Operations Research Transactions ›› 2025, Vol. 29 ›› Issue (3): 34-60.doi: 10.15960/j.cnki.issn.1007-6093.2025.03.002
Special Issue: 第九届中国运筹学会科学技术奖获奖者专辑
• Research Article • Previous Articles Next Articles
Received:
2025-04-10
Online:
2025-09-15
Published:
2025-09-09
Contact:
Shenglong HU
E-mail:hushenglong@nudt.edu.cn
CLC Number:
Shenglong HU. Uniqueness of tensor canonical polyadic decomposition[J]. Operations Research Transactions, 2025, 29(3): 34-60.
1 | Hitchcock F L . The expression of a tensor or a polyadic as a sum of products[J]. Journal of Mathematical Physics, 1927, 6, 164- 189. |
2 | Carroll J D , Chang J J . Analysis of individual differences in multidimensional scaling via an n-way generalization of "Eckart-Young" decomposition[J]. Psychometrika, 1927, 35, 283- 319. |
3 | Harshman R A . Foundations of the parafac procedure: Models and conditions for an "explanatory" multi-modal factor analysis[J]. UCLA Working Papers in Phonetics, 1970, 16, 1- 84. |
4 | Tucker L R . Some mathematical notes on three-mode factor analysis[J]. Psychometrika, 1966, 31, 273- 311. |
5 | Oseledets I V . Tensor-train decomposition[J]. SIAM Journal on Scientific Computing, 2011, 33 (5): 2295- 2317. |
6 | Hackbusch W . Tensor Spaces and Numerical Tensor Calculus[M]. Cham: Springer, 2019. |
7 | Cichocki A. Era of big data processing: A new approach via tensor networks and tensor decompositions [EB/OL]. [2025-04-09]. arXiv: 1403.2048. |
8 | Landsberg J M . Tensors: Geometry and Applications[M]. Providence: American Mathematical Society, 2012. |
9 | Bürgisser P , Clausen M , Shokrollahi M A . Algebraic Complexity Theory[M]. Heidelberg: Springer, 2013. |
10 | Comon P . Independent component analysis, a new concept?[J]. Signal Processing, 1994, 36, 287- 314. |
11 | Lathauwer L D , De Moor B , Vandewalle J . Independent component analysis and (simultaneous) third-order tensor diagonalization[J]. IEEE Transactions on Signal Processing, 2001, 49, 2262- 2271. |
12 | Anandkumar A, Ge R, Janzamin M. Learning overcomplete latent variable models through tensor methods [C]// Proceedings of Machine Learning Research, 2015, 40: 36-112. |
13 | Sanchez E , Kowalski B R . Tensorial resolution: A direct trilinear decomposition[J]. Journal of Chemometrics, 1990, 4, 29- 45. |
14 | Appellof C J , Davidson E R . Strategies for analyzing data from video fluorometric monitoring of liquid chromatographic effluents[J]. Analytical Chemistry, 1981, 53, 2053- 2056. |
15 | Allman E S , Matias C , Rhodes J A . Identifiability of parameters in latent structure models with many observed variables[J]. Annals of Statistics, 2009, 37 (6A): 3099- 3132. |
16 | Anandkumar A , Ge R , Hsu D , et al. Tensor decompositions for learning latent variable models[J]. Journal of Machine Learning Research, 2014, 15, 2773- 2832. |
17 | Orús R . A practical introduction to tensor networks: Matrix product states and projected entangled pair states[J]. Annals of Physics, 2014, 349, 117- 158. |
18 | Zhou G , Zhao Q , Zhang Y , et al. Linked component analysis from matrices to high-order tensors: Applications to biomedical data[J]. Proceedings of the IEEE, 2016, 104 (2): 310- 331. |
19 | Andersson J L , Beckmann C H M . Dimensionality reduction: Models and methods for exploratory data analysis[J]. NeuroImage, 2001, 13 (6): S176- S184. |
20 | Rendle S, Marinho L B, Nanopoulos A, et al. Learning optimal ranking with tensor factorization for tag recommendation [C]//Proceedings of the 15th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, 2009: 727-736. |
21 | Ji S , Xu W , Yang M , et al. 3D convolutional neural networks for human action recognition[J]. IEEE Transactions on Pattern Analysis & Machine Intelligence, 2013, 35 (1): 221- 231. |
22 | Sidiropoulos N D , Bro R , Giannakis G B . Parallel factor analysis in sensor array processing[J]. IEEE Transactions on Signal Processing, 2000, 48 (8): 2377- 2388. |
23 | Sidiropoulos N D , Giannakis G B , Bro R . Blind Parafac receivers for DS-CDMA systems[J]. IEEE Transactions on Signal Processing, 2000, 48 (3): 810- 823. |
24 | Klerk E D , Pasechnik D V , Schrijver A . Reduction of symmetric semidefinite programs using the regular $\ast$-representation[J]. Mathematical Programming, 2007, 109 (2-3): 613- 624. |
25 | Klerk E D , Sotirov R . Exploiting group symmetry in semidefinite programming relaxations of the quadratic assignment problem[J]. Mathematical Programming, 2010, 122 (2): 225- 246. |
26 | Kofidis E , Regalia P A . Tensor approximation and signal processing applications[J]. Contemporary Mathematics, 2001, 280, 103- 134. |
