  The optimal fusion of estimates in a Distributed Kalman Filter (DKF) requires
tracking of the complete network error covariance, problematic in terms of
memory and communication. A scalable alternative is to fuse estimates under
unknown correlations, doing the update by solving an optimisation problem.
Unfortunately, this problem is NP-hard, forcing relaxations that lose
optimality guarantees. 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
particular tight Semidefinite Programming (SDP) relaxation which allows to
certify, locally and in real time, if the relaxed solution is the actual
optimum. In that case, we prove optimality in the Mean Square Error (MSE)
sense. Additionally, we demonstrate the global asymptotic stability of the
estimator. CO-DKF outperforms other state-of-the-art DKF algorithms, specially
in sparse, highly noisy setups.
