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In this paper, we consider the geometric landscape connection of the widelystudied manifold and factorization formulations in low-rank positivesemidefinite (PSD) and general matrix optimization. We establish a sandwichrelation on the spectrum of Riemannian and Euclidean Hessians at first-orderstationary points (FOSPs). As a result of that, we obtain an equivalence on theset of FOSPs, second-order stationary points (SOSPs) and strict saddles betweenthe manifold and the factorization formulations. In addition, we show thesandwich relation can be used to transfer more quantitative geometricproperties from one formulation to another. Similarities and differences in thelandscape connection under the PSD case and the general case are discussed. Tothe best of our knowledge, this is the first geometric landscape connectionbetween the manifold and the factorization formulations for handling rankconstraints, and it provides a geometric explanation for the similar empiricalperformance of factorization and manifold approaches in low-rank matrixoptimization observed in the literature. In the general low-rank matrixoptimization, the landscape connection of two factorization formulations(unregularized and regularized ones) is also provided. By applying thesegeometric landscape connections, in particular, the sandwich relation, we areable to solve unanswered questions in literature and establish stronger resultsin the applications on geometric analysis of phase retrieval, well-conditionedlow-rank matrix optimization, and the role of regularization in factorizationarising from machine learning and signal processing.

Nonconvex Factorization and Manifold Formulations are Almost Equivalent in Low-rank Matrix Optimization