Optimal Estimation on the Graph Cycle Space

This paper addresses the problem of estimating the states of a group of agents from noisy measurements of pairwise differences between agents´ states. The agents can be viewed as nodes in a graph and the relative measurements between agents as the graph´s edges. We propose a new distributed algorithm that exploits the existence of cycles in the graph to compute the best linear state estimates. For large graphs, the new algorithm significantly reduces the total number of message exchanges that are needed to obtain an optimal estimate. We show that the new algorithm is guaranteed to converge for planar graphs and provide explicit formulas for its convergence rate for regular lattices.