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Optimal State Estimation in the Presence of Non-Gaussian Uncertainty via Wasserstein Distance Minimization

Himanshu Prabhat, Raktim Bhattacharya

TL;DR

This work introduces a distribution-agnostic state-estimation framework that minimizes the $2$-Wasserstein distance between the posterior error distribution $p_{e^+}$ and the Dirac delta $ obreak\delta(e^+)$. The approach yields a Wasserstein filter that naturally recovers the Kalman filter for linear Gaussian systems and derives a Gaussian Sum Filter (GSF) with explicit suboptimality bounds when the prior is a Gaussian Mixture. It further proposes a nonlinear extension, nGSF, which improves estimation accuracy via local optimization. Overall, the transport-based formulation provides a principled, non-particle-based method for fusing information under non-Gaussian uncertainty and can operate on general state manifolds.

Abstract

This paper presents a novel distribution-agnostic Wasserstein distance-based estimation framework. The goal is to determine an optimal map combining prior estimate with measurement likelihood such that posterior estimation error optimally reaches the Dirac delta distribution with minimal effort. The Wasserstein metric is used to quantify the effort of transporting from one distribution to another. We hypothesize that minimizing the Wasserstein distance between the posterior error and the Dirac delta distribution results in optimal information fusion and posterior state uncertainty. Framework validation is demonstrated by the successful recovery of the classical Kalman filter for linear systems with Gaussian uncertainties. Notably, the proposed Wasserstein filter does not rely on particle representation of uncertainty. Furthermore, the classical result for the Gaussian Sum Filter (GSF) is retrieved from the Wasserstein framework. This approach analytically exhibits the suboptimality of GSF and enables the use of nonlinear optimization techniques to enhance the accuracy of the Gaussian sum estimator.

Optimal State Estimation in the Presence of Non-Gaussian Uncertainty via Wasserstein Distance Minimization

TL;DR

This work introduces a distribution-agnostic state-estimation framework that minimizes the -Wasserstein distance between the posterior error distribution and the Dirac delta . The approach yields a Wasserstein filter that naturally recovers the Kalman filter for linear Gaussian systems and derives a Gaussian Sum Filter (GSF) with explicit suboptimality bounds when the prior is a Gaussian Mixture. It further proposes a nonlinear extension, nGSF, which improves estimation accuracy via local optimization. Overall, the transport-based formulation provides a principled, non-particle-based method for fusing information under non-Gaussian uncertainty and can operate on general state manifolds.

Abstract

This paper presents a novel distribution-agnostic Wasserstein distance-based estimation framework. The goal is to determine an optimal map combining prior estimate with measurement likelihood such that posterior estimation error optimally reaches the Dirac delta distribution with minimal effort. The Wasserstein metric is used to quantify the effort of transporting from one distribution to another. We hypothesize that minimizing the Wasserstein distance between the posterior error and the Dirac delta distribution results in optimal information fusion and posterior state uncertainty. Framework validation is demonstrated by the successful recovery of the classical Kalman filter for linear systems with Gaussian uncertainties. Notably, the proposed Wasserstein filter does not rely on particle representation of uncertainty. Furthermore, the classical result for the Gaussian Sum Filter (GSF) is retrieved from the Wasserstein framework. This approach analytically exhibits the suboptimality of GSF and enables the use of nonlinear optimization techniques to enhance the accuracy of the Gaussian sum estimator.
Paper Structure (8 sections, 7 theorems, 47 equations, 3 figures, 2 algorithms)

This paper contains 8 sections, 7 theorems, 47 equations, 3 figures, 2 algorithms.

Table of Contents

  1. INTRODUCTION
  2. MATHEMATICAL PRELIMINARIES
  3. WASSERSTEIN FILTER
  4. Linear Measurement with Gaussian Uncertainty
  5. Mixture of Gaussian (MoG) Prior
  6. EXAMPLE
  7. CONCLUSIONS
  8. ACKNOWLEDGMENTS

Key Result

Proposition 1

The Wasserstein distance $W_2$ between two multivariate Gaussians $\mathcal{N}(\mu_1, \Sigma_1)$ and $\mathcal{N}(\mu_2, \Sigma_2)$ in $\mathcal{R}^n$ is given by

Figures (3)

  • Figure 1: Initial and propagated state uncertainty after 1 time step (0.5 seconds).
  • Figure 2: Position ($x_1$) estimates with filter sample time: 0.5s.
  • Figure 3: Velocity ($x_2$) estimates with filter sample time: 0.5s.

Theorems & Definitions (16)

  • Definition 1
  • Remark 1
  • Proposition 1
  • Corollary 1
  • proof
  • Proposition 2
  • proof
  • Corollary 2
  • proof
  • Theorem 1
  • ...and 6 more