Discrete Double-Bracket Flows for Isotropic-Noise Invariant Eigendecomposition

Abstract

We study eigendecomposition on $SO(n)$ under streaming observations $C_k = C_{\mathrm{sig}} + \sigma_k^2 I + E_k$, where the isotropic background $\sigma_k^2 I$ may be time-varying and arbitrarily large. Standard algorithms couple their stability to $\lVert C_k \rVert_2 \approx \sigma^2$, forcing step sizes, contraction rates, and iteration counts to degrade with the noise floor. We observe that $\sigma^2 I$ 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 $\Omega = [A, \operatorname{diag}(A)]$ 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 $\eta_{\max} = 1/L_C$ depend exclusively on the trace-free signal $C_e$ -- achieving pointwise, pathwise $\sigma^2$-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 {\L}ojasiewicz argument, and extend the framework to top-$k$ eigentracking on the Stiefel manifold $\operatorname{St}(k,n)$ at cost $k$ matrix-vector products per step.