Coupling-Informed Transport Maps for Bayesian Filtering in Nonlinear Dynamical Systems

TL;DR

This paper introduces a likelihood-free transport filtering method leveraging couplings and block-triangular structures to accurately approximate non-Gaussian posteriors in nonlinear dynamical systems.

Contribution

It proposes a novel, training-free transport filter using gradient flows that avoids non-convex optimization and particle collapse, with extensions to high-dimensional problems.

Findings

Accurately approximates non-Gaussian filtering posteriors.

Avoids particle collapse in nonlinear filtering.

Demonstrates superior performance over conventional methods.

Abstract

A likelihood-free transport filtering method is proposed based on the couplings between state and observation variables. By exploiting a block-triangular structure in the transport map, the analysis step of filtering is reformulated as the minimization of the maximum mean discrepancy (MMD) between the true joint measure and its transport-based approximation. To circumvent the non-convexity in the MMD optimization, we introduce a training-free transport filter method via gradient flows, which leads to an analytic computation for the transport map that implies the steepest descent direction of the MMD. The proposed approach accurately approximates non-Gaussian filtering posteriors and avoids particle collapse. We provide a convergence analysis for the expectation of the MMD between the approximated posterior and the truth posterior. Finally, we extend the method to high-dimensional problems through domain localization. Numerical examples demonstrate the superior performance of our approach over conventional filtering methods in nonlinear, non-Gaussian scenarios.