On the convergence of orthogonalization-free conjugate gradient method for extreme eigenvalues of Hermitian matrices: a Riemannian optimization interpretation

TL;DR

This paper interprets an orthogonalization-free conjugate gradient method for extreme eigenvalues of large Hermitian matrices as a Riemannian optimization problem, proving its convergence and demonstrating efficiency in large-scale scenarios.

Contribution

It establishes a Riemannian optimization framework for an orthogonalization-free conjugate gradient method, providing convergence analysis and practical insights.

Findings

Method converges globally to a stationary point.

Efficient for large matrices with nearly uniformly distributed top eigenvalues.

Numerical tests confirm practical effectiveness.

Abstract

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.