Multilevel Toeplitz Matrices Generated by Tensor-Structured Vectors and Convolution with Logarithmic Complexity

By Vladimir A. Kazeev, Boris N. Khoromskij, E. E. Tyrtyshnikov. 2013

100 citations recorded by the source corpus.

Subfields: Computational Mathematics, Computational Theory and Mathematics, Hardware and Architecture

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Citations in the corpus

A literature survey of low‐rank tensor approximation techniques (2013, positive): “For the following matrices, QTT representations have been discussed: Toeplitz matrices [136, 216], (inverse) Laplace operators [139, 205], linear diffusion operators [62, 138].”
QTT Representation of the Hartree and Exchange Operators in Electronic Structure Calculations (2011, unknown): “: 37 2011”
Møller-Plesset (MP2) energy correction using tensor factorization of the grid-based two-electron integrals (2014, methodology): “The QTT approximation method allows us to represent a class of matrices in low QTT rank format [44,45] and to implement the multidimensional FFT and convolution transforms with logarithmic complexity scaling, O(log n) [46,47].”
Low-Rank Explicit QTT Representation of the Laplace Operator and Its Inverse (2012, methodology): “On the other hand, the notation and technique we introduce to deal with explicit QTT representations on paper proves to be useful for other tensors, which are related to particular algorithms: the presented approach was used in [17] to examine the low-rank QTT structure of multilevel Toeplitz matric”
Two-level Tucker-TT-QTT format for optimized tensor calculus (2012, methodology): “To use all the benefits from tensor decompositions for low-dimensional data, we use the QTT format [26, 27, 28, 29, 44, 30, 45, 40].”
Tensor Networks for Dimensionality Reduction and Large-scale Optimization: Part 2 Applications and Future Perspectives (2017, background): “The convolution tensor can then be represented in a QTT format of rank-2 [Kazeev et al., 2013] with core tensors C(2) = ¨ ¨ ¨ = C(D) = S, C(1) = S(1, :, :, :, :), and the last core tensor C(D+1) = [ 0 1 1 0 ] which is of size 2ˆ 1ˆ 1ˆ 2ˆ 1.”
Superfast Fourier Transform Using QTT Approximation (2012, methodology): “The multidimensional convolution of QTT-structured data can be computed also directly using the analytical QTT decomposition of a multilevel Toeplitz matrix [25].”
Tensor Networks for Dimensionality Reduction and Large-scale Optimization: Part 1 Low-Rank Tensor Decompositions (2016, unknown): “…as large as 10 50 entries, down to the aﬀordable levels of 10 7 or even less entries (Os-eledets and Tyrtyshnikov, 2009; Dolgov and Khoromskij, 2013; Kazeev et al. , 2013a, 2014; Kressner et al. , 2014a; Vervliet et al. , 2014; Dolgov and Khoromskij, 2015; Liao et al. , 2015; Bolten et al. , 2016).”