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Max Entropy Moment Kalman Filter for Polynomial Systems with Arbitrary Noise
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Proposes MEM-KF, a Kalman-type Bayes filter for polynomial systems that represents beliefs and noise as moment-constrained max-entropy distributions, propagating moments and recovering densities via convex optimization.
Problem
Optimal Bayes filtering for nonlinear polynomial systems with arbitrary non-Gaussian noise, where representing beliefs, handling non-Gaussian noise, and marginalizing past states are hard.
Approach
Represent beliefs and noise as max-entropy distributions (exponential of polynomials) matched to a finite set of moments; predict by propagating moments through the extended polynomial dynamics instead of symbolic marginalization, update via MAP with MED-modeled measurements, and extract point estimates via SDP moment relaxation.
Key results
- On a linear system with bimodal Bernoulli-based noise, MEM-KF with r=4 yields lower point-estimate covariance than BLUE for N>=5 measurements (e.g., N=10: 3.03e-2 vs 1.63e-1).
- On SE(2) localization with unknown data association, MEM-KF consistently outperforms EKF, UKF, InEKF and UKFM in both association scenarios, per the paper.
- Particle filter with sufficiently many particles (N>=1e3) reaches similar error to MEM-KF, which has slightly larger median error.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Increase matched moment order from r=2 (BLUE) to r=4 (ablation) | MEM-KF update with noise model [ref] · N=5 measurements | covariance: 3.26e^{-1} -> 1.87e^{-1} | Helped |
| Increase matched moment order from r=2 (BLUE) to r=4 (ablation) | MEM-KF update with noise model [ref] · N=1 measurement | covariance: 1.583 -> 3.379 | Hurt |
| Increase matched moment order (ablation) | MEM-KF update with noise model [ref] · N=5 and N=10 measurements | covariance: 3.26e^{-1} -> 1.87e^{-1}; 1.63e^{-1} -> 3.03e^{-2} | Helped |
| Increase matched moment order from r=2 (BLUE) to r=4 (ablation) | MEM-KF update with noise model [ref], second block · N=1 measurement | covariance: 2.073 -> 4.471 | Hurt |
| Increase matched moment order from r=2 (BLUE) to r=4 (ablation) | MEM-KF update with noise model [ref], second block · N=10 measurements | covariance: 2.08e^{-1} -> 1.28e^{-2} | Helped |
| Use higher-order moments rather than the r=2 Gaussian approximation (ablation) | MEM-KF update · Accumulated non-Gaussian-noise measurements | covariance | Mixed |
| Represent the association noise with MEM-KF rather than the EKF, UKF, InEKF, or UKFM | Robot localization with unknown data association on SE(2) · Equal-probability and biased landmark observations | trajectory estimation error | Helped |
| Increase particle count (ablation) | Particle filter for localization with unknown data association · Robot localization on SE(2) | position error | Helped |
| Use MEM-KF instead of a particle filter with sufficient particles | Robot localization with unknown data association · Robot localization on SE(2) | position error | No clear effect |
Surprising findings
- With small N the multimodal belief introduces bias, so point-estimate covariance can be higher than BLUE despite higher likelihood.
Limitations
- Numerical integration for gradients and Hessians scales exponentially with state dimension, limiting real-time use.
- Point estimate does not represent multimodal belief; small-N bias is higher than BLUE.
- Requires polynomial process and measurement models and smooth densities.
Experimental setup
Datasets: synthetic samples from nonlinear and Bernoulli-based noise distributions SE(2) localization simulation with four landmarks
Benchmarks: synthetic distribution approximation linear update with non-Gaussian noise SE(2) localization with unknown data association
Metrics: covariance of point estimate normalized log-likelihood position error (2-norm)
Baselines: EKF UKF InEKF UKFM BLUE particle filter
Builds on: Kalman Filter Lasserre moment relaxation Extended Kalman Filter Unscented Kalman Filter
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