Convergence Analysis of Gaussian Belief Propagation under High-Order Factorization and Asynchronous Scheduling
Abstract: 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-sum ability and convex decomposition, are derived under pair wise 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 highorder 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. quasiasynchronous scheduling demonstrates better convergence. Numerical examples and applications are presented to corroborate the newly-established theoretical results. Existing system: Although Gaussian BP under high-order factorization has been applied in many applications the convergence of Gaussian BP is not well understood. In general, the Gaussian BP message variances are known to converge . For the convergence of Gaussian BP beliefs, existing works mostly rely on numerical simulations, asymptotical analysis, or restricting the factor graph to be a single loop connected to multiple chains or trees. Moreover, all these analyses only consider the synchronous scheduling, where each round of message update has to be completed for the whole network before starting the next one. Obviously, there are many distributed computing systems where some nodes are slow or reluctant to update their messages, thus stagnating the message update of other nodes. Proposed system: Pdf, the high-order convex decomposition always exists, while this is not true for the convex decomposition. With the conventional convex decomposition being a special case of the proposed high-order convex decomposition, all the subsequent results would hold in the convex decomposition model. Besides guaranteeing existence, the high-order convex decomposition model finds wide-spread applications, including distributed uplink macro-diversity , solving systems of linear equations , distributed downlink beam forming ], data detection, and distributed inference . Detailed examples will be further illustrated in Section V. Advantages: Under asynchronous scheduling, each BP message may only be updated at a subset of iterations rather than all iterations.
Moreover, when one message is being updated, the most recent values of the neighboring messages used for the updating may not be available. In particular, in asynchronous scheduling, the updating time of the message. Disadvantages: Recently, investigates the convergence of Gaussian BP under high-order factorization in linear Gaussian model, where it is proved that Gaussian BP under high-order factorization is guaranteed to converge in the factor graph containing only one single loop connected to multiple chains or trees. However, only considers the synchronous scheduling. In contrast, in the following, we focus on asynchronous scheduling. However, for loopy graphs, the BP beliefs may fail to converge. For the existing convergence conditions of Gaussian BP, including diagonal dominance, walk-sum ability convex decomposition, and the necessary and sufficient condition, they are only valid under pair wise factorization. For high-order factorization, these conditions cannot be applied. Modules:
convergence analysis of gaussian bp under high-order convex decomposition and quasi-asynchronous scheduling : Under totally asynchronous scheduling, the convergence condition of the BP belief variances in Theorem 2 is necessary and sufficient, but the convergence condition of the BP belief means in Theorem 3 is only sufficient. Since the convergence condition of the BP belief means in Theorem 3 holds for any totally asynchronous schedule, this condition might be too conservative for most of the totally asynchronous schedules. To this end, we consider a subclass of totally asynchronous
scheduling, which is called quasi-asynchronous scheduling, and its necessary and sufficient convergence condition of the BP belief means can be obtained.