Geometric methods on low-rank matrix and tensor manifolds

  • In this chapter we present numerical methods for low-rank matrix and tensor problems that explicitly make use of the geometry of rank constrained matrix and tensor spaces. We focus on two types of problems: The first are optimization problems, like matrix and tensor completion, solving linear systems and eigenvalue problems. Such problems can be solved by numerical optimization for manifolds, called Riemannian optimization methods. We will explain the basic elements of differential geometry in order to apply such methods efficiently to rank constrained matrix and tensor spaces. The second type of problem is ordinary differential equations, defined on matrix and tensor spaces. We show how their solution can be approximated by the dynamical low-rank principle, and discuss several numerical integrators that rely in an essential way on geometric properties that are characteristic to sets of low rank matrices and tensors.

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Author:André UschmajewGND, Bart Vandereycken
Frontdoor URL
Parent Title (English):Handbook of variational methods for nonlinear geometric data
Place of publication:Cham
Editor:Philipp Grohs, Martin Holler, Andreas Weinmann
Type:Part of a Book
Year of first Publication:2020
Publishing Institution:Universität Augsburg
Release Date:2023/03/23
First Page:261
Last Page:313
Institutes:Mathematisch-Naturwissenschaftlich-Technische Fakultät
Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik
Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik / Lehrstuhl für Mathematical Data Science
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik
Licence (German):CC-BY 4.0: Creative Commons: Namensnennung (mit Print on Demand)