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

Message-Passing Algorithms for Quadratic Minimization