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Learning with tree-based tensor formats : Application to uncertainty quantification in vibroacoustics

Abstract : Many problems require the evaluation of complex parametrized models for many instances of the parameters, particularly for uncertainty quantification. When the model is costly to evaluate, it is usually approximated by another model cheaper to evaluate. The aim of this thesis is to develop statistical learning methods using model classes of functions in treebased tensor formats for the approximation of highdimensional functions, both for supervised and unsupervised learning tasks. These model classes, which are rank-structured functions parametrized by a tree-structured network of low-order tensors, can be interpreted as deep neural networks with particular architecture and activation functions. The approximation is obtained by empirical risk minimization over the set of functions in tree-based tensor format. For a high-dimensional function, or when little information on the function is available, the model class has to be carefully selected. We propose stable learning algorithms that adapt the tree and ranks and select the model based on crossvalidation estimates. Furthermore, some functions might only exhibit a low-rank structure after a suitable change of variables. For such cases, we propose adaptive learning algorithms with model classes combining tree-based tensor formats and changes of variables. The proposed algorithms are applied to uncertainty quantification in vibroacoustics. This thesis is included in the Joint Laboratory of Marine Technology between Naval Group, Centrale Nantes and Université de Nantes, and in the Eval-PI project.
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Submitted on : Thursday, February 27, 2020 - 3:24:09 PM
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Erwan Grelier. Learning with tree-based tensor formats : Application to uncertainty quantification in vibroacoustics. Numerical Analysis [math.NA]. École centrale de Nantes, 2019. English. ⟨NNT : 2019ECDN0070⟩. ⟨tel-02493056⟩

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