27 | Kolda T G , Bader B W . Tensor decompositions and applications[J]. SIAM Review, 2009, 51, 455- 500. |
28 | Comon P , Luciani X , De Almeida A L F . Tensor decompositions, alternating least squares and other tales[J]. Journal of Chemometrics, 2009, 23 (7-8): 393- 405. |
29 | De Lathauwer L. A short introduction to tensor-based methods for factor analysis and blind source separation [C]// Proceedings of the 7th International Symposium on Image and Signal Processing and Analysis, 2011: 558-563. |
30 | Acar E , Yener B . Unsupervised multiway data analysis: A literature survey[J]. IEEE Transactions on Knowledge and Data Engineering, 2008, 21 (1): 6- 20. |
31 | Comon P. Tensor decompostions: State of the art and applications [EB/OL]. [2025-04-09]. arXiv: 0905.0454. |
32 | Bro R. PARAFAC. Tutorial and applications [J]. Chemometrics and Intelligent Laboratory Systems, 1997, 38(2): 149-171. |
33 | McCullagh P . Tensor Methods in Statistics[M]. London: Chapman and Hall, 1987. |
34 | Lim L H . Tensors and hypermatrices[J]. Handbook of Linear Algebra, 2013, 231- 260. |
35 | Lim L H . Tensors in computations[J]. Acta Numerica, 2021, 30, 555- 764. |
36 | Cichocki A , Zdunek R , Phan A H . Nonnegative Matrix and Tensor Factorizations: Applications to Exploratory Multi-Way Data Analysis and Blind Source Separation[M]. New York: Wiley, 2009. |
37 | Grasedyck L , Kressner D , Tobler C . A literature survey of low-rank tensor approximation techniques[J]. GAMM-Mitteilungen, 2013, 36 (1): 53- 78. |
38 | Friedland S, Tammali V. Low-rank approximation of tensors [M]// Benner P, Bollhöfer M, Kressner D, et al. (eds. ) Numerical Algebra, Matrix Theory, Differential-Algebraic Equations and Control Theory: Festschrift in Honor of Volker Mehrmann, 2015: 377-411. |
39 | Uschmajew A, Vandereycken B. Geometric methods on low-rank matrix and tensor manifolds [M]//Philipp G, Martin H, Andreas W. (eds. ) Handbook of Variational Methods for Nonlinear Geometric Data, Springer. 2020: 261-313. |
40 | Sidiropoulos N D , De Lathauwer L , Fu X , et al. Tensor decomposition for signal processing and machine learning[J]. IEEE Transactions on Signal Processing, 2017, 65 (13): 3551- 3582. |
41 | Cichocki A , Mandic D , De Lathauwer L , et al. Tensor decompositions for signal processing applications: From two-way to multiway component analysis[J]. IEEE Signal Processing Magazine, 2015, 32 (2): 145- 163. |
42 | Berge J M F. Least squares optimization in multivariate analysis [R]. Leiden: DSWO Press, 1993. |
43 | Pajarola R, Suter S K, Ballester-Ripoll R, et al. Tensor approximation for multidimensional and multivariate data [M]// Özarslan E, Schultz T, Zhang E, et al. (eds. ) Anisotropy Across Fields and Scales, Cham: Springer, 2021: 73-98. |
44 | Hitchcock F L . Multiple invariants and generalized rank of a p-way matrix or tensor[J]. Journal of Mathematical Physics, 1928, 7 (1-4): 39- 79. |
45 | Möcks J . Topographic components model for event-related potentials and some biophysical considerations[J]. IEEE Transactions on Biomedical Engineering, 1988, 35 (6): 482- 484. |
46 | 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, 95- 138. |
47 | Kruskal J B . More factors than subjects, tests and treatments: An indeterminacy theorem for canonical decomposition and individual differences scaling[J]. Psychometrika, 1976, 41 (3): 281- 293. |
48 | Strassen V . Gaussian elimination is not optimal[J]. Numerische Mathematik, 1969, 13 (4): 354- 356. |
49 | Landsberg J . The border rank of the multiplication of 2$\times$2 matrices is seven[J]. Journal of the American Mathematical Society, 2006, 19 (2): 447- 459. |
50 | Li Y , Hu S L , Wang J , et al. An introduction to the computational complexity of matrix multiplication[J]. Journal of the Operations Research Society of China, 2020, 8, 29- 43. |
51 | Bini D , Capovani M , Romani F , et al. O(n2:7799) complexity for $n\times n$ approximate matrix multiplication[J]. Information Processing Letters, 1979, 8 (5): 234- 235. |
52 | Santamaria I . Handbook of blind source separation: Independent component analysis and applications (Common P, Jutten C; 2010 [Book Review])[J]. IEEE Signal Processing Magazine, 2013, 30 (2): 133- 134. |
53 | Comon P , Mourrain B . Decomposition of quantics in sums of powers of linear forms[J]. Signal Processing, 1996, 53 (2-3): 93- 107. |
54 | Liu X Q , Sidiropoulos N D . Cramér-Rao lower bounds for low-rank decomposition of multidimensional arrays[J]. IEEE Transactions on Signal Processing, 2001, 49 (9): 2074- 2086. |
