Riemannian Algorithms on the Stiefel and the Fixed-Rank Manifold

Diplôme Thèse Riemannian Algorithms on the Stiefel and the Fixed-Rank Manifold eng 2020-11-24 author Sutti Marco 849207 director Vandereycken Bart 829601 10.13097/archive-ouverte/unige:146438 urn:nbn:ch:unige-1464385 Sc. 5514 Several applications in optimization, image, and signal processing deal with data belonging to matrix manifolds. These are manifolds in the sense of classical Riemannian geometry, where variables are matrices. This thesis is divided into four main parts, and in all of them, we make extensive use of Riemannian geometry. In the first part, we deal with the problem of finding the distance between two points on the Stiefel manifold. We describe and specialize shooting methods to the Stiefel manifold, discuss their limitations, and provide some numerical examples. In the second part, we study another method for finding geodesics: the leapfrog algorithm introduced by L. Noakes. We propose a convergence proof of leapfrog as a nonlinear Gauss-Seidel method. In the third part, we give an overview of the Hager-Zhang line search, and we generalize this technique to Riemannian manifolds. In the final part, we use the manifold of fixed-rank matrices in the context of certain large-scale variational problems arising from the discretization of elliptic PDEs, where the optimization variable is rank-constrained. Riemannian manifolds Geodesics Stiefel manifold Nonlinear Gauss–Seidel Shooting methods Low-rank matrices Optimization on manifolds Multilevel optimization Retraction-based optimization Line search Roundoff error Dewey 510 Swiss National Science Foundation 163212 CH 25 Section de mathématiques 11 false 847 Analyse numérique 15d6d894-9757-4413-a197-d0d41cf05f7f Thesis Public Marco_Sutti_these_20201210.pdf 4503764 application/pdf openAccess Faculté des sciences / Section de mathématiques

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Sutti, Marco