On the Projective Geometry of Kalman Filter

TL;DR

This paper investigates the geometric properties of the Kalman filter's convergence by linking Riccati map contraction to non-expansiveness in probability distribution space, aiming to enhance convergence analysis in graphical models.

Contribution

It introduces a novel geometric perspective connecting Riccati map contraction with Hilbert metric non-expansiveness in filtering.

Findings

Establishes a relationship between Riccati map contraction and Hilbert metric properties.

Provides a foundation for improved convergence analysis in graphical model filtering.

Suggests potential for more general filtering algorithm analysis.

Abstract

Convergence of the Kalman filter is best analyzed by studying the contraction of the Riccati map in the space of positive definite (covariance) matrices. In this paper, we explore how this contraction property relates to a more fundamental non-expansiveness property of filtering maps in the space of probability distributions endowed with the Hilbert metric. This is viewed as a preliminary step towards improving the convergence analysis of filtering algorithms over general graphical models.