关于高阶张量的秩-(L_r,1,1)分解方法
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摘要
主要研究了高阶张量的秩-(L_r,1,1)分解方法。首先,引入高阶张量的秩-(L_r,1,1)分解概念,得到一个关于高阶张量的秩-(L_r,1,1)分解的基本唯一性的充分条件;然后,给出了求解高阶张量的秩-(L_r,1,1)分解的交替最小二乘算法;最后给出了2个数值实例。数值实例结果表明所提出的方法的有效性。
This paper concentrates on the rank-(L_r, 1, 1) decomposition methods of a tensor with higher order. Firstly, the concept of rank-(L_r, 1, 1) decomposition of high-order tensors is introduced, and a sufficient condition on the basic uniqueness of the rank-(L_r, 1, 1) decomposition of higher-order tensors is obtained. Then, an alternating least-squares based numerical algorithm for the rank-(L_r, 1, 1) decomposition of tensors with high order is given and the least squares algorithm is used for numerical examples. Finally, two numerical examples are presented to illustrate the effectiveness of the proposed method.
This paper concentrates on the rank-(L_r, 1, 1) decomposition methods of a tensor with higher order. Firstly, the concept of rank-(L_r, 1, 1) decomposition of high-order tensors is introduced, and a sufficient condition on the basic uniqueness of the rank-(L_r, 1, 1) decomposition of higher-order tensors is obtained. Then, an alternating least-squares based numerical algorithm for the rank-(L_r, 1, 1) decomposition of tensors with high order is given and the least squares algorithm is used for numerical examples. Finally, two numerical examples are presented to illustrate the effectiveness of the proposed method.
引文
[1] Lathauwer L D, Moor B D, Vandewalle J. A multilinear singular value decomposition [J]. SIAM Journal on Matrix Analysis and Applications, 2000, 21(4): 1253-1278.
[2] Carroll J, Chang J. Analysis of individual differences in multidimensional scaling via an N-way generalization of Eckart-Young decomposition [J]. Psychometrika, 1970, 35(3): 283-319.
[3] Harshman R A. Foundations of the PARAFAC procedure: Model and conditions for an explanatory multi-mode factor analysis [J]. UCLA Working Papers in Phonetics, 1970, 16: 184.
[4] Lathauwer L D. Decompositions of a higher-order tensor in block terms Part I: Lemmas for partitioned matrices [J]. SIAM Journal on Matrix Analysis and Applications, 2008, 30(3): 1022-1032.
[5] Smilde A, Bro R, Geladi P. Multi-Way Analysis: Applications in the Chemical Sciences [M]. West Sussex: Wiley, 2004.
[6] Savas B, Eldn L. Krylov subspace methods for tensor computations [J]. Linear Algebra and Its Applications, 2013, 438(2): 891-918.
[7] Lathauwer L D. Decompositions of a higher-order tensor in block terms Part Ⅲ: Alternating least squares algorithms [J]. SIAM Journal on Matrix Analysis and Applications, 2008, 30(3): 1067-1083.
[8] Kruskal J B. Three-way arrays: Rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statistics [J]. Linear Algebra and Its Applications, 1977, 18(2): 95-138.
[9] Kiers H. Towards a standardized notation and terminology in multiway analysis [J]. Journal of Chemometrics, 2000, 14(3): 105-122.
[10] Boutry G, Elad M, Golub G H, et al. The generalized eigenvalue problem for nonsquare pencils using a minimal perturbation approach [J]. SIAM Journal on Matrix Analysis and Applications, 2005, 27(2): 582-601.
[11] Elad M, Milanfar P, Golub G H. Shape from moments an estimation theory perspective [J]. IEEE Transactions on Signal Processing, 2004, 52(7): 1814-1829.
[2] Carroll J, Chang J. Analysis of individual differences in multidimensional scaling via an N-way generalization of Eckart-Young decomposition [J]. Psychometrika, 1970, 35(3): 283-319.
[3] Harshman R A. Foundations of the PARAFAC procedure: Model and conditions for an explanatory multi-mode factor analysis [J]. UCLA Working Papers in Phonetics, 1970, 16: 184.
[4] Lathauwer L D. Decompositions of a higher-order tensor in block terms Part I: Lemmas for partitioned matrices [J]. SIAM Journal on Matrix Analysis and Applications, 2008, 30(3): 1022-1032.
[5] Smilde A, Bro R, Geladi P. Multi-Way Analysis: Applications in the Chemical Sciences [M]. West Sussex: Wiley, 2004.
[6] Savas B, Eldn L. Krylov subspace methods for tensor computations [J]. Linear Algebra and Its Applications, 2013, 438(2): 891-918.
[7] Lathauwer L D. Decompositions of a higher-order tensor in block terms Part Ⅲ: Alternating least squares algorithms [J]. SIAM Journal on Matrix Analysis and Applications, 2008, 30(3): 1067-1083.
[8] Kruskal J B. Three-way arrays: Rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statistics [J]. Linear Algebra and Its Applications, 1977, 18(2): 95-138.
[9] Kiers H. Towards a standardized notation and terminology in multiway analysis [J]. Journal of Chemometrics, 2000, 14(3): 105-122.
[10] Boutry G, Elad M, Golub G H, et al. The generalized eigenvalue problem for nonsquare pencils using a minimal perturbation approach [J]. SIAM Journal on Matrix Analysis and Applications, 2005, 27(2): 582-601.
[11] Elad M, Milanfar P, Golub G H. Shape from moments an estimation theory perspective [J]. IEEE Transactions on Signal Processing, 2004, 52(7): 1814-1829.