Deterministic Optimal Transport-based Gaussian Mixture Particle Filtering for Verifiable Applications

TL;DR

This work tackles the randomness inherent in resampling for mixture-model particle filters by introducing a deterministic resampling approach based on optimal transport and Fibonacci-grid sampling. The method, termed the Pineapple Filter, deterministically constructs representative particles from a Gaussian mixture and uses an optimal transport map to select the final resampled set, avoiding stochastic variability. Empirical results on Lorenz '63 show substantially reduced particle requirements and improved uncertainty representation, while a cis-lunar NRHO tracking scenario demonstrates robust performance where standard filters fail. The approach offers verifiable, resource-efficient Bayesian inference for mission-critical applications and lays groundwork for deterministic Gaussian-sum updates and non-Gaussian extensions.

Abstract

Mixture-model particle filters such as the ensemble Gaussian mixture filter require a resampling procedure in order to converge to exact Bayesian inference. Canonically, stochastic resampling is performed, which provides useful samples with no guarantee of usefulness for a finite ensemble. We propose a new resampling procedure based on optimal transport that deterministically selects optimal resampling points. We show on a toy 3-variable problem that it significantly reduces the amount of particles required for useful state estimation. Finally, we show that this filter improves the state estimation of a seldomly-observed space object in an NRHO around the moon.

Figures (6)

• Figure 1: Illustration of the optimal transport framework. The blue top bars with diagonal stripes represent the discrete origin distribution, the red left bars represent the destination distribution, and the purple shaded squares in the middle represent the transport plan.
• Figure 2: A visual representation of the Fibonacci grid Gaussian sampling procedure for $N=5$, $N=25$, and $N=201$ samples from left to right, respectively. The large point represents the mean of the standard 2-dimensional Gaussian (zero mean) and the other points represent deterministic samples therefrom. The shaded outlines represent one, two, and three standard deviations, ordered radially outwards from the mean.
• Figure 3: A representation of a Gaussian mixture distribution that looks like a pineapple. The left most figure represents the unweighted Gaussian mixture model from which we wish to resample, with the contour lines describing one, two, and three sigma bounds respectively. The top part of the figure represents the stochastic resampling procedure
• Figure 4: A visual representation of the three variable Lorenz '63 attractor.
• Figure 5: Number of particles ($N$) versus spatio-temporal RMSE for the Lorenz '63 problem. The left figure plots the mean of the RMSE across the 192 Monte Carlo trials, while the right figure plots three standard deviations.