arXiv subjects

ZhiMing Li

Publications and source records attributed to ZhiMing Li.

3 recordsLinked to original sources

arXiv2026

Higher-Order Equilibrium Tracking for EM-Compressible Online Estimation

We study online estimation in latent-variable models by recasting the problem as tracking a moving empirical equilibrium. Standard online EM and stochastic approximation analyses primarily study convergence toward the population parameter and typically do not isolate the empirical batch optimum from the online tracking error at finite horizon. Our framework decomposes the online estimate into the frozen batch equilibrium at the current running statistic and a tracking lag that captures the algorithm's delay behind this moving target. We prove a batch-to-online transfer theorem: provided $\lVert e_T \rVert_{L^{2}} = o(T^{-1/2})$, the online estimator inherits the batch central limit theorem and the sharp first-order risk constant. Our key observation is that the empirical optimum evolves on a smooth equilibrium manifold indexed by the running statistic. An $m$-th order equilibrium-jet predictor combined with an order-$\nu$ frozen corrector yields localized tracking rates $O(T^{-\nu(m+1)})$. We formalize EM-compressibility and EM-jet$^R$-compressibility as the structural conditions that make the equilibrium response and the Newton corrector evaluable from a retained streaming statistic. The theory is instantiated in latent linear Gaussian covariance estimation, where the first-order scheme operates on a compressed $d \times d$ statistic with explicit finite-sample risk envelopes and a certified restart rule.

cs.LG↗

arXiv2026

K-GMRF: Kinetic Gauss-Markov Random Field for First-Principles Covariance Tracking on Lie Groups

Tracking non-stationary covariance matrices is fundamental to vision yet hindered by existing estimators that either neglect manifold constraints or rely on first-order updates, incurring inevitable phase lag during rapid evolution. We propose K-GMRF, an online, training-free framework for covariance tracking that reformulates the problem as forced rigid-body motion on Lie groups. Derived from the Euler-Poincar\'e equations, our method interprets observations as torques driving a latent angular velocity, propagated via a structure-preserving symplectic integrator. We theoretically prove that this second-order dynamics achieves zero steady-state error under constant rotation, strictly superior to the proportional lag of first-order baselines. Validation across three domains demonstrates robust tracking fidelity: (i) on synthetic ellipses, K-GMRF reduces angular error by 30x compared to Riemannian EMA while maintaining stability at high speeds; (ii) on SO(3) stabilization with 20% dropout, it decreases geodesic error from 29.4{\deg} to 9.9{\deg}; and (iii) on OTB motion-blur sequences, it improves loU from 0.55 to 0.74 on BlurCar2 with a 96% success rate. As a fully differentiable symplectic module, K-GMRF provides a plug-and-play geometric prior for data-constrained scenarios and an interpretable layer within modern deep architectures.

cs.CV↗

arXiv2026

Discrete Double-Bracket Flows for Isotropic-Noise Invariant Eigendecomposition

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.

cs.LG↗