Discrete Double-Bracket Flows for Isotropic-Noise Invariant Eigendecomposition

Abstract

We study eigendecomposition on $SO(n)$ under streaming observations $C_k=C_{\mathrm{sig}}+\ddot{I}{ƒ}_k^{2}I+E_k$, where the isotropic background $\ddot{I}{ƒ}_k^{2}I$ may be time-varying and arbitrarily large. Standard algorithms couple their stability to $\Vert C_k{\Vert }_2\approx \ddot{I}{ƒ}^2$, forcing step sizes, contraction rates, and iteration counts to degrade with the noise floor. We observe that $\ddot{I}{ƒ}^{2}I$ lies in the center of the matrix algebra and therefore *should never enter* the eigenspace dynamics. We construct a discrete double-bracket flow whose skew-symmetric generator $\hat{I}\text{c◯}=[A,\mathrm{diag}(A)]$ operates in the tangent Lie algebra $\mathfrak{so}(n)$, where scalar multiples of the identity vanish by antisymmetry. The resulting trajectory, Lyapunov function, and maximal stable step size $\hat{I}{\cdot }_{\max }=1/L_C$ depend exclusively on the trace-free signal $C_e$ -- achieving pointwise, pathwise $\ddot{I}{ƒ}^2$-invariance. We establish input-to-state stability with a noise ball governed solely by trace-free perturbations, prove global convergence via strict-saddle geometry and a discrete Łojasiewicz argument, and extend the framework to top-$k$ eigentracking on the Stiefel manifold $\mathrm{St}(k,n)$ at cost $k$ matrix-vector products per step.

75 pages, 9 figures