Distributed estimation from relative measurements of heterogeneous and uncertain quality

TL;DR

This paper introduces novel centralized and distributed algorithms for estimating node states from relative measurements affected by Gaussian mixture noise, demonstrating robustness and convergence through theoretical proofs and numerical experiments.

Contribution

It develops two new algorithms, LS-EM and Distributed LS-EM, for robust estimation from heterogeneous, uncertain measurements, with proven convergence and practical validation.

Findings

Algorithms are robust against various noise types.

Distributed LS-EM performs well even with noise parameter errors.

Both algorithms outperform classical solutions in experiments.

Abstract

This paper studies the problem of estimation from relative measurements in a graph, in which a vector indexed over the nodes has to be reconstructed from pairwise measurements of differences between its components associated to nodes connected by an edge. In order to model heterogeneity and uncertainty of the measurements, we assume them to be affected by additive noise distributed according to a Gaussian mixture. In this original setup, we formulate the problem of computing the Maximum-Likelihood (ML) estimates and we design two novel algorithms, based on Least Squares regression and Expectation-Maximization (EM). The first algorithm (LS- EM) is centralized and performs the estimation from relative measurements, the soft classification of the measurements, and the estimation of the noise parameters. The second algorithm (Distributed LS-EM) is distributed and performs estimation and soft classification of the measurements, but requires the knowledge of the noise parameters. We provide rigorous proofs of convergence of both algorithms and we present numerical experiments to evaluate and compare their performance with classical solutions. The experiments show the robustness of the proposed methods against different kinds of noise and, for the Distributed LS-EM, against errors in the knowledge of noise parameters.