Discrete Double-Bracket Flows for Isotropic-Noise Invariant Eigendecomposition

arXiv:2602.13759v2 Announce Type: replace Abstract: We study eigendecomposition on SO(n) under streaming observations C_k = C_{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.

Source: arXiv cs.LG | 2026-05-12