Abstract
In this article we introduce a black box type algorithm for the approximation of tensors A in high dimension d. The algorithm adaptively determines the positions of entries of the tensor that have to be computed or read, and using these (few) entries it constructs a low rank tensor approximation X that minimizes the ℓ 2-distance between A and X at the chosen positions. The full tensor A is not required, only the evaluation of A at a few positions. The minimization problem is solved by Newton’s method, which requires the computation and evaluation of the Hessian. For efficiency reasons the positions are located on fiber-crosses of the tensor so that the Hessian can be assembled and evaluated in a data-sparse form requiring a complexity of \(\mathcal{O}(Pd)\) , where P is the number of fiber-crosses and d the order of the tensor.
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References
Beylkin, G., Mohlenkamp, M.J.: Algorithms for numerical analysis in high dimensions. SIAM J. Sci. Comput. 26(6), 2133–2159 (2005)
Beylkin, G., Garcke, J., Mohlenkamp, M.: Multivariate regression and machine learning with sums of separable functions. SIAM J. Sci. Comput. 31, 1840–1857 (2009)
Börm, S., Grasedyck, L.: Hybrid cross approximation of integral operators. Numer. Math. 101, 221–249 (2005)
Chinnamsetty, S.R., Espig, M., Khoromskij, B.N., Hackbusch, W., Flad, H.J.: Tensor product approximation with optimal rank in quantum chemistry. J. Chem. Phys. 127(8) (2007)
De Lathauwer, L., De Moor, B., Vandewalle, J.: A multilinear singular value decomposition. SIAM J. Matrix Anal. Appl. 21(4), 1253–1278 (2000)
de Silva, V., Lim, L.-H.: Tensor rank and the ill-posedness of the best low-rank approximation problem. Technical Report SCCM-06-06, Stanford University (2006)
Espig, M.: Effiziente Bestapproximation mittels Summen von Elementartensoren in hohen Dimensionen. PhD thesis, Universität Leipzig (2008)
Flad, H.-J., Khoromskij, B.N., Savostyanov, D., Tyrtyshnikov, E.E.: Verification of the cross 3D algorithm on quantum chemistry data. Russ. J. Numer. Anal. Math. Model. 23, 329–344 (2008)
Grasedyck, L.: Existence and computation of low Kronecker-rank approximations for large linear systems of tensor product structure. Computing 72, 247–265 (2004)
Khoromskij, B.N., Khoromskaia, V.: Multigrid accelerated approximation of function related multi-dimensional arrays. Preprint 40/2008, Max Planck Institute for Mathematics in the Sciences (2008)
Kolda, T.G.: Orthogonal tensor decompositions. SIAM J. Matrix Anal. Appl. 23(1), 243–255 (2001)
Oseledets, I.V., Savost’yanov, D.V.: Minimization methods for approximating tensors and their comparison. Comput. Math. Math. Phys. 46(10), 1641–1650 (2006)
Oseledets, I.V., Tyrtyshnikov, E.E., Savost’yanov, D.V.: Tucker dimensionality reduction of three-dimensional arrays in linear time. SIAM J. Matrix Anal. Appl. (2008, to appear)
Paatero, P.: A weighted non-negative least squares algorithm for three-way ‘PARAFAC’ factor analysis. Chemometrics Intel. Lab. Syst. 38, 223–242 (1997)
Ten Berge, J.M.F.: Kruskal’s polynomial for 2×2×2 arrays and a generalization to 2×n×n. Psychometrika 56, 631–636 (1991)
Young, F.W., de Leeuw, J., Takane, Y.: Additive structure in qualitative data: an alternating least squares method with optimal scaling features. Psychometrika 41, 471–503 (1976)
Zhang, T., Golub, G.H.: Rank-one approximation to high order tensors. SIAM J. Matrix Anal. Appl. 23(2), 534–550 (2001)
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Communicated by Christoph Schwab.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Espig, M., Grasedyck, L. & Hackbusch, W. Black Box Low Tensor-Rank Approximation Using Fiber-Crosses. Constr Approx 30, 557–597 (2009). https://doi.org/10.1007/s00365-009-9076-9
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DOI: https://doi.org/10.1007/s00365-009-9076-9