The geometry of low-rank Kalman filters

TL;DR

This paper explores the geometric properties of low-rank Kalman filters, focusing on the Riccati flow on fixed-rank matrix manifolds and its contraction behavior under a specialized metric.

Contribution

It introduces a geometric framework for low-rank Kalman filters and analyzes their contraction properties on the manifold of fixed-rank positive semidefinite matrices.

Findings

Low-rank Riccati flow exhibits contraction under a new metric.

The geometry of low-rank filters parallels that of full-rank filters in certain aspects.

The study provides insights into stability and convergence of low-rank Kalman filters.

Abstract

An important property of the Kalman filter is that the underlying Riccati flow is a contraction for the natural metric of the cone of symmetric positive definite matrices. The present paper studies the geometry of a low-rank version of the Kalman filter. The underlying Riccati flow evolves on the manifold of fixed rank symmetric positive semidefinite matrices. Contraction properties of the low-rank flow are studied by means of a suitable metric recently introduced by the authors.