A continuous analogue of the tensor-train decomposition
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Publication:1987792
DOI10.1016/J.CMA.2018.12.015zbMATH Open1440.65054arXiv1510.09088OpenAlexW2962947392WikidataQ128708841 ScholiaQ128708841MaRDI QIDQ1987792
Author name not available (Why is that?)
Publication date: 16 April 2020
Published in: (Search for Journal in Brave)
Abstract: We develop new approximation algorithms and data structures for representing and computing with multivariate functions using the functional tensor-train (FT), a continuous extension of the tensor-train (TT) decomposition. The FT represents functions using a tensor-train ansatz by replacing the three-dimensional TT cores with univariate matrix-valued functions. The main contribution of this paper is a framework to compute the FT that employs adaptive approximations of univariate fibers, and that is not tied to any tensorized discretization. The algorithm can be coupled with any univariate linear or nonlinear approximation procedure. We demonstrate that this approach can generate multivariate function approximations that are several orders of magnitude more accurate, for the same cost, than those based on the conventional approach of compressing the coefficient tensor of a tensor-product basis. Our approach is in the spirit of other continuous computation packages such as Chebfun, and yields an algorithm which requires the computation of "continuous" matrix factorizations such as the LU and QR decompositions of vector-valued functions. To support these developments, we describe continuous versions of an approximate maximum-volume cross approximation algorithm and of a rounding algorithm that re-approximates an FT by one of lower ranks. We demonstrate that our technique improves accuracy and robustness, compared to TT and quantics-TT approaches with fixed parameterizations, of high-dimensional integration, differentiation, and approximation of functions with local features such as discontinuities and other nonlinearities.
Full work available at URL: https://arxiv.org/abs/1510.09088
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