Optimal State Estimation with Measurements Corrupted by Laplace Noise

TL;DR

This paper develops a randomized method for optimal state estimation in linear systems with measurements corrupted by Laplace noise, leveraging Gaussian and Rayleigh distributions to approximate the least mean square error estimate.

Contribution

It introduces a novel randomized approach that approximates the optimal estimate under Laplace noise by rewriting it as scaled Gaussian noise, and analyzes its probability of accuracy.

Findings

The proposed estimator outperforms traditional linear estimators in Laplace noise scenarios.

The method's accuracy improves with the number of parallel Kalman filters used.

Comparison shows the approach is competitive with particle filters and MAP estimates.

Abstract

Optimal state estimation for linear discrete-time systems is considered. Motivated by the literature on differential privacy, the measurements are assumed to be corrupted by Laplace noise. The optimal least mean square error estimate of the state is approximated using a randomized method. The method relies on that the Laplace noise can be rewritten as Gaussian noise scaled by Rayleigh random variable. The probability of the event that the distance between the approximation and the best estimate is smaller than a constant is determined as function of the number of parallel Kalman filters that is used in the randomized method. This estimator is then compared with the optimal linear estimator, the maximum a posteriori (MAP) estimate of the state, and the particle filter.