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It is well known that the convergence of Gaussian belief propagation (BP) is not guaranteed in loopy graphs. The classical convergence conditions, including diagonal dominance, walk-summability, and convex decomposition, are derived under pairwise factorizations of the joint Gaussian distribution. However, many applications run Gaussian BP under high-order factorizations, making the classical results not applicable. In this paper, the convergence of Gaussian BP under high-order factorization and asynchronous scheduling is investigated. In particular, three classes of asynchronous scheduling are considered. The first one is the totally asynchronous scheduling, and a sufficient convergence condition is derived. Since the totally asynchronous scheduling represents a broad class of asynchronous scheduling, the derived convergence condition might not be tight for a particular asynchronous schedule. Consequently, the second class of asynchronous scheduling, called quasi-asynchronous scheduling, is considered. Being a subclass of the totally asynchronous scheduling, quasi-asynchronous scheduling possesses a simpler structure, which facilitates the derivation of the necessary and sufficient convergence condition. To get a deeper insight into the quasi-asynchronous scheduling, a third class of asynchronous scheduling, named independent and identically distributed (i.i.d.) quasi-asynchronous scheduling, is further proposed, and the convergence is analyzed in the probabilistic sense. Compared to the synchronous scheduling, it is found that Gaussian BP under the i.i.d. quasi-asynchronous scheduling demonstrates better convergence. Numerical examples and applications are presented to corroborate the newly established theoretical results.
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