Message-Passing Algorithms for Quadratic Minimization

Gaussian belief propagation (GaBP) is an iterative algorithm for computing the mean of a multivariate Gaussian distribution, or equivalently, the minimum of a multivariate positive definite quadratic function. Sufficient conditions, such as walk-summability, that guarantee the convergence and correctness of GaBP are known, but GaBP may fail to converge to the correct solution given an arbitrary positive definite quadratic function. As was observed in previous work, the GaBP algorithm fails to converge if the computation trees produced by the algorithm are not positive definite. In this work, we will show that the failure modes of the GaBP algorithm can be understood via graph covers, and we prove that a parameterized generalization of the min-sum algorithm can be used to ensure that the computation trees remain positive definite whenever the input matrix is positive definite. We demonstrate that the resulting algorithm is closely related to other iterative schemes for quadratic minimization such as the Gauss-Seidel and Jacobi algorithms. Finally, we observe, empirically, that there always exists a choice of parameters such that the above generalization of the GaBP algorithm converges

Cooperative Synchronization in Wireless Networks

Synchronization is a key functionality in wireless network, enabling a wide variety of services. We consider a Bayesian inference framework whereby network nodes can achieve phase and skew synchronization in a fully distributed way. In particular, under the assumption of Gaussian measurement noise, we derive two message passing methods (belief propagation and mean field), analyze their convergence behavior, and perform a qualitative and quantitative comparison with a number of competing algorithms. We also show that both methods can be applied in networks with and without master nodes. Our performance results are complemented by, and compared with, the relevant Bayesian Cram\'er-Rao bounds

Distributed Convergence Verification for Gaussian Belief Propagation

Gaussian belief propagation (BP) is a computationally efficient method to approximate the marginal distribution and has been widely used for inference with high dimensional data as well as distributed estimation in large-scale networks. However, the convergence of Gaussian BP is still an open issue. Though sufficient convergence conditions have been studied in the literature, verifying these conditions requires gathering all the information over the whole network, which defeats the main advantage of distributed computing by using Gaussian BP. In this paper, we propose a novel sufficient convergence condition for Gaussian BP that applies to both the pairwise linear Gaussian model and to Gaussian Markov random fields. We show analytically that this sufficient convergence condition can be easily verified in a distributed way that satisfies the network topology constraint.Comment: accepted by Asilomar Conference on Signals, Systems, and Computers, 2017, Asilomar, Pacific Grove, CA. arXiv admin note: text overlap with arXiv:1706.0407