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

TL;DR

The paper addresses the challenge of extracting the dominant invariant subspace of a general square matrix using Oja's flow on the Stiefel manifold. It proves global exponential convergence for almost all initial conditions, even beyond symmetric positive-definite matrices, by introducing a stabilizing spectral shift and rigorously characterizing the domain of attraction. It also develops practical tools to adjust subspace dimensionality and demonstrates applications to model reduction and low-rank controller design, preserving key system properties and enabling stable, scalable control of large-scale linear systems. The results extend the Oja flow's applicability to general matrices and offer a theoretically solid, computationally tractable approach for dominant-mode identification, model reduction, and stabilization in control contexts.

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.

Figures (6)

• Figure 1: Sphere representing $\mathrm{St}(1,3)$. The red markers represent $\mathcal{U}_{1}=\{ \pm \psi _{1}\}$ and the blue line represents the unit circle $\mathrm{St}(1,3) \cap \mathrm{span}\{ \psi _{2}, \psi _{3} \}$ in Example \ref{['exmp:St13']}.
• Figure 2: Plot of $|U(t)^{\top} U(t) -1|$ for each time $t$ by the forward Euler method with normalization (black solid line), without normalization (red dashed line), with $A+2I_{3}$ (blue chain line), and with $A+4I_{3}$ (green dotted line) under the conditions in Example \ref{['exmp:St13']} with $U(0) = (\psi _{2} + \psi _{3})/\| \psi _{2} + \psi _{3} \|$.
• Figure 3: Plot of $|U(t)^{\top} U(t) -1|$ for each time $t$ by the forward Euler method without normalization (red dashed line), with $A+2I_{3}$ (blue chain line), and with $A+4I_{3}$ (green dotted line) under the conditions in Example \ref{['exmp:St13']} with $U(0) = \frac{11}{10}(\psi _{2} + \psi _{3})/\| \psi _{2} + \psi _{3} \|$.
• Figure 4: Plot of $\| U(t) U(t)^{\top} - \psi _{1} \psi _{1}^{\top} \| _{\rm ind}$ for each time $t$ using the forward Euler method with $A+2I_{3}$ (blue chain line) under the conditions in Example \ref{['exmp:St13']} with $U(0) = (\psi _{1} + \psi _{2} + \psi _{3})/\| \psi _{1} + \psi _{2} + \psi _{3} \|$. The black line shows the upper bound of the convergence rate.
• Figure 5: Bode diagrams of $P(s)$, $P_{\bar{V}}(s)$, and $P_{\rm rd}(s)$.

Theorems & Definitions (40)

• Proposition 1: yan1994global
• Example 2
• Proposition 3: yan1994global
• Proposition 4: TsuzukiOhki2024TsuzukiOhki2024akuo2025asymptotic
• Example 5: Visualization of the set $\mathcal{V}_{r}$
• Lemma 6: sasagawa1982finite
• Lemma 7
• proof
• Lemma 8
• proof