2020/02/14 by Nick Alger, Peng Chen, Alger, Nick +3 · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #Computational Physics and Python Applications #Tensor decomposition and applications #Model Reduction and Neural Networks
paper · pdf · doi:10.48550/arxiv.2002.06244
We present a method for converting tensors into tensor train format based on\nactions of the tensor as a vector-valued multilinear function. Existing methods\nfor constructing tensor trains require access to "array entries" of the tensor\nand are therefore inefficient or computationally prohibitive if the tensor is\naccessible only through its action, especially for high order tensors. Our\nmethod permits efficient tensor train compression of large high order\nderivative tensors for nonlinear mappings that are implicitly defined through\nthe solution of a system of equations. Array entries of these derivative\ntensors are not directly accessible, but actions of these tensors can be\ncomputed efficiently via a procedure that we discuss. Such tensors are often\namenable to tensor train compression in theory, but until now no efficient\nalgorithm existed to convert them into tensor train format. We demonstrate our\nmethod by compressing a Hilbert tensor of size 41 \× 42 \× 43 \×\n44 \× 45, and by forming high order (up to 5^\th order\nderivatives/6^\th order tensors) Taylor series surrogates of the\nnoise-whitened parameter-to-output map for a stochastic partial differential\nequation with boundary output.\n