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      Principal component analysis and optimal weighted least-squares method for training tree tensor networks



            One of the most challenging tasks in computational science is the approximation of high-dimensional functions. Most of the time, only a few information on the functions is available, and approximating high-dimensional functions requires exploiting low-dimensional structures of these functions. In this work, the approximation of a function u is built using point evaluations of the function, where these evaluations are selected adaptively. Such problems are encountered when the function represents the output of a black-box computer code, a system or a physical experiment for a given value of a set of input variables. This algorithm relies on an extension of principal components analysis (PCA) to multivariate functions in order to estimate the tensors $v_{\alpha}$. In practice, the PCA is realized on sample-based projections of the function u, using interpolation or least-squares regression. Least-squares regression can provide a stable projection but it usually requires a high number of evaluations of u, which is not affordable when one evaluation is very costly. In [1] the authors proposed an optimal weighted least-squares method, with a choice of weights and samples that garantee an approximation error of the order of the best approximation error using a minimal number of samples. We here present an extension of this methodology for the approximation in tree-based format, where optimal weighted least-squares method is used for the projection onto tensor product spaces. This approach will be compared with a strategy using standard least-squares method or interpolation (as proposed in [2]).


            Author and article information

            ScienceOpen Posters
            27 April 2018
            [1 ]Centrale Nantes, LMJL UMR 6629, CEA/DAM/DIF, F-91297, Arpajon
            [2 ]Centrale Nantes, LMJL UMR 6629
            [3 ]CEA/DAM/DIF, F-91297, Arpajon
            [* ]Correspondence: cecile.haberstich@ 123456ec-nantes.fr
            Copyright © 2018

            This work has been published open access under Creative Commons Attribution License CC BY 4.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Conditions, terms of use and publishing policy can be found at www.scienceopen.com.

            Applied mathematics,Applications,Statistics,Data analysis,Mathematics,Mathematical modeling & Computation
            PCA,optimal weighted least-squares method,Tree-based tensor,low-rank


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