Global Convergence of Oja's Component Flow for General Square Matrices and Its Applications

Abstract

In this study, the global convergence properties of the Oja flow, a continuous-time algorithm for principal component extraction, was established for general square matrices. The Oja flow is a matrix differential equation on the Stiefel manifold designed to extract a dominant subspace. Although its analysis has traditionally been restricted to symmetric positive-definite matrices, where it acts as a gradient flow, recent applications have extended its use to general matrices. In this non-symmetric case, the flow extracts the invariant subspace corresponding to the eigenvalues with the largest real parts. However, prior convergence results have been purely local, leaving the global behavior as an open problem. The findings of this study fill this gap by providing a comprehensive global convergence analysis, establishing that the flow converges exponentially for almost all initial conditions. We also propose a modification to the algorithm that enhances its numerical stability. As an application of this theory, we developed novel methods for model reduction of linear dynamical systems and the synthesis of low-rank stabilizing controllers. The study advances the theoretical understanding of the Oja flow and demonstrates its potential as a reliable and versatile tool for analyzing and controlling complex linear systems.