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 a flat 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和特征向量。尤其对于大规模问题，无正交化方法更受青睐，这类方法无需在每次迭代中显式计算正交向量即可求解极端特征值的特征空间。针对前p个特征值，最简单的无正交化方法是利用求解无约束Burer-Monteiro公式的算法，寻找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方法等价于具有平坦度量的商流形上的黎曼共轭梯度方法，因此其全局收敛至驻点的性质可被证明。数值实验表明，若最大k个特征值近似均匀分布，该方法对计算大规模矩阵的最大k个特征值具有高效性。