In many applications, it is desired to obtain extreme eigenvalues and eigenvectors of large Hermitian matrices by efficient and compact algorithms. In particular, orthogonalization-free methods are preferred for large-scale problems for finding eigenspaces of extreme eigenvalues without explicitly computing orthogonal vectors in each iteration. For the top $p$ eigenvalues, the simplest orthogonalization-free method is to find the best rank-$p$ approximation to a positive semi-definite Hermitian matrix by algorithms solving the unconstrained Burer-Monteiro formulation. We show that the nonlinear conjugate gradient method for the unconstrained Burer-Monteiro formulation is equivalent to a Riemannian conjugate gradient method on a quotient manifold with the Bures-Wasserstein metric, thus its global convergence to a stationary point can be proven. Numerical tests suggest that it is efficient for computing the largest $k$ eigenvalues for large-scale matrices if the largest $k$ eigenvalues are nearly distributed uniformly.

翻译：在许多应用中，需要通过高效且紧凑的算法获取大型埃尔米特矩阵的极端特征值与特征向量。对于大规模问题，免正交化方法尤为适用，这类方法能在无需每次迭代显式计算正交向量的情况下，求解极端特征值的特征空间。针对前p个特征值，最简单的免正交化方法是通过求解无约束Burer-Monteiro公式的算法，找到半正定埃尔米特矩阵的最佳秩p近似。我们证明，无约束Burer-Monteiro公式的非线性共轭梯度法等价于具有Bures-Wasserstein度量的商流形上的黎曼共轭梯度法，从而可证明其全局收敛到驻点。数值测试表明，若前k个最大特征值近似均匀分布，该方法对大规模矩阵的最大k个特征值计算具有高效性。