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，需要利用高效紧凑算法获得大型埃尔米特矩阵的极端特征值和特征向量。特别地，对于大规模问题，无正交化方法因无需在每次迭代中显式计算正交向量而成为求解极端特征值特征空间的首选。针对前p个特征值，最简洁的无正交化方法是通过求解无约束Burer-Monteiro公式来获得半正定埃尔米特矩阵的最佳秩p逼近。我们证明，针对无约束Burer-Monteiro公式的非线性共轭梯度法等价于在具有Bures-Wasserstein度量的商流形上的黎曼共轭梯度法，因此其全局收敛至驻点可被证明。数值实验表明：当最大k个特征值近似均匀分布时，该方法对大型矩阵的最大k个特征值计算具有高效性。