📝 SummarXiv

Abstract

This paper establishes the global convergence properties of the Oja flow, a continuous-time algorithm for principal component extraction, for general square matrices. The Oja flow is a matrix differential equation on the Stiefel manifold designed to extract a dominant subspace. While 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. This paper fills 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 develop novel methods for the model reduction of linear dynamical systems and the synthesis of low-rank stabilizing controllers.