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Journal of Scientific Computing
Published in: Journal of Scientific Computing 1/2023

01-01-2023

Rank Properties and Computational Methods for Orthogonal Tensor Decompositions

Author: Chao Zeng

Published in: Journal of Scientific Computing | Issue 1/2023

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Abstract

The orthogonal decomposition factorizes a tensor into a sum of an orthogonal list of rank-one tensors. The corresponding rank is called orthogonal rank. We present several properties of orthogonal rank, which are different from those of tensor rank in many aspects. For instance, a subtensor may have a larger orthogonal rank than the whole tensor. To fit the orthogonal decomposition, we propose an algorithm based on the augmented Lagrangian method. The gradient of the objective function has a nice structure, inspiring us to use gradient-based optimization methods to solve it. We guarantee the orthogonality by a novel orthogonalization process. Numerical experiments show that the proposed method has a great advantage over the existing methods for strongly orthogonal decompositions in terms of the approximation error.
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Footnotes
1
Strongly orthogonal decomposition has a different definition in Ref. [17].
 
2
Such tensors exist. See [9, Lemma 4.7] for an example.
 
3
The hyperspectral image data have been used in [36] and available at https://​rslab.​ut.​ac.​ir/​data.
 
4
The video data are from the video trace library [29] and available at http://​trace.​eas.​asu.​edu/​yuv/​.
 
5
A Matlab implementation, adapted by Dianne P. O’Leary, is available at http://​www.​cs.​umd.​edu/​users/​oleary/​software/​.
 
