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Statistical Learning in Tree-Based Tensor Format

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      Abstract

      Tensor methods are widely used tools for the approximation of high dimensional functions. Such problems are encountered in uncertainty quantification and statistical learning, where the high dimensionality imposes to use specific techniques, such as rank-structured approximations [1].

      In this work, we introduce a statistical learning algorithm for the approximation in tree-based tensor format, which are tensor networks whose graphs are dimension partition trees. This tensor format includes the Tucker format, the Tensor-Train format, as well as the more general Hierarchical tensor formats [4]. It can be interpreted as a deep neural network with a particular architecture [2].

      The proposed algorithm uses random evaluations of a function to provide a tree-based tensor approximation, with adaptation of the tree-based rank by using a heuristic criterion based on the higher-order singular values to select the ranks to increase, and of the approximation spaces of the leaves of the tree.

      We then present a learning algorithm for the approximation under the form u(x) ≈ v(z_1,...,z_m) where v is a tensor in tree-based format and the z_i = g_i(x), 1 ≤ i ≤ m, are new variables. A strategy based on the projection pursuit regression [3] is proposed to compute the mappings g_i and increase the effective dimension m.

      The methods are illustrated on different examples to show their efficiency and adaptability as well as the power of representation of the tree-based tensor format, possibly combined with changes of variables.

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      Affiliations
      [1 ]Centrale Nantes, GeM (UMR CNRS 6183)
      [2 ]Université de Nantes, GeM (UMR CNRS 6183)
      [3 ]Centrale Nantes, LMJL (UMR CNRS 6629)
      [* ]Correspondence: erwan.grelier@ 123456ec-nantes.fr
      Journal
      ScienceOpen Posters
      ScienceOpen
      27 April 2018
      10.14293/P2199-8442.1.SOP-MATH.JHXASX.v1
      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.

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