Speeding up backpropagation of gradients through the Kalman filter via closed-form expressions

TL;DR

This paper introduces novel closed-form expressions for differentiating scalar functions of Kalman filter outputs, significantly reducing computational costs and enabling faster neural network-Kalman filter integration.

Contribution

It provides a backward gradient-based approach for differentiating Kalman filter outputs, improving efficiency over traditional sensitivity equations.

Findings

Drastic reduction in computational cost for gradient calculations

Exact closed-form expressions derived for differentiation

Potential speed-ups in neural network and Kalman filter integration

Abstract

In this paper we provide novel closed-form expressions enabling differentiation of any scalar function of the Kalman filter's outputs with respect to all its tuning parameters and to the measurements. The approach differs from the previous well-known sensitivity equations in that it is based on a backward (matrix) gradient calculation, that leads to drastic reductions of the overall computational cost. It is our hope that practitioners seeking numerical efficiency and reliability will benefit from the concise and exact equations derived in this paper and the methods that build upon them. They may notably lead to speed-ups when interfacing a neural network with a Kalman filter.