Literature
1.
Acar, E., Dunlavy, D.M., Kolda, T.G.: A scalable optimization approach for fitting canonical tensor decompositions. J. Chemom. 25(2), 67–86 (2011)CrossRef
2.
Bader, B.W., Kolda, T.G. et al.: MATLAB Tensor Toolbox Version 3.0-dev. Available online, Oct. (2017)
3.
Bertsekas, D.P.: Constrained Optimization and Lagrange Multiplier Methods. Academic press, Cambridge (1982)MATH
4.
Carroll, J.D., Chang, J.-J.: Analysis of individual differences in multidimensional scaling via an N-way generalization of “Eckart-Young’’ decomposition. Psychometrika 35(3), 283–319 (1970)CrossRefMATH
5.
Chen, J., Saad, Y.: On the tensor SVD and the optimal low rank orthogonal approximation of tensors. SIAM J. Matrix Anal. Appl. 30(4), 1709–1734 (2008)MathSciNetCrossRefMATH
6.
Comon, P.: Independent component analysis, A new concept? Signal Process. 36(3), 287–314 (1994)CrossRefMATH
7.
Conn, A.R., Gould, N., Sartenaer, A., Toint, P.L.: Convergence properties of an augmented Lagrangian algorithm for optimization with a combination of general equality and linear constraints. SIAM J. Optim. 6(3), 674–703 (1996)MathSciNetCrossRefMATH
8.
De Lathauwer, L., De Moor, B., Vandewalle, J.: A multilinear singular value decomposition. SIAM J. Matrix Anal. Appl. 21(4), 1253–1278 (2000)MathSciNetCrossRefMATH
9.
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(3), 1084–1127 (2008)MathSciNetCrossRefMATH
10.
De Sterck, H., Howse, A.J.: Nonlinearly preconditioned L-BFGS as an acceleration mechanism for alternating least squares with application to tensor decomposition. Num. Linear Algebra Appl. 25(6), e2202 (2018)MathSciNetCrossRefMATH
11.
Eckart, C., Young, G.: The approximation of one matrix by another of lower rank. Psychometrika 1(3), 211–218 (1936)CrossRefMATH
12.
Espig, M., Hackbusch, W.: A regularized Newton method for the efficient approximation of tensors represented in the canonical tensor format. Numer. Math. 122(3), 489–525 (2012)MathSciNetCrossRefMATH
13.
Guan, Y., Chu, D.: Numerical computation for orthogonal low-rank approximation of tensors. SIAM J. Matrix Anal. Appl. 40(3), 1047–1065 (2019)MathSciNetCrossRefMATH
14.
Harshman, R.A. et al.: Foundations of the PARAFAC procedure: models and conditions for an “explanatory” multimodal factor analysis. (1970)
15.
16.
17.
18.
Kolda, T.G.: A counterexample to the possibility of an extension of the Eckart-Young low-rank approximation theorem for the orthogonal rank tensor decomposition. SIAM J. Matrix Anal. Appl. 24(3), 762–767 (2003)MathSciNetCrossRefMATH
19.
20.
Krijnen, W.P., Dijkstra, T.K., Stegeman, A.: On the non-existence of optimal solutions and the occurrence of “degeneracy’’ in the CANDECOMP/PARAFAC model. Psychometrika 73(3), 431–439 (2008)MathSciNetCrossRefMATH
21.
Kruskal, J.B.: Three-way arrays: rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statistics. Linear Algebra Appl. 18(2), 95–138 (1977)MathSciNetCrossRefMATH
22.
Li, Z., Nakatsukasa, Y., Soma, T., Uschmajew, A.: On orthogonal tensors and best rank-one approximation ratio. SIAM J. Matrix Anal. Appl. 39(1), 400–425 (2018)MathSciNetCrossRefMATH
23.
24.
Martin, C.D.M., Van Loan, C.F.: A Jacobi-type method for computing orthogonal tensor decompositions. SIAM J. Matrix Anal. Appl. 30(3), 1219–1232 (2008)MathSciNetCrossRefMATH
25.
More, J.J., Thuente, D.J.: Line search algorithms with guaranteed sufficient decrease. ACM Trans. Math. Softw. 20(3), 286–307 (1994)MathSciNetCrossRefMATH
26.
Nazih, M., Minaoui, K., Comon, P.: Using the proximal gradient and the accelerated proximal gradient as a canonical polyadic tensor decomposition algorithms in difficult situations. Signal Process. 171, 107472 (2020)CrossRef
27.
Nocedal, J., Wright, S.: Numerical Optimization. Springer Science & Business Media, New York (2006)MATH
28.
Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer Science & Business Media, New York (2009)MATH
29.
Seeling, P., Reisslein, M.: Video transport evaluation with H. 264 video traces. IEEE Commun. Surv. Tutor. 14(4), 1142–1165 (2011)CrossRef
30.
Sidiropoulos, N.D., Bro, R.: On the uniqueness of multilinear decomposition of N-way arrays. J. Chemometr. J. Chemometr. Soc. 14(3), 229–239 (2000)CrossRef
31.
Sørensen, M., De Lathauwer, L., Comon, P., Icart, S., Deneire, L.: Canonical polyadic decomposition with a columnwise orthonormal factor matrix. SIAM J. Matrix Anal. Appl. 33(4), 1190–1213 (2012)MathSciNetCrossRefMATH
32.
Sterck, H.D.: A nonlinear GMRES optimization algorithm for canonical tensor decomposition. SIAM J. Sci. Comput. 34(3), A1351–A1379 (2012)MathSciNetCrossRefMATH
33.
Sun, W., Yuan, Y.-X.: Optimization Theory and Methods: Nonlinear Programming. Springer Optimization and Its Applications. Springer Science & Business Media, New York (2010)
34.
Wang, L., Chu, M.T., Yu, B.: Orthogonal low rank tensor approximation: alternating least squares method and its global convergence. SIAM J. Matrix Anal. Appl. 36(1), 1–19 (2015)MathSciNetCrossRefMATH
35.
Yang, Y.: The epsilon-alternating least squares for orthogonal low-rank tensor approximation and its global convergence. SIAM J. Matrix Anal. Appl. 41(4), 1797–1825 (2020)MathSciNetCrossRefMATH
36.
Zhu, F., Wang, Y., Fan, B., Xiang, S., Meng, G., Pan, C.: Spectral unmixing via data-guided sparsity. IEEE Trans. Image Process. 23(12), 5412–5427 (2014)MathSciNetCrossRefMATH
Metadata
Title
Rank Properties and Computational Methods for Orthogonal Tensor Decompositions
Author
Chao Zeng
Publication date
01-01-2023
Publisher
Springer US
Published in
Journal of Scientific Computing / Issue 1/2023
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI
https://doi.org/10.1007/s10915-022-02054-9

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