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

• 成果类型: Article
• 署名作者: Luo, Yuetian; Li, Xudong; Zhang, Anru R.
• 署名单位: University of Chicago; Fudan University; Duke University; Duke University; Duke University; Duke University
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• 发表日期: 2024
• 页码: 782-825
• 关键词: monotonic transformations; maximal monotonicity; Weak monotonicity; Operators

摘要

In this paper, we propose a general procedure for establishing the geometric land-scape 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 struc-ture on the landscape connection are discussed. We also observe an algorithmic connection between two geometries with some specific Riemannian metrics in fixed-rank matrix optimiza-tion: 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 uni-fied 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 con-nections of Riemannian optimization under different Riemannian geometries and provide a few new theoretical insights to unanswered questions in the literature