提出新算法，让特征分解不受背景噪声影响，稳定高效。
Discrete Double-Bracket Flows for Isotropic-Noise Invariant Eigendecomposition
- 用双括号流构造不依赖噪声的特征分解方法
- 最大步长可达1/L_C，与噪声大小无关
- 适合高噪声下实时追踪特征值的场景
研究在流式观测 $C_k = C_{\mathrm{sig}} + σ_k^2 I + E_k$ 下 $SO(n)$ 上的特征分解问题，其中各向同性背景噪声 $σ_k^2 I$ 可随时间变化且任意大。传统算法稳定性受 $\lVert C_k \rVert_2 \approx σ^2$ 影响，导致步长、收敛速率和迭代次数随噪声底噪恶化。我们发现 $σ^2 I$ 位于矩阵代数中心，应不参与特征空间演化。构建了一种离散双括号流，其反对称生成元 $Ω= [A, \operatorname{diag}(A)]$ 定义在李代数 $\mathfrak{so}(n)$ 中，恒等矩阵项因反对称性消失。轨迹、李雅普诺夫函数及最大稳定步长 $η_{\max} = 1/L_C$ 仅依赖于无迹信号 $C_e$，实现逐点与路径上的 $σ^2$-不变性。建立输入到状态稳定性，噪声界仅由无迹扰动决定；通过严格鞍点几何与离散 Łojasiewicz 不等式证明全局收敛；并将框架扩展至 $\operatorname{St}(k,n)$ 上的 top-$k$ 特征追踪，每步仅需 $k$ 次矩阵-向量乘法。
原文摘要 · Abstract (English)
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.
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