On Geometric Connections of Embedded and Quotient Geometries in Riemannian Fixed-Rank Matrix Optimization

Yuetian Luo (), Xudong Li () and Anru R. Zhang ()
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Yuetian Luo: Data Science Institute, University of Chicago, Chicago, Illinois 60637
Xudong Li: School of Data Science, Fudan University, Shanghai 200433, China
Anru R. Zhang: Departments of Biostatistics and Bioinformatics, Computer Science, Mathematics, and Statistical Science, Duke University, Durham, North Carolina 27701

Mathematics of Operations Research, 2024, vol. 49, issue 2, 782-825

Abstract: In this paper, we propose a general procedure for establishing the geometric landscape connections of a Riemannian optimization problem under the embedded and quotient geometries. By applying the general procedure to the fixed-rank positive semidefinite (PSD) and general matrix optimization, we establish an exact Riemannian gradient connection under two geometries at every point on the manifold and sandwich inequalities between the spectra of Riemannian Hessians at Riemannian first-order stationary points (FOSPs). These results immediately imply an equivalence on the sets of Riemannian FOSPs, Riemannian second-order stationary points (SOSPs), and strict saddles of fixed-rank matrix optimization under the embedded and the quotient geometries. To the best of our knowledge, this is the first geometric landscape connection between the embedded and the quotient geometries for fixed-rank matrix optimization, and it provides a concrete example of how these two geometries are connected in Riemannian optimization. In addition, the effects of the Riemannian metric and quotient structure on the landscape connection are discussed. We also observe an algorithmic connection between two geometries with some specific Riemannian metrics in fixed-rank matrix optimization: there is an equivalence between gradient flows under two geometries with shared spectra of Riemannian Hessians. A number of novel ideas and technical ingredients—including a unified treatment for different Riemannian metrics, novel metrics for the Stiefel manifold, and new horizontal space representations under quotient geometries—are developed to obtain our results. The results in this paper deepen our understanding of geometric and algorithmic connections of Riemannian optimization under different Riemannian geometries and provide a few new theoretical insights to unanswered questions in the literature.

Keywords: 58C05; 90C06; 90C22; Geometric landscape connection; algorithmic connection; Riemannian optimization; fixed-rank matrix optimization; embedded geometry; quotient geometry (search for similar items in EconPapers)
Date: 2024
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Persistent link: https://EconPapers.repec.org/RePEc:inm:ormoor:v:49:y:2024:i:2:p:782-825

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