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Concept# Positive definiteness

Summary

In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular:

- Positive-definite bilinear form
- Positive-definite function
- Positive-definite function on a group
- Positive-definite functional
- Positive-definite kernel
- Positive-definite matrix
- Positive-definite quadratic form

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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.