Convergence analysis of the information matrix in Gaussian belief propagation

Gaussian belief propagation (BP) has been widely used for distributed estimation in large-scale networks such as the smart grid, communication networks, and social networks, where local measurements/observations are scattered over a wide geographical area. However, the convergence of Gaus- sian BP is still an open issue. In this paper, we consider the convergence of Gaussian BP, focusing in particular on the convergence of the information matrix. We show analytically that the exchanged message information matrix converges for arbitrary positive semidefinite initial value, and its dis- tance to the unique positive definite limit matrix decreases exponentially fast.Comment: arXiv admin note: substantial text overlap with arXiv:1611.0201

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

Structure-Based Bayesian Sparse Reconstruction

Sparse signal reconstruction algorithms have attracted research attention due to their wide applications in various fields. In this paper, we present a simple Bayesian approach that utilizes the sparsity constraint and a priori statistical information (Gaussian or otherwise) to obtain near optimal estimates. In addition, we make use of the rich structure of the sensing matrix encountered in many signal processing applications to develop a fast sparse recovery algorithm. The computational complexity of the proposed algorithm is relatively low compared with the widely used convex relaxation methods as well as greedy matching pursuit techniques, especially at a low sparsity rate.Comment: 29 pages, 15 figures, accepted in IEEE Transactions on Signal Processing (July 2012

Convergence Analysis of the Variance in Gaussian Belief Propagation

It is known that Gaussian belief propagation (BP) is a low-complexity algorithm for (approximately) computing the marginal distribution of a high dimensional Gaussian distribu- tion. However, in loopy factor graph, it is important to determine whether Gaussian BP converges. In general, the convergence conditions for Gaussian BP variances and means are not nec- essarily the same, and this paper focuses on the convergence condition of Gaussian BP variances. In particular, by describing the message-passing process of Gaussian BP as a set of updating functions, the necessary and sufficient convergence conditions of Gaussian BP variances are derived under both synchronous and asynchronous schedulings, with the converged variances proved to be independent of the initialization as long as it is chosen from the proposed set. The necessary and sufficient convergence condition is further expressed in the form of a semi-definite programming (SDP) optimization problem, thus can be verified more efficiently compared to the existing convergence condition based on compu- tation tree. The relationship between the proposed convergence condition and the existing one based on computation tree is also established analytically. Numerical examples are presented to corroborate the established theories.published_or_final_versio

Decentralized Turbo Bayesian ompressed Sensing with application to UWB Systems

In many situations, there exist plenty of spatial and temporal redundancies in original signals. Based on this observation, a novel Turbo Bayesian Compressed Sensing (TBCS) algorithm is proposed to provide an efficient approach to transfer and incorporate this redundant information for joint sparse signal reconstruction. As a case study, the TBCS algorithm is applied in Ultra-Wideband (UWB) systems. A space-time TBCS structure is developed for exploiting and incorporating the spatial and temporal a priori information for space-time signal reconstruction. Simulation results demonstrate that the proposed TBCS algorithm achieves much better performance with only a few measurements in the presence of noise, compared with the traditional Bayesian Compressed Sensing (BCS) and multitask BCS algorithms