Abstract
In this paper, we obtain explicit representations of several multivariate functions in the Tensor Train (TT) format and explicit TT-representations of tensors that stem from the tensorization of univariate functions on grids. Previous results contained only estimates on the number of parameters (tensor ranks), and this paper fills this gap by providing explicit low-parametric representations for these functions and tensors.
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Notes
Canonical ranks depend on the field: for the complex field the rank is equal to 2, but for the real field the upper bound is d, which is assumed to be tight.
References
Beylkin, G., Mohlenkamp, M.J.: Numerical operator calculus in higher dimensions. Proc. Natl. Acad. Sci. USA 99, 10246–10251 (2002)
Beylkin, G., Mohlenkamp, M.J.: Algorithms for numerical analysis in high dimensions. SIAM J. Sci. Comput. 26, 2133–2159 (2005)
Bieri, M., Schwab, C.: Sparse high order FEM for elliptic sPDEs. Comput. Methods Appl. Mech. Eng. 198, 1149–1170 (2009)
Buhmann, M.: Multivariate cardinal interpolation with radial-basis functions. Constr. Approx. 6, 225–255 (1990)
Buhmann, M.: Radial basis functions. Acta Numer. 9, 1–38 (2000)
Bungartz, H.-J., Griebel, M., Röschke, D., Zenger, C.: Pointwise convergence of the combination technique for Laplace’s equation. East-West J. Numer. Math. 2, 21–45 (1994)
Caroll, J.D., Chang, J.J.: Analysis of individual differences in multidimensional scaling via n-way generalization of Eckart–Young decomposition. Psychometrika 35, 283–319 (1970)
de Lathauwer, L., de Moor, B., Vandewalle, J.: A multilinear singular value decomposition. SIAM J. Matrix Anal. Appl. 21, 1253–1278 (2000)
de Silva, V., Lim, L.-H.: Tensor rank and the ill-posedness of the best low-rank approximation problem. SIAM J. Matrix Anal. Appl. 30, 1084–1127 (2008)
Espig, M., Grasedyck, L., Hackbusch, W.: Black box low tensor rank approximation using fibre-crosses. Constr. Approx. 30, 557–597 (2009)
Falco, A., Hackbusch, W.: On minimal subspaces in tensor representation. Preprint 70, MPI MIS, Leipzig (2010)
Garcke, J., Griebel, M., Thess, M.: Data mining with sparse grids. Computing 67, 225–253 (2001)
Grasedyck, L.: Polynomial approximation in hierarchical Tucker format by vector-tensorization. DFG-SPP1324 Preprint 43, Philipps-Univ., Marburg (2010)
Hackbusch, W., Khoromskij, B.N.: Low-rank Kronecker-product approximation to multi-dimensional nonlocal operators. I. Separable approximation of multi-variate functions. Computing 76, 177–202 (2006)
Hackbusch, W., Kühn, S.: A new scheme for the tensor representation. J. Fourier Anal. Appl. 15, 706–722 (2009)
Harshman, R.A.: Foundations of the PARAFAC procedure: models and conditions for an explanatory multimodal factor analysis. UCLA Working Papers in Phonetics, 16, pp. 1–84 (1970)
Hastad, J.: Tensor rank is NP-complete. J. Algorithms 11, 644–654 (1990)
Khoromskaia, V.: Numerical solution of the Hartree–Fock equation by multilevel tensor-structured methods. PhD thesis, TU, Berlin (2010)
Khoromskij, B.N.: Tensor-structured preconditioners and approximate inverse of elliptic operators in ℝd. Constr. Approx. 599–620 (2009)
Khoromskij, B.N.: Fast and accurate tensor approximation of multivariate convolution with linear scaling in dimension. J. Comput. Appl. Math. 234, 3122–3139 (2010)
Khoromskij, B.N.: \(\mathcal{O}(d \log n)\)—quantics approximation of N−d tensors in high-dimensional numerical modeling. Constr. Approx. 34, 257–280 (2011)
Khoromskij, B.N., Khoromskaia, V.: Multigrid accelerated tensor approximation of function related multidimensional arrays. SIAM J. Sci. Comput. 31, 3002–3026 (2009)
Khoromskij, B.N., Khoromskaia, V., Chinnamsetty, S.R., Flad, H.-J.: Tensor decomposition in electronic structure calculations on 3D Cartesian grids. J. Comput. Phys. 228, 5749–5762 (2009)
Kolda, T.G., Bader, B.W.: Tensor decompositions and applications. SIAM Rev. 51, 455–500 (2009)
Oseledets, I.V.: Lower bounds for separable approximations of the Hilbert kernel. Mat. Sb. 198, 137–144 (2007)
Oseledets, I.V.: Approximation of matrices with logarithmic number of parameters. Dokl. Math. 428, 23–24 (2009)
Oseledets, I.V.: Compact matrix form of the d-dimensional tensor decomposition. Preprint 2009-01, INM RAS, Moscow (2009)
Oseledets, I.V.: Approximation of 2d×2d matrices using tensor decomposition. SIAM J. Matrix Anal. Appl. 31, 2130–2145 (2010)
Oseledets, I.V.: Tensor-train decomposition. SIAM J. Sci. Comput. 33, 2295–2317 (2011)
Oseledets, I.V., Tyrtyshnikov, E.E.: Breaking the curse of dimensionality, or how to use SVD in many dimensions. SIAM J. Sci. Comput. 31, 3744–3759 (2009)
Oseledets, I.V., Tyrtyshnikov, E.E.: TT-cross approximation for multidimensional arrays. Linear Algebra Appl. 432, 70–88 (2010)
Östlund, S., Rommer, S.: Thermodynamic limit of density matrix renormalization. Phys. Rev. Lett. 75, 3537–3540 (1995)
Sloan, I., Wozniakowski, H.: When are quasi-Monte Carlo algorithms efficient for high dimensional integrals. J. Complex. 14, 1–33 (1998)
Tucker, L.R.: Some mathematical notes on three-mode factor analysis. Psychometrika 31, 279–311 (1966)
Vidal, G.: Efficient classical simulation of slightly entangled quantum computations. Phys. Rev. Lett. 91, 147902 (2003)
Wang, X., Sloan, I.H.: Why are high-dimensional finance problems often of low effective dimension? SIAM J. Sci. Comput. 27, 159–183 (2006)
Acknowledgements
I am thankful to the anonymous referees. Their comments helped to improve the paper a lot.
Supported by RFBR grants, 12-01-00546-a, 12-01-33013-mol-a-ved, 11-01-12137-ofi-m-2011, 11-01-00549-a, by Rus. Gov. Contracts Π1112, 14.740.11.0345, 14.740.11.1067, 16.740.12.0727 by Rus. President grant MK-140.2011.1, by Priority Research Program OMN-3, by Dmitriy Zimin Dynasty Foundation. Part of this work was done during the stay of the author in Max-Planck Institute for Mathematics in Sciences, Leipzig.
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Communicated by Wolfgang Dahmen.
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Oseledets, I.V. Constructive Representation of Functions in Low-Rank Tensor Formats. Constr Approx 37, 1–18 (2013). https://doi.org/10.1007/s00365-012-9175-x
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DOI: https://doi.org/10.1007/s00365-012-9175-x