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Mat. Sb., 2003, Volume 194, Number 6, Pages 147–160 (Mi msb747)  

This article is cited in 48 scientific papers (total in 48 papers)

Tensor approximations of matrices generated by asymptotically smooth functions

E. E. Tyrtyshnikov

Institute of Numerical Mathematics, Russian Academy of Sciences

Abstract: For a broad class of matrices (discrete analogues of typical integral operators) their approximability by a sum of direct products of matrices of smaller size is demonstrated. Estimates of the number of terms (the tensor rank) and the corresponding error are obtained. It is shown that, as a method of data compression, tensor approximations provide superlinear compression.

DOI: https://doi.org/10.4213/sm747

Full text: PDF file (275 kB)
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English version:
Sbornik: Mathematics, 2003, 194:6, 941–954

Bibliographic databases:

UDC: 512.64+517.5
MSC: Primary 15A45, 15A72; Secondary 65F05, 65F30
Received: 31.10.2002

Citation: E. E. Tyrtyshnikov, “Tensor approximations of matrices generated by asymptotically smooth functions”, Mat. Sb., 194:6 (2003), 147–160; Sb. Math., 194:6 (2003), 941–954

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Ford J.M., Oseledetst I.V., Tyrtyshnikov E.E., “Matrix approximations and solvers using tensor products and non-standard wavelet transforms related to irregular grids”, Russian J. Numer. Anal. Math. Modelling, 19:2 (2004), 185–204  crossref  mathscinet  zmath  isi  elib
    2. Ford J.M., Tyrtyshnikov E.E., “Solving linear systems using wavelet compression combined with Kronecker product approximation”, Numer. Algorithms, 40:2 (2005), 125–135  crossref  mathscinet  zmath  adsnasa  isi  elib
    3. Tian Yongge, “Some rank equalities and inequalities for Kronecker products of matrices”, Linear Multilinear Algebra, 53:6 (2005), 445–454  crossref  mathscinet  zmath  isi  elib
    4. Beylkin G., Mohlenkamp M.J., “Algorithms for numerical analysis in high dimensions”, SIAM J. Sci. Comput., 26:6 (2005), 2133–2159  crossref  mathscinet  zmath  isi  elib
    5. Gavrilyuk I.P., Hackbusch W., Khoromskij B.N., “Hierarchical tensor-product approximation to the inverse and related operators for high-dimensional elliptic problems”, Computing, 74:2 (2005), 131–157  crossref  mathscinet  zmath  isi  elib
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    7. Schneider R., Weber T., “Wavelets for density matrix computation in electronic structure calculation”, Appl. Numer. Math., 56:10-11 (2006), 1383–1396  crossref  mathscinet  zmath  isi  elib
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    16. Hackbusch W., Khoromskij B.N., “Tensor-product approximation to multidimensional integral operators and Green's functions”, SIAM J. Matrix Anal. Appl., 30:3 (2008), 1233–1253  crossref  mathscinet  isi  elib
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    18. Hackbusch W., Khoromskij B.N., Tyrtyshnikov E.E., “Approximate iterations for structured matrices”, Numer. Math., 109:3 (2008), 365–383  crossref  mathscinet  zmath  isi  elib
    19. S. A. Goreinov, “On cross approximation of multi-index arrays”, Dokl. Math., 77:3 (2008), 404–406  crossref  mathscinet  zmath  isi  elib  elib
    20. Oseledets I., “The integral operator with logarithmic kernel has only one positive eigenvalue”, Linear Algebra Appl., 428:7 (2008), 1560–1564  crossref  mathscinet  zmath  isi  elib
    21. Oseledets I.V., Tyrtyshnikov E.E., “Breaking the curse of dimensionality, or how to use SVD in many dimensions”, SIAM J. Sci. Comput., 31:5 (2009), 3744–3759  crossref  mathscinet  zmath  isi  elib
    22. Oseledets I.V., Savostyanov D.V., Tyrtyshnikov E.E., “Cross approximation in tensor electron density computations”, Numer. Linear Algebra Appl., 2009  crossref  mathscinet  isi
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    24. Oseledets I., Tyrtyshnikov E., Zamarashkin N., “Matrix inversion cases with size-independent tensor rank estimates”, Linear Algebra Appl., 431:5-7 (2009), 558–570  crossref  mathscinet  zmath  isi  elib
    25. Tyrtyshnikov E, “Preservation of Linear Constraints in Approximation of Tensors”, Numerical Mathematics-Theory Methods and Applications, 2:4 (2009), 421–426  mathscinet  zmath  isi
    26. Oseledets I., Tyrtyshnikov E., “TT-cross approximation for multidimensional arrays”, Linear Algebra Appl., 432:1 (2010), 70–88  crossref  mathscinet  zmath  isi  elib
    27. I. V. Oseledets, “Approximation of $2^d\times2^d$ Matrices Using Tensor Decomposition”, SIAM J Matrix Anal Appl, 31:4 (2010), 2130  crossref  mathscinet  zmath  isi
    28. Schneider J., “Error estimates for two-dimensional cross approximation”, Journal of Approximation Theory, 162:9 (2010), 1685–1700  crossref  mathscinet  zmath  isi  elib
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    37. Vladimir Kazeev, Oleg Reichmann, Christoph Schwab, “Low-rank tensor structure of linear diffusion operators in the TT and QTT formats”, Linear Algebra and its Applications, 2013  crossref  mathscinet  isi
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