Convexity-like Structures in Linear Algebra and Applications in Algorithmic Computation

Diplôme Thèse Convexity-like Structures in Linear Algebra and Applications in Algorithmic Computation eng 2025 2025 author Alimisis Foivos 954998 Section de mathématiques (MATH) director Vandereycken Bart 829601 Section de mathématiques (MATH) 10.13097/archive-ouverte/unige:183353 urn:nbn:ch:unige-1833535 Sc. 5886 This thesis primarily addresses the following problem: why can certain nonconvex optimization problems be solved efficiently? We focus on two important problems in linear algebra and uncover a convexity-like structure that may answer the aforementioned question. This convexity-like structure does not hold in a Euclidean space but rather “geodesically” on a Riemannian manifold. Thus, the main components of this thesis are linear algebra, optimization, and differential geometry. Since it is challenging for a reader to be familiar with all these prerequisites, we strive to present each topic as independently as possible. The structure analyzed in this thesis facilitates numerous applications in the field of numerical linear algebra. Consequently, a significant portion of the thesis is dedicated to these applications. They include eigenvalue problems in a distributed setting, a new analysis for preconditioned eigenvalue solvers, as well as the development and analysis of new eigenvalue solvers for the most elementary cases. This should certainly be of interest to readers with a background in numerical linear algebra. On the other hand, readers with an optimization background may view this thesis as an illustration of the importance of the aforementioned structure (weak-quasi-strong convexity). Insights into such structures have appeared not only in linear algebra but also in deep learning. In the final section, we show that this structure is indeed special for optimization in general, as it is in some sense necessary for linear convergence of gradient descent with respect to the error defined by the distances of iterates to the set of optima. Readers interested in differential geometry may see this thesis as a good example of the power of optimization on Riemannian manifolds. Although differential geometry is not the central focus of this thesis, all the structures discussed hold over curved spaces. This serves as an excellent example of the primary goal of optimization on Riemannian manifolds: solving typically non-convex problems within geometries where they satisfy a certain convexity structure. Convexity Weak-quasi-convexity Symmetric eigenvalue problem Polar decomposition Optimization Dewey 510 Swiss National Science Foundation Analysis and implementation of low-rank tensor algorithms 192363 CH Projects 25 Section de mathématiques 11 false c7cb7770-2ad5-4ef3-9dfa-fa17455622c9 Thesis Public phd_thesis_Foivos_Alimisis.pdf 4806200 application/pdf 9f50ba5d4602a0089681e72dd95bf2f02eaf977717ffae58ee31ab6b1cbe4e5a 0198faba-7142-48dc-92ad-ea3258585763 Imprimatur Public imprimatur.pdf 217110 application/pdf e9fb48fccaa34aeddcb9443bf82721b26cea28d0768196bedd48fef0d762d477 openAccess Faculté des sciences / Section de mathématiques

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