Discovering the Kalman-Bucy-Koopman Filter

This paper introduces the Kalman-Bucy-Koopman (KBK) filter, a novel framework for nonlinear state estimation grounded in Koopman operator spectral theory.

The nonlinear estimation problem is formulated as a maximum-likelihood (Mortensen) estimator whose solution is characterized by a Hamilton-Jacobi (HJ) partial differential equation.

The proposed KBK filter provides a spectral, operator-theoretic realization of this nonlinear filtering problem by parameterizing the HJ value function in terms of principal Koopman eigenfunctions.

This transformation converts the nonlinear estimation problem into a Riccati-type evolution in Koopman coordinates, yielding a linear-operator analogue of the classical Kalman-Bucy filter while preserving nonlinear structure in the original state variables.

We develop a path-integral formulation for computing principal Koopman eigenfunctions and introduce a dynamics-informed, characteristics-inspired basis construction for their approximation.

Theoretical error bounds are derived for value-function and state-estimation approximations.

Simulation results demonstrate improved performance over the extended Kalman filter and illustrate the ability of the KBK framework to operate in data-driven settings without explicit model linearization.