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Principal component analysis for the approximation of high-dimensional functions in tree-based tensor formats Nouy, Anthony


We present an algorithm for the approximation of high-dimensional functions using tree-based low-rank approximation formats (tree tensor networks). A multivariate function is here considered as an element of a Hilbert tensor space of functions defined on a product set equipped with a probability measure. The algorithm only requires evaluations of functions on a structured set of points which is constructed adaptively. The algorithm is a variant of higher-order singular value decomposition which constructs a hierarchy of subspaces associated with the different nodes of a dimension partition tree and a corresponding hierarchy of interpolation operators. Optimal subspaces are estimated using empirical principal component analysis of interpolations of partial random evaluations of the function. The algorithm is able to provide an approximation in any tree-based format with either a prescribed rank or a prescribed relative error, with a number of evaluations of the order of the storage complexity of the approximation format. Reference : A. Nouy. Higher-order principal component analysis for the approximation of tensors in tree-based low rank formats. arxiv preprint arXiv:1705.00880, 2017.

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