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For smooth optimization problems with a Hermitian positive semi-definitefixed-rank constraint, we consider three existing approaches including thesimple Burer--Monteiro method, and Riemannian optimization over quotientgeometry and the embedded geometry. These three methods can be all representedvia quotient geometry with three Riemannian metrics $g^i(\cdot, \cdot)$$(i=1,2,3)$. By taking the nonlinear conjugate gradient method (CG) as anexample, we show that CG in the factor-based Burer--Monteiro approach isequivalent to Riemannian CG on the quotient geometry with the Bures-Wassersteinmetric $g^1$. Riemannian CG on the quotient geometry with the metric $g^3$ isequivalent to Riemannian CG on the embedded geometry. For comparing the threeapproaches, we analyze the condition number of the Riemannian Hessian near theminimizer under the three different metrics. Under certain assumptions, thecondition number from the Bures-Wasserstein metric $g^1$ is significantlydifferent from the other two metrics.  Numerical experiments show that the Burer--Monteiro CG method has obviouslyslower asymptotic convergence rate when the minimizer is rank deficient, whichis consistent with the condition number analysis.

Riemannian optimization using three different metrics for Hermitian PSD fixed-rank constraints: an extended version