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January
2021, 17(1): 357-368.
doi: 10.3934/jimo.2019115
The point-wise convergence of shifted symmetric higher order power method
| 1. | School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China |
| 2. | School of Mathematics and Statistics, Kashi University, Kashi 844006, China |
Shifted symmetric higher-order power method (SS-HOPM) is an effective method of computing tensor eigenpairs. However the point-wise convergence of SS-HOPM has not been proven yet. In this paper, we provide a solid proof of the point-wise convergence of SS-HOPM via Łojasiewicz inequality. In particular, we establish a mapping from the sequence generated by the algorithm to a specially defined sequence. Using Łojasiewicz inequality, we prove the convergence of the new sequence, then the original sequence is convergent based on the relation of two sequences.
Keywords:
Z-eigenvalue,
shifted symmetric higher-order power method,
point-wise convergence,
Łojasiewicz inequality.
Mathematics Subject Classification:
Primary: 15A18, 90C26; Secondary: 15A69.
Citation:
Gang Luo, Qingzhi Yang. The point-wise convergence of shifted symmetric higher order power method. Journal of Industrial and Management Optimization,
2021, 17
(1)
: 357-368.
doi: 10.3934/jimo.2019115
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References:
| [1] |
A. Uschmajew,
A new convergence proof for the higher-order power method and generalizations, Pac. J. Optim., 11 (2015), 309-321.
|
| [2] |
A. T. Erdogan,
On the convergence of ICA algorithms with symmetric orthogonalization, IEEE Trans. Signal Process., 57 (2009), 2209-2221.
doi: 10.1109/TSP.2009.2015114. |
| [3] |
D. Cartwright and B. Sturmfels,
The number of eigenvalues of a tensor, Linear Alg. Appl., 438 (2013), 942-952.
doi: 10.1016/j.laa.2011.05.040. |
| [4] |
E. Kofidis and P. A. Regalia,
On the best rank-1 approximation of higher-order supersymmetric tensors, SIAM J. Matrix Anal. Appl., 23 (2001), 863-884.
doi: 10.1137/S0895479801387413. |
| [5] |
G. H. Golub and C. F. Van Loan, Matrix Computations, Fourth edition, Johns Hopkins Studies in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, MD, 2013. |
| [6] |
L. De Lathauwer, B. De Moor and J. Vandewalle,
On the best rank-1 and rank-($r_1$, $r_2$, ..., $r_N$) approximation of higher-order tensors, SIAM J. Matrix Anal. Appl., 21 (2000), 1324-1342.
doi: 10.1137/S0895479898346995. |
| [7] |
L. Lim, Singular values and eigenvalues of tensors: A variational approach, 1st IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, 2005., IEEE, (2005), 129-132. |
| [8] |
L. Q. Qi,
Eigenvalues of a real supersymmetric tensor, J. Symb. Comput., 40 (2005), 1302-1324.
doi: 10.1016/j.jsc.2005.05.007. |
| [9] |
L. Q. Qi and K. L. Teo,
Multivariate polynomial minimization and its application in signal processing, J. Glob. Optim., 26 (2013), 419-433.
doi: 10.1023/A:1024778309049. |
| [10] |
L. Q. Qi, W. Y. Sun and Y. J. Wang,
Numerical multilinear algebra and its applications, Front. Math. China, 2 (2007), 501-526.
doi: 10.1007/s11464-007-0031-4. |
| [11] |
M. Ng, L. Q. Qi and G. L. Zhou,
Finding the largest eigenvalue of a nonnegative tensor, SIAM J. Matrix Anal. Appl., 31 (2009), 1090-1099.
doi: 10.1137/09074838X. |
| [12] |
P.-A. Absil, R. Manhony and B. Andrews,
Convergence of the iterates of descent methods for analytic cost functions, SIAM J. Optim., 16 (2005), 531-547.
doi: 10.1137/040605266. |
| [13] |
P. A. Regalia and E. Kofidis,
Monotonic convergence of fixed-point algorithms for ICA, IEEE Trans. Neural Netw., 14 (2003), 943-949.
doi: 10.1109/TNN.2003.813843. |
| [14] |
Q. Ni, L. Q. Qi and F. Wang,
An eigenvalue method for testing positive definiteness of a multivariate form, IEEE Trans. Automat. Contr., 53 (2008), 1096-1107.
doi: 10.1109/TAC.2008.923679. |
| [15] |
R. Schneider and A. Uschmajew,
Convergence results for projected line-search methods on varieties of low-rank matrices via Łojasiewicz inequality, SIAM J. Optim., 25 (2015), 622-646.
doi: 10.1137/140957822. |
| [16] |
S. Łojasiewicz, Ensembles semi-analytiques, Lectures Notes, IHES Bures-sur-Yvette, (1965). |
| [17] |
T. G. Kolda and J. R. Mayo,
Shifted power method for computing tensor eigenpairs, SIAM J. Matrix Anal. Appl., 32 (2011), 1095-1124.