55 | Bro R, Sidiropoulos N D, Giannakis G B. A fast least squares algorithm for separating trilinear mixtures [C]//International Workshop on Independent Component Analysis and Blind Separation, 1999: 3. |
56 | Rhodes J A , Sullivant S . Identifiability of large phylogenetic mixture models[J]. Bulletin of Mathematical Biology, 2012, 74, 212- 231. |
57 | Guo X J, Miron S, Brie D. Identifiability of the parafac model for polarized source mixture on a vector sensor array [C]// 2008 IEEE International Conference on Acoustics, Speech and Signal Processing, 2008: 2401-2404. |
58 | Kruskal J B. Rank, decomposition, and uniqueness for 3-way and N-way arrays [M]// Coppi R, Bolasco S (eds. ) Multiway Data Analysis, Amsterdam: North-Holland Publishing Co., 1989: 7-18. |
59 | Strassen V . Rank and optimal computation of generic tensors[J]. Linear Algebra and Its Applications, 1983, 52, 645- 685. |
60 | Harshman R A , Lundy M E . The Parafac model for three-way factor analysis and multidimensional scaling[J]. Research Methods for Multimode Data Analysis, 1984, 46, 122- 215. |
61 | Harshman R A , Lundy M E . PARAFAC: Parallel factor analysis[J]. Computational Statistics and Data Analysis, 1994, 18 (1): 39- 72. |
62 | Bini D , Lotti G , Romani F . Approximate solutions for the bilinear form computational problem[J]. SIAM Journal on Computing, 1980, 9 (4): 692- 697. |
63 | De Lathauwer L , De Moor B , Vandewalle J . A multilinear singular value decomposition[J]. SIAM Journal on Matrix Analysis and Applications, 2000, 21 (4): 1253- 1278. |
64 | Novikov A, Podoprikhin D, Osokin A, et al. Tensorizing neural networks [C]//Proceedings of the 29th International Conference on Neural Information Processing Systems, 2015, 1: 442-450. |
65 | Che M , Wei Y . Randomized algorithms for the approximations of tucker and the tensor train decompositions[J]. Advances in Computational Mathematics, 2019, 45 (1): 395- 428. |
66 | Edelman A , Arias T A , Smith S T . The geometry of algorithms with orthogonality constraints[J]. SIAM Journal on Matrix Analysis and Applications, 1998, 20 (2): 303- 353. |
67 | Smith S T , Howington D A . GPU-accelerated Tucker factorization for sparse tensors[J]. Journal of Parallel and Distributed Computing, 2017, 109, 42- 52. |
68 | De Lathauwer L , De Moor B , Vandewalle J . On the best rank-1 and rank-($r_1, r_2, \cdots, r_n$) approximation of higher-order tensors[J]. SIAM Journal on Matrix Analysis and Applications, 2000, 21, 1324- 1342. |
69 | Hu S, Sun D, Toh K C. Quantifying low rank approximations of third order symmetric tensors [J/OL]. [2025-04-09]. Mathematical Programming. https://doi.org/10.1007/s10107-024-02165-1 |
70 | Hu S , Wang Y , Zhou J . A DCA-Newton method for quartic minimization over the sphere[J]. Advances in Computational Mathematics, 2023, 49, 53. |
71 | Nie J . Generating polynomials and symmetric tensor decompositions[J]. Foundations of Computational Mathematics, 2017, 17, 423- 465. |
72 | Nie J . Low rank symmetric tensor approximations[J]. SIAM Journal on Matrix Analysis and Applications, 2017, 38, 1517- 1540. |
73 | Leurgans S E , Ross R T , Abel R B . A decomposition for three-way arrays[J]. SIAM Journal on Matrix Analysis and Applications, 1993, 14 (4): 1064- 1083. |
74 | Ten Berge J M F , Sidiropoulos N D . On uniqueness in Candecomp/Parafac[J]. Psychometrika, 2002, 67 (3): 399- 409. |
75 | Harshman R A . Determination and proof of minimum uniqueness conditions for Parafac1[J]. UCLA Working Papers in Phonetics, 1972, 22, 111- 117. |
76 | Lim L H , Comon P . Multiarray signal processing: Tensor decomposition meets compressed sensing[J]. Comptes Rendus Mecanique, 2010, 338 (6): 311- 320. |
77 | Jiang T , Sidiropoulos N D . Kruskal's permutation lemma and the identification of Candecomp/Parafac and bilinear models with constant modulus constraints[J]. IEEE Transactions on Signal Processing, 2004, 52 (9): 2625- 2636. |
78 | Stegeman A , Sidiropoulos N D . On Kruskal's uniqueness condition for the Candecomp/Parafac decomposition[J]. Linear Algebra and Its Applications, 2007, 420 (2-3): 540- 552. |
79 | Rhodes J A . A concise proof of Kruskal's theorem on tensor decomposition[J]. Linear Algebra and Its Applications, 2010, 432 (7): 1818- 1824. |
80 | Stegeman A , Ten Berge J M F . Kruskal's condition for uniqueness in Candecomp/Parafac when ranks and k-ranks coincide[J]. Computational Statistics & Data Analysis, 2006, 50 (1): 210- 220. |
81 | Sidiropoulos N D , Bro R . On the uniqueness of multilinear decomposition of n-way arrays[J]. Journal of Chemometrics, 2000, 14, 229- 239. |
