Riemannian Newton Method for the Multivariate Eigenvalue Problem

The multivariate eigenvalue problem (MEP) which originally arises from the canonical correlation analysis is an important generalization of the classical eigenvalue problem. Recently, the MEP also finds applications in many other areas and continues to receive interest. However, the existing algorithms for the MEP are the generalization of the power iteration for the classical eigenvalue problem and converge slowly. In this paper, we propose a Riemannian Newton method for the MEP, which is a generalization of the classical Rayleigh quotient iteration (RQI). Under a mild condition, the local quadratic convergence can be guaranteed. We also develop the inexact implementation by employing some Krylov subspace method and establishing the preconditioning technique to obtain an inexact Riemannian Newton step efficiently. Preliminary but promising numerical experiments are reported which show a good convergence performance in terms of the proposed method’s speed and global convergence.