KAUST DepartmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Physical Sciences and Engineering (PSE) Division
Earth Science and Engineering Program
Applied Mathematics and Computational Science Program
KAUST Grant NumberCRG3-2156
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AbstractOptimal dimensionality reduction methods are proposed for the Bayesian inference of a Gaussian linear model with additive noise in presence of overabundant data. Three different optimal projections of the observations are proposed based on information theory: the projection that minimizes the Kullback–Leibler divergence between the posterior distributions of the original and the projected models, the one that minimizes the expected Kullback–Leibler divergence between the same distributions, and the one that maximizes the mutual information between the parameter of interest and the projected observations. The first two optimization problems are formulated as the determination of an optimal subspace and therefore the solution is computed using Riemannian optimization algorithms on the Grassmann manifold. Regarding the maximization of the mutual information, it is shown that there exists an optimal subspace that minimizes the entropy of the posterior distribution of the reduced model; a basis of the subspace can be computed as the solution to a generalized eigenvalue problem; an a priori error estimate on the mutual information is available for this particular solution; and that the dimensionality of the subspace to exactly conserve the mutual information between the input and the output of the models is less than the number of parameters to be inferred. Numerical applications to linear and nonlinear models are used to assess the efficiency of the proposed approaches, and to highlight their advantages compared to standard approaches based on the principal component analysis of the observations.
CitationGiraldi L, Le Maître OP, Hoteit I, Knio OM (2018) Optimal projection of observations in a Bayesian setting. Computational Statistics & Data Analysis. Available: http://dx.doi.org/10.1016/j.csda.2018.03.002.
SponsorsThis work is supported by King Abdullah University of Science and Technology Awards CRG3-2156 and OSR-2016-RPP-3268.