Two-level Tucker-TT-QTT format for optimized tensor calculus

Sergey Dolgov and Boris N. Khoromskij

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Submission date: 04. Apr. 2012
Pages: 31
published in: SIAM journal on matrix analysis and applications, 34 (2013) 2, p. 593-623 
DOI number (of the published article): 10.1137/120882597
Bibtex
with the following different title: Two-level QTT-Tucker format for optimized tensor calculus
MSC-Numbers: 65N22, 65F50, 15A69, 33F05, 65F10, 65F30, 65N35
Keywords and phrases: Tensor formats, QTT-format, Tucker format, multilinear algebra, DMRG/ALS, higher dimensions, tensor methods, constructive tensor representations
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Abstract:
We propose a combined tensor format, which encapsulates the benefits of Tucker, Tensor Train (TT) and Quantized TT (QTT) formats. The structure is composed of subtensors in TT representations, so the approximation problem is proven to be stable. We describe all important algebraic and optimization operations, which are recast to the TT routines. Several examples on explicit function and operator representations are provided. The asymptotic storage complexity is at most cubic in the rank parameter, that is larger than for the QTT format, but the numerical examples manifest, that the ranks in the two-level format increase usually slower with the approximation accuracy than the QTT ones. In particular, we observe, that high rank peaks, which usually occur in the QTT representation, are significantly relaxed. Thus the reduced costs can be achieved.