Abstract — We consider the problem of estimating vectorvalued variables from noisy “relative ” measurements. The measurement model can be expressed in terms of a graph, whose nodes correspond to the variables being estimated and the edges to noisy measurements of the difference between the two variables. We take the value of one particular variable as a reference and consider the optimal estimator for the differences between the remaining variables and the reference. This type of measurement model appears in several sensor network problems, such as sensor localization and time synchronization. Two algorithms are proposed to compute the optimal estimate in a distributed, iterative manner. The first algorithm implements the Jacobi method to iteratively compute the optimal estimate, assuming all communication is perfect. The second algorithm is robust to temporary communication failures, and converges to the optimal estimate when certain mild conditions on the failure rate are satisfied. It also employs an initialization scheme to improve accuracy in spite of the slow convergence of the Jacobi method. I
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