Discrete Double-Bracket Flows for Isotropic-Noise Invariant Eigendecomposition
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arXiv
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| Format: | Preprint |
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2026
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| author | Li, ZhiMing Feng, JiaHe |
| author_facet | Li, ZhiMing Feng, JiaHe |
| contents | We study eigendecomposition on $SO(n)$ under streaming observations $C_k = C_{\mathrm{sig}} + σ_k^2 I + E_k$, where the isotropic background $σ_k^2 I$ may be time-varying and arbitrarily large. Standard algorithms couple their stability to $\lVert C_k \rVert_2 \approx σ^2$, forcing step sizes, contraction rates, and iteration counts to degrade with the noise floor. We observe that $σ^2 I$ 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 $Ω= [A, \operatorname{diag}(A)]$ 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 $η_{\max} = 1/L_C$ depend exclusively on the trace-free signal $C_e$ -- achieving pointwise, pathwise $σ^2$-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 $\operatorname{St}(k,n)$ at cost $k$ matrix-vector products per step. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_13759 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Discrete Double-Bracket Flows for Isotropic-Noise Invariant Eigendecomposition Li, ZhiMing Feng, JiaHe Machine Learning Numerical Analysis Optimization and Control 15A18, 65F15, 65K05, 90C26 G.1.3; G.1.6 We study eigendecomposition on $SO(n)$ under streaming observations $C_k = C_{\mathrm{sig}} + σ_k^2 I + E_k$, where the isotropic background $σ_k^2 I$ may be time-varying and arbitrarily large. Standard algorithms couple their stability to $\lVert C_k \rVert_2 \approx σ^2$, forcing step sizes, contraction rates, and iteration counts to degrade with the noise floor. We observe that $σ^2 I$ 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 $Ω= [A, \operatorname{diag}(A)]$ 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 $η_{\max} = 1/L_C$ depend exclusively on the trace-free signal $C_e$ -- achieving pointwise, pathwise $σ^2$-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 $\operatorname{St}(k,n)$ at cost $k$ matrix-vector products per step. |
| title | Discrete Double-Bracket Flows for Isotropic-Noise Invariant Eigendecomposition |
| topic | Machine Learning Numerical Analysis Optimization and Control 15A18, 65F15, 65K05, 90C26 G.1.3; G.1.6 |
| url | https://arxiv.org/abs/2602.13759 |