Supervisory Measurement-Guided Noise Covariance Estimation: Discussing Forward and Reverse Differentiation

TL;DR

This paper introduces a bilevel optimization framework for estimating noise covariances in linear Gaussian systems, utilizing forward and reverse differentiation for efficient gradient computation and improved state estimation accuracy.

Contribution

It formulates noise covariance estimation as a bilevel problem with a novel factorization, enabling efficient gradient computation via forward and reverse differentiation methods.

Findings

Efficient algorithms for noise covariance estimation are developed.

Comparison of forward and reverse differentiation in terms of complexity.

The proposed approach improves state estimation accuracy in Gaussian systems.

Abstract

Reliable state estimation depends on accurately modeled noise covariances, which are difficult to determine in practice. This paper formulates the noise covariance estimation as a bilevel optimization problem that factorizes the joint likelihood of primary and supervisory measurements to reconcile information exploitation with computational tractability. The factorization converts the nested Bayesian dependency into a Markov-chain structure, allowing efficient computation. At the lower level, a Kalman filter with state augmentation performs such computation. Meanwhile, closed-form forward and reverse differentiation provide efficient gradients for the upper-level updates, and we compare the two models' space and time complexities to inform their practical selection. The upper level subsequently refines the noise covariances to guide the lower-level estimation. Taken together, the proposed algorithms offer a systematic and computationally efficient approach to noise covariance estimation in linear Gaussian systems.