# GMKF: Generalized Moment Kalman Filter for Polynomial Systems with Arbitrary Noise

arXiv: https://arxiv.org/abs/2403.04712 · HTML: https://mlsanity.com/paper/2403.04712/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Mar 2024.

## Built on by (most cited first, 8 of 8)

- [Learning Correspondence for Deformable Objects](https://mlsanity.com/paper/2405.08996/) (2024)
- [CHyLL: Learning Continuous Neural Representations of Hybrid Systems](https://mlsanity.com/paper/2512.10117/) (2025)
- [H2OFlow: Grounding Human-Object Affordances with 3D Generative Models and Dense Diffused Flows](https://mlsanity.com/paper/2510.21769/) (2025)
- [Embedding Hybrid Systems into Continuous Latent Vector Fields](https://mlsanity.com/paper/2606.10596/) (2026)
- [Towards Improving Learning from Demonstration Algorithms via MCMC Methods](https://mlsanity.com/paper/2405.02243/) (2024)
- [Theoretical Analysis for Expectation-Maximization-Based Multi-Model 3D Registration](https://mlsanity.com/paper/2405.08991/) (2024)
- [The Score Kalman Filter](https://mlsanity.com/paper/2605.16644/) (2026)
- [Learning Continuous Neural Representation of Stochastic Hybrid Systems](https://mlsanity.com/paper/2609.38893/) (2026)

## Abstract (arXiv)

This paper develops a new filtering approach for state estimation in polynomial systems corrupted by arbitrary noise, which commonly arise in robotics. We first consider a batch setup where we perform state estimation using all data collected from the initial to the current time. We formulate the batch state estimation problem as a Polynomial Optimization Problem (POP) and relax the assumption of Gaussian noise by specifying a finite number of moments of the noise. We solve the resulting POP using a moment relaxation and prove that under suitable conditions on the rank of the relaxation, (i) we can extract a provably optimal estimate from the moment relaxation, and (ii) we can obtain a belief representation from the dual (sum-of-squares) relaxation. We then turn our attention to the filtering setup and apply similar insights to develop a GMKF for recursive state estimation in polynomial systems with arbitrary noise. The GMKF formulates the prediction and update steps as POPs and solves them using moment relaxations, carrying over a possibly non-Gaussian belief. In the linear-Gaussian case, GMKF reduces to the standard Kalman Filter. We demonstrate that GMKF performs well under highly non-Gaussian noise and outperforms common alternatives, including the Extended and Unscented Kalman Filter, and their variants on matrix Lie group.