QTT-rank-one vectors with QTT-rank-one and full-rank Fourier images

Dmitry Savostyanov

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Submission date: 18. Jul. 2011
Pages: 11
published in: Linear algebra and its applications, 436 (2012) 9, p. 3215-3224 
DOI number (of the published article): 10.1016/j.laa.2011.11.008
Bibtex
MSC-Numbers: 15A23, 15A69, 65F99, 65T50
Keywords and phrases: Multidimensional arrays, Quantics Tensor Train, Fourier transform, data-sparse formats
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Abstract:
Quantics tensor train (QTT), a new data-sparse format for one– and multi–dimensional vectors, is based on a bit representation of mode indices followed by a separation of variables. A radix-2 reccurence, that lays behind the famous FFT algorithm, can be efficiently applied to vectors in the QTT format. If input and all intermediate vectors of the FFT algorithm have moderate QTT ranks, the resulted QTT-FFT algorithm outperforms the FFT for large vectors. It is instructive to describe a class of such vectors explicitly. We find all vectors that have QTT ranks one on input, intermediate steps and output of the FFT algorithm. We also give an example of QTT-rank-one vector that has the Fourier image with full QTT ranks. By numerical experiments we show that for certain rank-one vectors with full-rank Fourier images, the practical ε–ranks remain moderate for large mode sizes.