82 | Ma X S , Hu S L , Wang J . Efficient low-rank regularization-based algorithms combining advanced techniques for solving tensor completion problems with application to color image recovering[J]. Journal of Computational and Applied Mathematics, 2023, 423, 114947. |
83 | De Lathauwer L . A link between the canonical decomposition in multilinear algebra and simultaneous matrix diagonalization[J]. SIAM Journal on Matrix Analysis and Applications, 2006, 28 (3): 642- 666. |
84 | Stegeman A , Ten Berge J M F , De Lathauwer L . Sufficient conditions for uniqueness in Candecomp/Parafac and Indscal with random component matrices[J]. Psychometrika, 2006, 71 (2): 219- 229. |
85 | Horn R A , Johnson C R . Matrix Analysis[M]. Cambridge: Cambridge University Press, 2012. |
86 | Domanov I , De Lathauwer L . Canonical polyadic decomposition of third-order tensors: Reduction to generalized eigenvalue decomposition[J]. SIAM Journal on Matrix Analysis and Applications, 2014, 35 (2): 636- 660. |
87 | Stegeman A . On uniqueness conditions for Candecomp/Parafac and Indscal with full column rank in one mode[J]. Linear Algebra And Its Applications, 2009, 431 (1-2): 211- 227. |
88 | Guo X J , Miron S , Brie D , et al. A candecomp/parafac perspective on uniqueness of DOA estimation using a vector sensor array[J]. IEEE Transactions on Signal Processing, 2011, 59 (7): 3475- 3481. |
89 | Stegeman A . On uniqueness of the n-th order tensor decomposition into rank-1 terms with linear independence in one mode[J]. SIAM Journal on Matrix Analysis and Applications, 2010, 31 (5): 2498- 2516. |
90 | Domanov I , De Lathauwer L . Canonical polyadic decomposition of third-order tensors: Relaxed uniqueness conditions and algebraic algorithm[J]. Linear Algebra and Its Applications, 2017, 513, 342- 375. |
91 | De Lathauwer L . Blind separation of exponential polynomials and the decomposition of a tensor in rank-$(L_r, L_r, 1)$ terms[J]. SIAM Journal on Matrix Analysis and Applications, 2011, 32 (4): 1451- 1474. |
92 | Krijnen W P . The Analysis of Three-Way Arrays by Constrained Parafac Methods[M]. Leiden: DSWO Press, 1993. |
93 | Sidiropoulos N D , Liu X Q . Identifiability results for blind beamforming in incoherent multipath with small delay spread[J]. IEEE Transactions on Signal Processing, 2001, 49 (1): 228- 236. |
94 | Ten Berge J M F. The k-rank of a Khatri-Rao product [R]. The Netherlands: Heijmans Institute of Psychological Research, University of Groningen, 2000. |
95 | Chiantini L , Mella M , Ottaviani G . One example of general unidentifiable tensors[J]. Journal of Algebraic Statistics, 2014, 5 (1): 64- 71. |
96 | Ballico E . Partially symmetric tensor rank: The description of the non-uniqueness case for low rank[J]. Linear and Multilinear Algebra, 2018, 66 (3): 608- 624. |
97 | Carroll J D , Pruzansky S , Kruskal J B . CANDELINC: A general approach to multidimensional analysis of many-way arrays with linear constraints on parameters[J]. Psychometrika, 1980, 45 (1): 3- 24. |
98 | Harshman R A. "How can I know if it's 'real'?" A catalog of diagnostics for use with three-mode factor analysis and multidimensional scaling [M]// Law H G, Snyder C W, Hattie J A, et al. (eds. ) Research Methods for Multimode Data Analysis, New York: Praeger, 1984: 566-591. |
99 | Bro R , Harshman R A , Sidiropoulos N D , et al. Modeling multi-way data with linearly dependent loadings[J]. Journal of Chemometrics, 2009, 23 (7-8): 324- 340. |
100 | De Almeida A L F , Favier G , Mota J C M . A constrained factor decomposition with application to MIMO antenna systems[J]. IEEE Transactions on Signal Processing, 2008, 56 (6): 2429- 2442. |
101 | Stegeman A , De Almeida A L F . Uniqueness conditions for constrained three-way factor decompositions with linearly dependent loadings[J]. SIAM Journal on Matrix Analysis and Applications, 2010, 31 (3): 1469- 1490. |
102 | Ten Berge J M F . Partial uniqueness in Candecomp/Parafac[J]. Journal of Chemometrics, 2004, 18 (1): 12- 16. |
103 | Guo X J , Miron S , Brie D , et al. Uni-mode and partial uniqueness conditions for Candecomp/Parafac of three-way arrays with linearly dependent loadings[J]. SIAM Journal on Matrix Analysis and Applications, 2012, 33 (1): 111- 129. |
104 | Zniyed Y, Miron S, Boyer R, et al. Uniqueness of tensor train decomposition with linear dependencies [C]//2019 IEEE 8th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, 2019: 460-464. |
105 | Zhang L , Huang T Z , Zhu Q F , et al. Uni-mode uniqueness conditions for Candecomp/Parafac decomposition of n-way arrays with linearly dependent loadings[J]. Linear Algebra and Its Applications, 2013, 439 (7): 1918- 1928. |
