Abstract
The paper introduces a hybrid approach to the CUR-type decomposition of tensors in the Tucker format. The idea of the hybrid algorithm is to write a tensor \({\mathscr {X}}\) as a product of a core tensor \({\mathscr {S}}\), a matrix C obtained by extracting mode-k fibers of \({\mathscr {X}}\), and matrices \(Z_j\), \(j=1,\ldots ,k-1,k+1,\ldots ,d\), chosen to minimize the approximation error. The approximation can easily be modified to preserve the fibers in more than one mode. The approximation error obtained this way is smaller than the one from the standard tensor CUR-type method. This difference increases as the tensor dimension increases. It also increases as the number of modes in which the original fibers are preserved decreases.
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References
Ayala, A., Claeys, X., Grigori, L.: Linear-time CUR approximation of BEM matrices. J. Comput. Appl. Math. 368, 112528 (2020)
Begović Kovač, E., Kressner, D.: Structure-preserving low multilinear rank approximation of antisymmetric tensors. SIAM J. Matrix Anal. Appl. 38(3), 967–983 (2017)
Boutsidis, C., Woodruff, D.: Optimal CUR matrix decompositions. In: Proceedings of the 46th Annual ACM Symposium on Theory of Computing, pp. 353–362. ACM (2014)
Caiafa, C.F., Cichocki, A.: Generalizing the column-row matrix decomposition to multi-way arrays. Linear Algebra Appl. 433, 557–573 (2010)
De Lathauwer, L., De Moor, B., Vandewalle, J.: A multilinear singular value decomposition. SIAM J. Matrix Anal. Appl. 21(4), 1253–1278 (2000)
Drineas, P., Mahoney, M.W., Muthukrishnan, S.: Relative-error CUR matrix decompositions. SIAM J. Matrix Anal. Appl. 30(2), 844–881 (2008)
Drmač, Z., Gugercin, S.: A new selection operator for the discrete empirical interpolation method-Improved a priori error bound and extensions. SIAM J. Sci. Comput. 38(2), A631–A648 (2016)
Friedland, S., Torokhti, A.: Generalized rank-constrained matrix approximations. SIAM J. Matrix Anal. Appl. 29(2), 656–659 (2007)
Ishteva, M., Absil, P.-A., Van Dooren, P.: Jacobi algorithm for the best low multilinear rank approximation of symmetric tensors. SIAM J. Matrix Anal. Appl. 34(2), 651–672 (2013)
Kolda, T.G., Bader, B.W.: Tensor decompositions and applications. SIAM Rev. 51(3), 455–500 (2009)
Mahoney, M.W., Drineas, P.: CUR matrix decompositions for improved data analysis. Proc. Natl. Acad. Sci. USA 106, 697–702 (2009)
Mahoney, M.W., Maggioni, M., Drineas, P.: Tensor-CUR decompositions for tensor-based data. SIAM J. Matrix Anal. Appl. 30(3), 957–987 (2008)
Oseledets, I.V., Savostianov, D.V., Tyrtyshnikov, E.E.: Tucker dimensionality reduction of three-dimensional arrays in linear time. SIAM J. Matrix Anal. Appl. 30(3), 939–956 (2008)
Sheehan, B.N., Saad, Y.: Higher order orthogonal iteration of tensors (HOOI) and its relation to PCA and GLRAM. In: Proceedings of the 2007 SIAM International Conference on Data Mining, pp. 355–365 (2007)
Saibaba, A.K.: HOID: Higher order interpolatory decomposition for tensors based on Tucker representation. SIAM J. Matrix Anal. Appl. 37(3), 1223–1249 (2016)
Sorensen, D.C., Embree, M.: A DEIM induced CUR factorization. SIAM J. Sci. Comput. 38(3), A1454–A1482 (2016)
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The author would like to thank Georgia Tech for the kind hospitality during the process of writing this paper. The author also thanks anonymous referees for their useful comments.
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Communicated by Marko Huhtanen.
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This work has been supported in part by Croatian Science Foundation under the project UIP-2019-04-5200.
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Begović Kovač, E. Hybrid CUR-type decomposition of tensors in the Tucker format. Bit Numer Math 62, 125–138 (2022). https://doi.org/10.1007/s10543-021-00876-x
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DOI: https://doi.org/10.1007/s10543-021-00876-x