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Variable Elimination in the Fourier Domain
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Learning highdimensional probability distributions using tree tensor networks
We consider the problem of the estimation of a highdimensional probability distribution using model classes of functions in treebased tensor formats, a particular case of tensor networks associated with a dimension partition tree. The distribution is assumed to admit a density with respect to a product measure, possibly discrete for handling the case of discrete random variables. After discussing the representation of classical model classes in treebased tensor formats, we present learning algorithms based on empirical risk minimization using a L^2 contrast. These algorithms exploit the multilinear parametrization of the formats to recast the nonlinear minimization problem into a sequence of empirical risk minimization problems with linear models. A suitable parametrization of the tensor in treebased tensor format allows to obtain a linear model with orthogonal bases, so that each problem admits an explicit expression of the solution and crossvalidation risk estimates. These estimations of the risk enable the model selection, for instance when exploiting sparsity in the coefficients of the representation. A strategy for the adaptation of the tensor format (dimension tree and treebased ranks) is provided, which allows to discover and exploit some specific structures of highdimensional probability distributions such as independence or conditional independence. We illustrate the performances of the proposed algorithms for the approximation of classical probabilistic models (such as Gaussian distribution, graphical models, Markov chain).
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