## Convergence Analysis of Distributed Inference with Vector-Valued Gaussian Belief Propagation

*Jian Du, Shaodan Ma, Yik-Chung Wu, Soummya Kar, JosÃ© M. F. Moura*; 18(172):1−38, 2018.

### Abstract

This paper considers inference over distributed linear Gaussian
models using factor graphs and Gaussian belief propagation
(BP). The distributed inference algorithm involves only local
computation of the information matrix and of the mean vector,
and message passing between neighbors. Under broad conditions,
it is shown that the message information matrix converges to a
unique positive definite limit matrix for arbitrary positive
semidefinite initialization, and it approaches an arbitrarily
small neighborhood of this limit matrix at an exponential rate.
A necessary and sufficient convergence condition for the belief
mean vector to converge to the optimal centralized estimator is
provided under the assumption that the message information
matrix is initialized as a positive semidefinite matrix.
Further, it is shown that Gaussian BP always converges when the
underlying factor graph is given by the union of a forest and a
single loop. The proposed convergence condition in the setup of
distributed linear Gaussian models is shown to be strictly
weaker than other existing convergence conditions and
requirements, including the Gaussian Markov random field based
walk-summability condition, and applicable to a large class of
scenarios.

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