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

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### Abstract

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

###### Affiliations
[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
###### Journal
ScienceOpen Posters
ScienceOpen
27 April 2018
10.14293/P2199-8442.1.SOP-MATH.WTUXCF.v1