz constant. 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 unit interval by Lipschitz curves. We present numerical examples for measures on the 2- and 3-dimensional torus, the 2-sphere, the rotation group on R3 and the Grassmannian of all 2-dimensional linear subspaces of R4. Our algorithm of choice is a conjugate gradient method on these manifolds, which incorporates second-order information. For efficient gradient and Hessian evaluations within the algorithm, we approximate the given measures by truncated Fourier series and use fast Fourier transform techniques on these manifolds.
Issue Date: 11-Feb-2021
Date Available: 14-Sep-2021
DDC Class: 510 Mathematik
Subject(s): approximation of measures
fourier methods
non-convex optimization
quadrature rules
sampling theory
Sponsor/Funder: TU Berlin, Open-Access-Mittel – 2021
Journal Title: Foundations of Computational Mathematics
Publisher: Springer Nature
Publisher Place: Heidelberg
Publisher DOI: 10.1007/s10208-021-09491-2
EISSN: 1615-3383
ISSN: 1615-3375
Appears in Collections:FG Angewandte Mathematik » Publications

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