Learning Kalman Policy for Singular Unknown Covariances via Riemannian Regularization

TL;DR

This paper introduces a Riemannian regularization approach for learning the steady-state Kalman gain from measurement data, effectively handling unknown and singular noise covariances with theoretical guarantees.

Contribution

It proposes a novel geometric regularization technique that improves the optimization landscape for data-driven Kalman filter learning under challenging noise conditions.

Findings

The method achieves non-asymptotic convergence guarantees.

It demonstrates robustness to stepsize choices in singular noise regimes.

Numerical results show effective and scalable learning of Kalman gains.

Abstract

Kalman filtering is a cornerstone of estimation theory, yet learning the optimal filter under unknown and potentially singular noise covariances remains a fundamental challenge. In this paper, we revisit this problem through the lens of control--estimation duality and data-driven policy optimization, formulating the learning of the steady-state Kalman gain as a stochastic policy optimization problem directly from measurement data. Our key contribution is a Riemannian regularization that reshapes the optimization landscape, restoring structural properties such as coercivity and gradient dominance. This geometric perspective enables the effective use of first-order methods under significantly relaxed conditions, including unknown and rank-deficient noise covariances. Building on this framework, we develop a computationally efficient algorithm with a data-driven gradient oracle, enabling scalable stochastic implementations. We further establish non-asymptotic convergence and error guarantees enabled by the Riemannian regularization, quantifying the impact of bias and variance in gradient estimates and demonstrating favorable scaling with problem dimension. Numerical results corroborate the effectiveness of the proposed approach and robustness to the choice of stepsize in challenging singular estimation regimes.