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

arXiv: https://arxiv.org/abs/2603.19601 · HTML: https://mlsanity.com/paper/2603.19601/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 34 references that are in the map.
- First posted on arXiv 20 Mar 2026.

## Builds on (most cited first, 9 of 9)

- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [New insights and perspectives on the natural gradient method](https://mlsanity.com/paper/1412.1193/) (2014)
- [A Riemannian Network for SPD Matrix Learning](https://mlsanity.com/paper/1608.04233/) (2016)
- [A micro Lie theory for state estimation in robotics](https://mlsanity.com/paper/1812.01537/) (2018)
- [Riemannian batch normalization for SPD neural networks](https://mlsanity.com/paper/1909.02414/) (2019)
- [Riemannian Local Mechanism for SPD Neural Networks](https://mlsanity.com/paper/2201.10145/) (2022)
- [The invariant extended Kalman filter as a stable observer](https://mlsanity.com/paper/1410.1465/) (2014)
- [Geometry-aware Similarity Learning on SPD Manifolds for Visual Recognition](https://mlsanity.com/paper/1608.04914/) (2016)
- [Lie Algebraic Unscented Kalman Filter for Pose Estimation](https://mlsanity.com/paper/2005.00385/) (2020)

## Abstract (arXiv)

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 29.4deg to 9.9deg; 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.