106 | Domanov I , De Lathauwer L . On the uniqueness of the canonical polyadic decomposition of third-order tensors-part Ⅰ: Basic results and uniqueness of one factor matrix[J]. SIAM Journal on Matrix Analysis and Applications, 2013, 34 (3): 855- 875. |
107 | De Lathauwer L . 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. |
108 | De Lathauwer L . Decompositions of a higher-order tensor in block terms-part Ⅱ: Definitions and uniqueness[J]. SIAM Journal on Matrix Analysis and Applications, 2008, 30 (3): 1033- 1066. |
109 | De Lathauwer L , Nion 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. |
110 | Shafarevich I R . Basic Algebraic Geometry[M]. Vol. 1 and 2 Berlin: Springer, 2013. |
111 | Chiantini L , Ottaviani G , Vannieuwenhoven N . An algorithm for generic and low-rank specific identifiability of complex tensors[J]. SIAM Journal on Matrix Analysis and Applications, 2014, 35 (4): 1265- 1287. |
112 | Domanov I , De Lathauwer L . Generic uniqueness conditions for the canonical polyadic decomposition and INDSCAL[J]. SIAM Journal on Matrix Analysis and Applications, 2015, 36 (4): 1567- 1589. |
113 | Domanov I , De Lathauwer L . On the uniqueness of the canonical polyadic decomposition of third-order tensors-part Ⅱ: Uniqueness of the overall decomposition[J]. SIAM Journal on Matrix Analysis and Applications, 2013, 34 (3): 876- 903. |
114 | Lickteig T . Typical tensorial rank[J]. Linear Algebra and Its Applications, 1985, 69, 95- 120. |
115 | Alexander J , Hirschowitz A . Polynomial interpolation in several variables[J]. Journal of Algebraic Geometry, 1995, 4 (4): 201- 222. |
116 | Friedland S . On the generic and typical ranks of 3-tensors[J]. Linear Algebra and Its Applications, 2012, 436 (3): 478- 497. |
117 | Comon P , Ten Berge J M F , De Lathauwer L , et al. Generic and typical ranks of multi-way arrays[J]. Linear Algebra and Its Applications, 2009, 430 (11-12): 2997- 3007. |
118 | Comon P , Golub G , Lim L H , et al. Symmetric tensors and symmetric tensor rank[J]. SIAM Journal on Matrix Analysis and Applications, 2008, 30 (3): 1254- 1279. |
119 | Blekherman G , Teitler Z . On maximum, typical and generic ranks[J]. Mathematische Annalen, 2015, 362, 1021- 1031. |
120 | Ten Berge J M F . The typical rank of tall three-way arrays[J]. Psychometrika, 2000, 65 (4): 525- 532. |
121 | Ten Berge J M F , Kiers H A L . Simplicity of core arrays in three-way principal component analysis and the typical rank of $p\times q\times2$ arrays[J]. Linear Algebra and Its Applications, 1999, 294 (1-3): 169- 179. |
122 | Abo H , Ottaviani G , Peterson C . Induction for secant varieties of Segre varieties[J]. Linear Algebra and Its Applications, 2009, 361 (2): 767- 792. |
123 | Ballico E , Bernardi A , Catalisano M V , et al. Grassmann secants, identifiability, and linear systems of tensors[J]. Linear Algebra and Its Applications, 2013, 438 (1): 121- 135. |
124 | Bocci C , Chiantini L . On the identifiability of binary Segre products[J]. Journal of Algebraic Geometry, 2013, 22 (1): 1- 11. |
125 | Bocci C , Chiantini L , Ottaviani G . Refined methods for the identifiability of tensors[J]. Annali di Matematica Pura ed Applicata, 2014, 193, 1691- 1702. |
126 | Chiantini L , Ciliberto C . On the k-th secant order of a projective variety[J]. Journal of the London Mathematical Society, 2006, 73 (2): 436- 454. |
127 | Ranestad K , Schreyer F O . Varieties of sums of powers[J]. Journal Fur Die Reine Und Angewandte Mathematik, 2000, 147- 182. |
128 | Chiantini L , Ciliberto C . On the dimension of secant varieties[J]. Journal of The European Mathematical Society, 2010, 12 (5): 1267- 1291. |
129 | Casarotti A , Mella M . From non-defectivity to identifiability[J]. Journal of The European Mathematical Society, 2022, 25 (3): 913- 931. |
130 | Ballico E . On the weak non-defectivity of Veronese embeddings of projective spaces[J]. Central European Journal of Mathematics, 2005, 3, 183- 187. |
131 | Chiantini L , Ciliberto C . Weakly defective varieties[J]. Transactions of the American Mathematical Society, 2002, 354 (1): 151- 178. |
132 | Mella M . Singularities of linear systems and the Waring problem[J]. Transactions of the American Mathematical Society, 2006, 358 (12): 5523- 5538. |
133 | Angelini E , Bocci C , Chiantini L . Real identifiability vs. complex identifiability[J]. Linear & Multilinear Algebra, 2018, 66 (6): 1257- 1267. |
134 | Chiantini L , Ottaviani G . On generic identifiability of 3-tensors of small rank[J]. SIAM Journal on Matrix Analysis and Applications, 2012, 33 (3): 1018- 1037. |
135 | Chiantini L , Ottaviani G , Vannieuwenhoven N . On generic identifiability of symmetric tensors of subgeneric rank[J]. Transactions of the American Mathematical Society, 2016, 369, 4021- 4042. |