doi: 10.1137/100801482. |
| [18] |
Y. J. Wang, L. Q. Qi and X. Z. Zhang,
A practical method for computing the largest $m$-eigenvalue of a fourth-order partially symmetric tensor, Numer. Linear Algebra Appl., 16 (2009), 589-601.
doi: 10.1002/nla.633. |
| [19] |
Y. Y. Xu and W. T. Yin,
A block coordinate descent method for regularized multiconvex optimization with applications to nonnegative tensor factorization and completion, SIAM J. Imaging Sci., 6 (2013), 1758-1789.
doi: 10.1137/120887795. |
show all references
References:
| [1] |
A. Uschmajew,
A new convergence proof for the higher-order power method and generalizations, Pac. J. Optim., 11 (2015), 309-321.
|
| [2] |
A. T. Erdogan,
On the convergence of ICA algorithms with symmetric orthogonalization, IEEE Trans. Signal Process., 57 (2009), 2209-2221.
doi: 10.1109/TSP.2009.2015114. |
| [3] |
D. Cartwright and B. Sturmfels,
The number of eigenvalues of a tensor, Linear Alg. Appl., 438 (2013), 942-952.
doi: 10.1016/j.laa.2011.05.040. |
| [4] |
E. Kofidis and P. A. Regalia,
On the best rank-1 approximation of higher-order supersymmetric tensors, SIAM J. Matrix Anal. Appl., 23 (2001), 863-884.
doi: 10.1137/S0895479801387413. |
| [5] |
G. H. Golub and C. F. Van Loan, Matrix Computations, Fourth edition, Johns Hopkins Studies in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, MD, 2013. |
| [6] |
L. De Lathauwer, B. De Moor and J. Vandewalle,
On the best rank-1 and rank-($r_1$, $r_2$, ..., $r_N$) approximation of higher-order tensors, SIAM J. Matrix Anal. Appl., 21 (2000), 1324-1342.
doi: 10.1137/S0895479898346995. |
| [7] |
L. Lim, Singular values and eigenvalues of tensors: A variational approach, 1st IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, 2005., IEEE, (2005), 129-132. |
| [8] |
L. Q. Qi,
Eigenvalues of a real supersymmetric tensor, J. Symb. Comput., 40 (2005), 1302-1324.
doi: 10.1016/j.jsc.2005.05.007. |
| [9] |
L. Q. Qi and K. L. Teo,
Multivariate polynomial minimization and its application in signal processing, J. Glob. Optim., 26 (2013), 419-433.
doi: 10.1023/A:1024778309049. |
| [10] |
L. Q. Qi, W. Y. Sun and Y. J. Wang,
Numerical multilinear algebra and its applications, Front. Math. China, 2 (2007), 501-526.
doi: 10.1007/s11464-007-0031-4. |
| [11] |
M. Ng, L. Q. Qi and G. L. Zhou,
Finding the largest eigenvalue of a nonnegative tensor, SIAM J. Matrix Anal. Appl., 31 (2009), 1090-1099.
doi: 10.1137/09074838X. |
| [12] |
P.-A. Absil, R. Manhony and B. Andrews,
Convergence of the iterates of descent methods for analytic cost functions, SIAM J. Optim., 16 (2005), 531-547.
doi: 10.1137/040605266. |
| [13] |
P. A. Regalia and E. Kofidis,
Monotonic convergence of fixed-point algorithms for ICA, IEEE Trans. Neural Netw., 14 (2003), 943-949.
doi: 10.1109/TNN.2003.813843. |
| [14] |
Q. Ni, L. Q. Qi and F. Wang,
An eigenvalue method for testing positive definiteness of a multivariate form, IEEE Trans. Automat. Contr., 53 (2008), 1096-1107.
doi: 10.1109/TAC.2008.923679. |
| [15] |
R. Schneider and A. Uschmajew,
Convergence results for projected line-search methods on varieties of low-rank matrices via Łojasiewicz inequality, SIAM J. Optim., 25 (2015), 622-646.
doi: 10.1137/140957822. |
| [16] |
S. Łojasiewicz, Ensembles semi-analytiques, Lectures Notes, IHES Bures-sur-Yvette, (1965). |
| [17] |
T. G. Kolda and J. R. Mayo,
Shifted power method for computing tensor eigenpairs, SIAM J. Matrix Anal. Appl., 32 (2011), 1095-1124.
doi: 10.1137/100801482. |
| [18] |
Y. J. Wang, L. Q. Qi and X. Z. Zhang,
A practical method for computing the largest $m$-eigenvalue of a fourth-order partially symmetric tensor, Numer. Linear Algebra Appl., 16 (2009), 589-601.
doi: 10.1002/nla.633. |
| [19] |
Y. Y. Xu and W. T. Yin,
A block coordinate descent method for regularized multiconvex optimization with applications to nonnegative tensor factorization and completion, SIAM J. Imaging Sci., 6 (2013), 1758-1789.
doi: 10.1137/120887795. |
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