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

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In this paper, we consider the geometric landscape connection of the widely studied manifold and factorization formulations in low-rank positive semidefinite (PSD) and general matrix optimization.

• We establish a sandwich relation on the spectrum of Riemannian and Euclidean Hessians at first-order stationary points (FOSPs).
• As a result of that, we obtain an equivalence on the set of FOSPs, second-order stationary points (SOSPs) and strict saddles between the manifold and the factorization formulations.
• In addition, we show the sandwich relation can be used to transfer more quantitative geometric properties from one formulation to another.