Matrix analysis
Tree consistency and bounds on the performance of the max-product algorithm and its generalizations
Statistics and Computing
Correctness of Local Probability Propagation in Graphical Models with Loops
Neural Computation
Walk-Sums and Belief Propagation in Gaussian Graphical Models
The Journal of Machine Learning Research
A Linear Programming Approach to Max-Sum Problem: A Review
IEEE Transactions on Pattern Analysis and Machine Intelligence
Approximate inference in gaussian graphical models
Approximate inference in gaussian graphical models
Convergence of min-sum message passing for quadratic optimization
IEEE Transactions on Information Theory
Fixing convergence of Gaussian belief propagation
ISIT'09 Proceedings of the 2009 IEEE international conference on Symposium on Information Theory - Volume 3
Graph covers and quadratic minimization
Allerton'09 Proceedings of the 47th annual Allerton conference on Communication, control, and computing
Convergent message passing algorithms: a unifying view
UAI '09 Proceedings of the Twenty-Fifth Conference on Uncertainty in Artificial Intelligence
Convergence of min-sum message-passing for convex optimization
IEEE Transactions on Information Theory
Norm-product belief propagation: primal-dual message-passing for approximate inference
IEEE Transactions on Information Theory
Loopy belief propagation and Gibbs measures
UAI'02 Proceedings of the Eighteenth conference on Uncertainty in artificial intelligence
On the optimality of solutions of the max-product belief-propagation algorithm in arbitrary graphs
IEEE Transactions on Information Theory
Signal-space characterization of iterative decoding
IEEE Transactions on Information Theory
Tree-based reparameterization framework for analysis of sum-product and related algorithms
IEEE Transactions on Information Theory
MAP estimation via agreement on trees: message-passing and linear programming
IEEE Transactions on Information Theory
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Gaussian belief propagation (GaBP) is an iterative algorithm for computing the mean (and variances) of a multivariate Gaussian distribution, or equivalently, the minimum of a multivariate positive definite quadratic function. Sufficient conditions, such as walk-summability, that guarantee the convergence and correctness of GaBP are known, but GaBP may fail to converge to the correct solution given an arbitrary positive definite covariance matrix. As was observed by Malioutov et al. (2006), the GaBP algorithm fails to converge if the computation trees produced by the algorithm are not positive definite. In this work, we will show that the failure modes of the GaBP algorithm can be understood via graph covers, and we prove that a parameterized generalization of the min-sum algorithm can be used to ensure that the computation trees remain positive definite whenever the input matrix is positive definite. We demonstrate that the resulting algorithm is closely related to other iterative schemes for quadratic minimization such as the Gauss-Seidel and Jacobi algorithms. Finally, we observe, empirically, that there always exists a choice of parameters such that the above generalization of the GaBP algorithm converges.