Discrete Double-Bracket Flows for Isotropic-Noise Invariant Eigendecomposition

TL;DR

This paper introduces a novel discrete double-bracket flow for eigendecomposition on SO(n) that remains stable under time-varying isotropic noise, improving robustness and convergence.

Contribution

It develops a noise-invariant eigendecomposition algorithm using a double-bracket flow that isolates signal from isotropic noise, with proven stability and convergence properties.

Findings

The proposed flow is invariant to isotropic noise in the eigendecomposition process.

Global convergence is established via strict-saddle geometry and a discrete Łojasiewicz argument.

The method extends to top-k eigentracking on the Stiefel manifold with computational efficiency.

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.