Truncating the loop series expansion for Belief Propagation

Recently, M. Chertkov and V.Y. Chernyak derived an exact expression for the partition sum (normalization constant) corresponding to a graphical model, which is an expansion around the Belief Propagation solution. By adding correction terms to the BP free energy, one for each "generalized loop" in the factor graph, the exact partition sum is obtained. However, the usually enormous number of generalized loops generally prohibits summation over all correction terms. In this article we introduce Truncated Loop Series BP (TLSBP), a particular way of truncating the loop series of M. Chertkov and V.Y. Chernyak by considering generalized loops as compositions of simple loops. We analyze the performance of TLSBP in different scenarios, including the Ising model, regular random graphs and on Promedas, a large probabilistic medical diagnostic system. We show that TLSBP often improves upon the accuracy of the BP solution, at the expense of increased computation time. We also show that the performance of TLSBP strongly depends on the degree of interaction between the variables. For weak interactions, truncating the series leads to significant improvements, whereas for strong interactions it can be ineffective, even if a high number of terms is considered.Comment: 31 pages, 12 figures, submitted to Journal of Machine Learning Researc

An exact smooth Gowdy-symmetric generalized Taub-NUT solution

In a recent paper (Beyer and Hennig, 2012 [9]), we have introduced a class of inhomogeneous cosmological models: the smooth Gowdy-symmetric generalized Taub-NUT solutions. Here we derive a three-parametric family of exact solutions within this class, which contains the two-parametric Taub solution as a special case. We also study properties of this solution. In particular, we show that for a special choice of the parameters, the spacetime contains a curvature singularity with directional behaviour that can be interpreted as a "true spike" in analogy to previously known Gowdy symmetric solutions with spatial T3-topology. For other parameter choices, the maximal globally hyperbolic region is singularity-free, but may contain "false spikes".Comment: 39 pages, 3 figure

Global sensitivity analysis of computer models with functional inputs

Global sensitivity analysis is used to quantify the influence of uncertain input parameters on the response variability of a numerical model. The common quantitative methods are applicable to computer codes with scalar input variables. This paper aims to illustrate different variance-based sensitivity analysis techniques, based on the so-called Sobol indices, when some input variables are functional, such as stochastic processes or random spatial fields. In this work, we focus on large cpu time computer codes which need a preliminary meta-modeling step before performing the sensitivity analysis. We propose the use of the joint modeling approach, i.e., modeling simultaneously the mean and the dispersion of the code outputs using two interlinked Generalized Linear Models (GLM) or Generalized Additive Models (GAM). The ``mean'' model allows to estimate the sensitivity indices of each scalar input variables, while the ``dispersion'' model allows to derive the total sensitivity index of the functional input variables. The proposed approach is compared to some classical SA methodologies on an analytical function. Lastly, the proposed methodology is applied to a concrete industrial computer code that simulates the nuclear fuel irradiation

A Self-learning Algebraic Multigrid Method for Extremal Singular Triplets and Eigenpairs

A self-learning algebraic multigrid method for dominant and minimal singular triplets and eigenpairs is described. The method consists of two multilevel phases. In the first, multiplicative phase (setup phase), tentative singular triplets are calculated along with a multigrid hierarchy of interpolation operators that approximately fit the tentative singular vectors in a collective and self-learning manner, using multiplicative update formulas. In the second, additive phase (solve phase), the tentative singular triplets are improved up to the desired accuracy by using an additive correction scheme with fixed interpolation operators, combined with a Ritz update. A suitable generalization of the singular value decomposition is formulated that applies to the coarse levels of the multilevel cycles. The proposed algorithm combines and extends two existing multigrid approaches for symmetric positive definite eigenvalue problems to the case of dominant and minimal singular triplets. Numerical tests on model problems from different areas show that the algorithm converges to high accuracy in a modest number of iterations, and is flexible enough to deal with a variety of problems due to its self-learning properties.Comment: 29 page