136 | Jiang T , Sidiropoulos N D , Ten Berge J M F . Almost-sure identifiability of multidimensional harmonic retrieval[J]. IEEE Transactions on Signal Processing, 2001, 49 (9): 1849- 1859. |
137 | Ni G Y , Li Y . A semidefinite relaxation method for partially symmetric tensor decomposition[J]. Mathematics of Operations Research, 2022, 47 (4): 2931- 2949. |
138 | Nie J W , Yang Z . Hermitian tensor decompositions[J]. SIAM Journal on Matrix Analysis and Applications, 2020, 41 (3): 1115- 1144. |
139 | Cardoso J F , Souloumiac A . Blind beamforming for non-Gaussian signals[J]. IEE Proceedings F (Radar and Signal Processing), 1993, 140, 362- 370. |
140 | Yang B , Li Y , Ni G Y , et al. Rank-R positive Hermitian approximation algorithms and positive Hermitian decompositions of Hermitian tensors[J]. Science China Mathematics, 2023, 53 (8): 1125- 1144. |
141 | Nie J W , Ye K . Hankel tensor decompositions and ranks[J]. SIAM Journal on Matrix Analysis and Applications, 2019, 40 (2): 486- 516. |
142 | Qi L Q . Hankel tensors: Associated Hankel matrices and Vandermonde decomposition[J]. Communications in Mathematical Sciences, 2014, 13 (1): 113- 125. |
143 | Uschmajew A . Local convergence of the alternating least squares algorithm for canonical tensor approximation[J]. SIAM Journal on Matrix Analysis and Applications, 2012, 33, 639- 652. |
144 | Zhang T , Golub G H . Rank-one approximation to high order tensors[J]. SIAM Journal on Matrix Analysis and Applications, 2001, 23, 534- 550. |
145 | Breiding P , Vannieuwenhoven N . The condition number of join decompositions[J]. SIAM Journal on Matrix Analysis and Applications, 2018, 39 (1): 287- 309. |
146 | Vannieuwenhoven N . Condition numbers for the tensor rank decomposition[J]. Linear Algebra and Its Applications, 2017, 535, 35- 86. |
147 | Nion D . A tensor framework for nonunitary joint block diagonalization[J]. IEEE Transactions on Signal Processing, 2011, 59 (10): 4585- 4594. |
148 | Sørensen T F , De Lathauwer L . New uniqueness conditions for the canonical polyadic decomposition of third-order tensors[J]. SIAM Journal on Matrix Analysis and Applications, 2015, 36 (4): 1381- 1403. |
149 | Yang M , Li W , Xiao M Q . On identifiability of higher order block term tensor decompositions of rank $l_r\otimes $ rank-1[J]. Linear Algebra and Its Applications, 2020, 68 (2): 223- 245. |
150 | Yang M . On partial and generic uniqueness of block term tensor decompositions[J]. Annali Dell'Universita' Di Ferrara, 2014, 60 (2): 465- 493. |
151 | Stegeman A . On uniqueness of the canonical tensor decomposition with some form of symmetry[J]. SIAM Journal on Matrix Analysis and Applications, 2011, 32 (2): 561- 583. |
152 | Sørensen T F , De Lathauwer L . Coupled canonical polyadic decompositions and (coupled) decompositions in multilinear rank-($l_{r, n}, l_{r, n}, 1 $) terms-Part Ⅰ: Uniqueness[J]. SIAM Journal on Matrix Analysis and Applications, 2015, 36 (2): 496- 522. |
153 | De Silva V , Lim L H . Tensor rank and the ill-posedness of the best low-rank approximation problem[J]. SIAM Journal on Matrix Analysis and Applications, 2008, 30, 1084- 1127. |
154 | Hu S , Ye K . Linear convergence of an alternating polar decomposition method for low rank orthogonal tensor approximation[J]. Mathematical Programming, 2023, 199, 1305- 1364. |
155 | Ye K, Hu S L. When geometry meets optimization theory: Partially orthogonal tensors [EB/OL]. [2025-04-09]. arXiv: 2201.04824. |
156 | Chen J , Saad Y . On the tensor SVD and the optimal low rank orthogonal approximation of tensors[J]. SIAM Journal on Matrix Analysis and Applications, 2009, 30, 1709- 1734. |
157 | Mohlenkamp M J . Musings on multilinear fitting[J]. Linear Algebra and Its Applications, 2013, 438, 834- 852. |
158 | Nie J , Wang L , Zheng Z . Higher order correlation analysis for multi-view learning[J]. Pacific Journal of Optimization, 2023, 19, 237- 255. |
159 | Luo Y , Tao D , Ramamohanarao K , et al. Tensor canonical correlation analysis for multi-view dimension reduction[J]. IEEE Transactions on Knowledge and Data Engineering, 2015, 27, 3111- 3124. |
160 | Mohlenkamp M J . The dynamics of swamps in the canonical tensor approximation problem[J]. SIAM Journal on Applied Dynamical Systems, 2019, 18, 1293- 1333. |
161 | Mohlenkamp M J , Young T R , Barany B . Transient dynamics of block coordinate descent in a valley[J]. International Journal of Numerical Analysis And Modeling, 2020, 17, 557- 591. |
