For smooth optimization problems with a Hermitian positive semi-definite fixed-rank constraint, we consider three existing approaches including the simple Burer--Monteiro method, and Riemannian optimization over quotient geometry and the embedded geometry. These three methods can be all represented via quotient geometry with three Riemannian metrics $g^i(\cdot, \cdot)$ $(i=1,2,3)$. By taking the nonlinear conjugate gradient method (CG) as an example, we show that CG in the factor-based Burer--Monteiro approach is equivalent to Riemannian CG on the quotient geometry with the Bures-Wasserstein metric $g^1$. Riemannian CG on the quotient geometry with the metric $g^3$ is equivalent to Riemannian CG on the embedded geometry. For comparing the three approaches, we analyze the condition number of the Riemannian Hessian near the minimizer under the three different metrics. Under certain assumptions, the condition number from the Bures-Wasserstein metric $g^1$ is significantly different from the other two metrics. Numerical experiments show that the Burer--Monteiro CG method has obviously slower asymptotic convergence rate when the minimizer is rank deficient, which is consistent with the condition number analysis.

翻译：针对具有厄米特正半定固定秩约束的光滑优化问题，本文探讨了三种现有方法，包括简单的Burer-Monteiro方法，以及基于商流形几何和嵌入几何的黎曼优化。这三种方法均可通过引入三种黎曼度量$g^i(\cdot, \cdot)$ $(i=1,2,3)$的商流形几何统一表示。以非线性共轭梯度法（CG）为例，我们证明基于因子的Burer-Monteiro方法中的CG等价于采用Bures-Wasserstein度量$g^1$的商流形上的黎曼CG；而采用度量$g^3$的商流形上的黎曼CG等价于嵌入几何上的黎曼CG。为比较三种方法，我们分析了在三种不同度量下黎曼Hessian矩阵在极小值点附近的条件数。在特定假设下，Bures-Wasserstein度量$g^1$对应的条件数与其他两种度量存在显著差异。数值实验表明，当极小值点秩亏时，Burer-Monteiro CG方法的渐近收敛速度明显较慢，这与条件数分析结果一致。