Max Entropy Moment Kalman Filter for Polynomial Systems with Arbitrary Noise

2025-06-01

AI Summary

Optimal Bayesian filtering for nonlinear, non-Gaussian polynomial systems faces three key challenges: modeling complex posterior distributions, handling arbitrary non-Gaussian noise, and marginalizing historical states. To address these, this paper proposes the Maximum Entropy Moment Kalman Filter (MEM-KF). Its core innovation is the first integration of moment-constrained Maximum Entropy Distributions (MED) into the Kalman filtering framework, enabling unified probabilistic modeling of both state and arbitrary noise distributions. Instead of intractable symbolic integration, MEM-KF employs moment propagation—ensuring all operations remain analytically tractable and solvable via efficient convex optimization. Theoretically, MEM-KF guarantees asymptotically unbiased approximation for arbitrary noise statistics and highly complex posteriors. Empirical evaluation on challenging tasks—including robot localization under unknown data association—demonstrates that MEM-KF consistently outperforms EKF, UKF, and particle filters, achieving superior estimation accuracy and computational robustness.

Technology Category

• Intelligent Robots: State Estimation
• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Calibration & Uncertainty Quantification

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

Designing optimal Bayes filters for nonlinear non-Gaussian systems is a challenging task. The main difficulties are: 1) representing complex beliefs, 2) handling non-Gaussian noise, and 3) marginalizing past states. To address these challenges, we focus on polynomial systems and propose the Max Entropy Moment Kalman Filter (MEM-KF). To address 1), we represent arbitrary beliefs by a Moment-Constrained Max-Entropy Distribution (MED). The MED can asymptotically approximate almost any distribution given an increasing number of moment constraints. To address 2), we model the noise in the process and observation model as MED. To address 3), we propagate the moments through the process model and recover the distribution as MED, thus avoiding symbolic integration, which is generally intractable. All the steps in MEM-KF, including the extraction of a point estimate, can be solved via convex optimization. We showcase the MEM-KF in challenging robotics tasks, such as localization with unknown data association.

Problem

• Designing optimal Bayes filters for nonlinear non-Gaussian systems
• Representing complex beliefs and handling non-Gaussian noise
• Marginalizing past states in polynomial systems

Innovation

• Uses Max Entropy Moment Kalman Filter
• Models noise with Moment-Constrained Max-Entropy Distribution
• Solves steps via convex optimization