162 | Lim L H. Singular values and eigenvalues of tensors: A variational approach [C]//Proceedings of the 1st IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, 2005: 129-132. |
163 | Stegeman A . Low-rank approximation of generic $p\times q\times 2 $ arrays and diverging components in the Candecomp/Parafac model[J]. SIAM Journal on Matrix Analysis and Applications, 2008, 30, 988- 1007. |
164 | Evert E , De Lathauwer L . Guarantees for existence of a best canonical polyadic approximation of a noisy low rank tensor[J]. SIAM Journal on Matrix Analysis and Applications, 2022, 43, 328- 369. |
165 | Evert E , De Lathauwer L . On best low rank approximation of positive definite tensors[J]. SIAM Journal on Matrix Analysis and Applications, 2023, 44, 867- 893. |
166 | Qi Y , Michałek M , Lim L H . Complex best r-term approximations almost always exist in finite dimensions[J]. Applied and Computational Harmonic Analysis, 2020, 49 (1): 180- 207. |
167 | Stegeman A . Degeneracy in Candecomp/Parafac and Indscal explained for several three-sliced arrays with a two-valued typical rank[J]. Psychometrika, 2007, 72, 601- 619. |
168 | Hoff P D . Separable covariance arrays via the Tucker product[J]. Bayesian Analysis, 2011, 6, 179- 196. |
169 | Cohen J E, Usevich K, Comon P. A tour of constrained tensor canonical polyadic decomposition [R]. Grenoble: GIPSA-lab, 2016. |
170 | Friedland S , Stawiska M . Some approximation problems in semi-algebraic geometry[J]. Banach Center Publications, 2015, 107, 133- 147. |
171 | Friedland S , Ottaviani G . The number of singular vector tuples and uniqueness of best rank-one approximation of tensors[J]. Foundations of Computational Mathematics, 2014, 14, 1209- 1242. |
172 | Dewaele N , Breiding P , Vannieuwenhoven N . The condition number of many tensor decompositions is invariant under tucker compression[J]. Numerical Algorithms, 2023, 94 (2): 1003- 1029. |
173 | Breiding P , Vannieuwenhoven N . On the average condition number of tensor rank decompositions[J]. IMA Journal of Numerical Analysis, 2020, 40 (3): 1908- 1936. |
174 | Beltrán C , Breiding P , Vannieuwenhoven N . The average condition number of most tensor rank decomposition problems is infinite[J]. Foundations of Computational Mathematics, 2023, 23 (2): 433- 491. |
175 | De Lathauwer L . First-order perturbation analysis of the best rank-$(r_1, r_2, r_3)$ approximation in multilinear algebra[J]. Journal of Chemometrics, 2004, 18, 2- 11. |
176 | Eldén L , Savas B . Perturbation theory and optimality conditions for the best multilinear rank approximation of a tensor[J]. SIAM Journal on Matrix Analysis and Applications, 2011, 32, 1422- 1450. |
177 | Luo Y , Raskutti G , Yuan M , et al. A sharp blockwise tensor perturbation bound for orthogonal iteration[J]. Journal of Machine Learning Research, 2021, 22, 1- 48. |
178 | Luo Z Y , Qi L Q . Optimality conditions for Tucker low-rank tensor optimization[J]. Computational Optimization and Applications, 2023, 86 (3): 1275- 1298. |
179 | Mu C , Hsu D , Goldfarb D . Successive rank-one approximations for nearly orthogonally decomposable symmetric tensors[J]. SIAM Journal on Matrix Analysis and Applications, 2015, 36, 1638- 1659. |
180 | Belkin M , Rademacher L , Voss J . Eigenvectors of orthogonally decomposable functions[J]. SIAM Journal on Computing, 2018, 47, 547- 615. |
181 | Auddy A , Yuan M . Perturbation bounds for (nearly) orthogonally decomposable tensors with statistical applications[J]. Information and Inference: A Journal of the IMA, 2023, 12, 1044- 1072. |
182 | Breiding P , Vannieuwenhoven N . The condition number of Riemannian approximation problems[J]. SIAM Journal on Optimization, 2021, 31 (1): 1049- 1077. |
183 | Cai Y F , Liu C Y . An algebraic approach to nonorthogonal general joint block diagonalization[J]. SIAM Journal on Matrix Analysis and Applications, 2017, 38 (1): 50- 71. |
184 | Cai Y F , Li R C . Perturbation analysis for matrix joint block diagonalization[J]. Linear Algebra and Its Applications, 2019, 581, 163- 197. |
185 | Cardoso J F. Multidimensional independent component analysis [C]//Proceedings of the 1998 IEEE International Conference on Acoustics, Speech and Signal Processing, 1998, 4: 1941-1944. |
186 | Russo F G . A note on joint diagonalizers of shear matrices[J]. Science Asia, 2012, 38, 401- 407. |
187 | Afsari B . Sensitivity analysis for the problem of matrix joint diagonalization[J]. SIAM Journal on Matrix Analysis and Applications, 2008, 30 (3): 1148- 1171. |
188 | Shi Q Z , Cai Y F , Xu S L . Some perturbation results for a normalized non-orthogonal joint diagonalization problem[J]. Linear Algebra and Its Applications, 2015, 484, 457- 476. |
