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.

翻译：本文提出了一般性流程，用于建立黎曼优化问题在嵌入几何与商几何下的几何景观关联。将该流程应用于固定秩半正定矩阵与一般矩阵优化问题后，我们在流形上的每一点建立了两种几何下黎曼梯度的精确关联，并在黎曼一阶稳定点处建立了黎曼海森矩阵谱之间的夹逼不等式。这些结果立即意味着固定秩矩阵优化在嵌入几何与商几何下的黎曼一阶稳定点集、黎曼二阶稳定点集以及严格鞍点集具有等价性。据我们所知，这是首次揭示固定秩矩阵优化中嵌入几何与商几何之间的几何景观关联，为黎曼优化中两种几何如何关联提供了具体范例。此外，本文还讨论了黎曼度量与商结构对景观关联的影响。我们还观察到固定秩矩阵优化中两种几何在特定黎曼度量下的算法关联：当两种几何的黎曼海森矩阵谱共享时，对应梯度流之间存在等价性。为获得上述结果，我们提出了一系列创新思想与技术要点，包括对不同黎曼度量的统一处理、施蒂费尔流形上的新型度量，以及商几何下新的水平空间表示。本文结果深化了对不同黎曼几何下黎曼优化几何与算法关联的理解，并为文献中若干未解问题提供了新的理论启示。