All-in-one: Certifiable Optimal Distributed Kalman Filter under Unknown Correlations.

The optimal fusion of estimates in a Distributed Kalman Filter (DKF) requires tracking of the complete network error covariance, which is a problem in terms of memory and communication bandwidth. A scalable alternative is to fuse estimates under unknown correlations, updating the local estimates and error covariance matrix as the solution of an optimisation problem. Unfortunately, this problem is NP-hard, forcing relaxations that lose optimality guarantees over the original problem. Motivated by this, we present the first Certifiable Optimal DKF (CO-DKF). Using only information from one-hop neighbours, CO-DKF solves the optimal fusion of estimates under unknown correlations by a tight Semidefinite Programming (SDP) relaxation. This particular relaxation allows to certify locally and in real time if the solution from the relaxed problem is the optimum of the original. In that case, we prove that CO-DKF is optimal in the Mean Square Error (MSE) sense. Additionally, we demonstrate that CO-DKF is a globally asymptotically stable estimator. Simulations show that CO-DKF outperforms other state-of-the-art DKF algorithms, specially in sparse, highly noisy setups.