Unconstrained minimization of quadratic functions via min-sum

Gaussian belief propagation is an iterative algorithm for computing the mean of a multivariate Gaussian distribution. Equivalently, the min-sum algorithm can be used to compute the minimum of a multivariate positive definite quadratic function. Although simple sufficient conditions that guarantee the convergence and correctness of these algorithms are known, the algorithms may fail to converge to the correct solution even when restricted to only positive definite quadratic functions. In this work, we propose a novel change to the typical factorization used in GaBP that allows us to construct a variant of GaBP that can solve the minimization problem for arbitrary positive semidefinite matrices while still preserving the distributed message passing nature of GaBP. We prove that the new factorization avoids the major pitfalls of the standard factorization, and we demonstrate empirically that the algorithm can be used to solve problems for which the standard GaBP algorithm would have failed. As quadratic minimization is equivalent to solving a system of linear equations, this work can be applied to solve large positive semidefinite linear systems in many application areas.