Interpretable Gradient Descent for Kalman Gain

TL;DR

This paper presents a new interpretable gradient descent method for estimating the Kalman gain in linear systems, providing convergence guarantees and insights into the role of observability conditions.

Contribution

It introduces a novel decomposition of the gradient of the innovation loss, enabling convergence analysis and interpretability of the gradient descent process for Kalman gain estimation.

Findings

Proves convergence of gradient descent to the optimal Kalman gain.

Identifies a non-standard observability condition critical for recovery.

Provides an interpretable geometric convergence rate.

Abstract

We derive a decomposition for the gradient of the innovation loss with respect to the filter gain in a linear time-invariant system, decomposing as a product of an observability Gramian and a term quantifying the ``non-orthogonality" between the estimation error and the innovation. We leverage this decomposition to give a convergence proof of gradient descent to the optimal Kalman gain, specifically identifying how recovery of the Kalman gain depends on a non-standard observability condition, and obtaining an interpretable geometric convergence rate.