Fast distributed smoothing of relative measurements

We consider the problem of estimation from noisy relative measurements in a network. In previous work, a distributed scheme for obtaining least-squares (LS) estimates was developed based on the Jacobi algorithm; in a synchronous version, the algorithm was shown to converge exponentially and bounds on the rate of convergence have been obtained. In this paper, we design and analyze a new class of distributed asynchronous smoothing algorithms based on a randomized version of Kaczmarz algorithm for solving linear systems. One of the proposed schemes applies Randomized Kaczmarz directly to the noisy linear system, whereas the other one operates on the normal equations for LS estimation. We analyze the expected convergence rate of the proposed algorithms depending solely on properties of the network topology. Inspired by the analytical insights, we propose a distributed smoothing algorithm, namely Randomized Kaczmarz Over-smoothing (RKO), which has demonstrated significant improvement over existing protocols in terms of both convergence speedup and energy savings.