189 | Grcar J F. Optimal sensitivity analysis of linear least squares [R]. Berkeley: Lawrence Berkeley National Laboratory, 2003. |
190 | Hu S , Li G . Convergence rate analysis for the higher order power method in best rank one approximations of tensors[J]. Numerische Mathematik, 2018, 140, 993- 1031. |
191 | Hu S L . Nondegeneracy of eigenvectors and singular vector tuples of tensors[J]. Science China Mathematics, 2022, 65 (12): 2483- 2492. |
192 | Nie J , Wang L . Semidefinite relaxations for best rank-1 tensor approximations[J]. SIAM Journal on Matrix Analysis and Applications, 2014, 35, 1155- 1179. |
193 | Hu S , Sun D , Toh K C . Best nonnegative rank-one approximations of tensors[J]. SIAM Journal on Matrix Analysis and Applications, 2019, 40, 1527- 1554. |
194 | Hu S . Certifying the global optimality of quartic minimization over the sphere[J]. Journal of the Operations Research Society of China, 2022, 10, 241- 287. |
195 | Hu S L , Yan Z F . Quadratic growth and linear convergence of a DCA method for quartic minimization over the sphere[J]. Journal of Optimization Theory and Applications, 2024, 201 (1): 378- 395. |
196 | Hu S L , Huang Z H , Qi L Q . Finding the extreme Z-eigenvalues of tensors via a sequential semidefinite programming method[J]. Numerical Linear Algebra with Applications, 2013, 20 (6): 972- 984. |
197 | Qi L Q , Wang F , Wang Y J . Z-eigenvalue methods for a global polynomial optimization problem[J]. Mathematical Programming, 2009, 118 (2): 301- 316. |
198 | Hu S L , Huang Z , Ni H , et al. Positive definiteness of diffusion kurtosis imaging[J]. Inverse Problems and Imaging, 2012, 6 (1): 57- 75. |
199 | He T T , Hu S L , Huang Z H . A low-rank tensor completion method via Strassen-Ottaviani flattening[J]. SIAM Journal on Imaging Sciences, 2024, 17 (4): 2242- 2276. |
200 | Frandsen A , Ge R . Optimization landscape of Tucker decomposition[J]. Mathematical Programming, 2022, 193, 687- 712. |
201 | Draisma J , Ottaviani G , Tocino A . Best rank-k approximations for tensors: Generalizing eckart-young[J]. Research in the Mathematical Sciences, 2018, 5, 27. |
202 | Yang Y N, Hu S L, De Lathauwer L, et al. Convergence study of block singular value maximization methods for rank-1 approximation to higher order tensors [EB/OL]. [2025-04-01]. https://ftp.esat.kuleuven.be/pub/pub/pub/sista/yyang/study.pdf |
203 | Levin E. Towards optimization on varieties [D]. Princeton: Princeton University, 2020. |
204 | Hu S L, Ye K. Generic linear convergence for algorithms of non-linear least squares over smooth varieties [EB/OL]. [2025-04-09]. arXiv: 2503.06877. |
205 | Levin E , Kileel J , Boumal N . Finding stationary points on bounded-rank matrices: A geometric hurdle and a smooth remedy[J]. Mathematical Programming, 2023, 199 (1): 831- 864. |
206 | Levin E , Kileel J , Boumal N . The effect of smooth parametrizations on nonconvex optimization landscapes[J]. Mathematical Programming, 2025, 209 (1): 63- 111. |
207 | Ge R, Huang F, Jin C, et al. Escaping from saddle points-online stochastic gradient for tensor decomposition [C]// Proceedings of the 28th Conference on Learning Theory, 2015: 797-842. |
208 | So A M C . Deterministic approximation algorithms for sphere constrained homogeneous polynomial optimization problems[J]. Mathematical Programming, 2011, 129, 357- 382. |
209 | Song Z, Woodruff D P, Zhong P. Relative error tensor low rank approximation [C]// Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, 2019: 2772-2789. |
210 | Ge R , Ma T Y . On the optimization landscape of tensor decompositions[J]. Mathematical Programming, 2022, 193 (2): 713- 759. |
211 | Roemer F , Haardt M . A semi-algebraic framework for approximate CP decompositions via simultaneous matrix diagonalizations (secsi)[J]. Signal Processing, 2013, 93, 2722- 2738. |
212 | Marmin A, Castella M, Pesquet J C. Globally optimizing owing to tensor decomposition [C]//IEEE European Signal Processing Conference, 2020: 990-994. |
213 | Omberg L , Golub G H , Alter O . A tensor higher-order singular value decomposition for integrative analysis of DNA microarray data from different studies[J]. Proceedings of the National Academy of Sciences of the United States of America, 2007, 104, 18371- 18376. |
214 | Bhaskara A , Charikar M , Vijayaraghavan A . Uniqueness of tensor decompositions with applications to polynomial identifiability[J]. Journal of Machine Learning Research, 2014, 35, 742- 778. |
215 | Beltrán C , Breiding P , Vannieuwenhoven N . Pencil-based algorithms for tensor rank decomposition are not stable[J]. SIAM Journal on Matrix Analysis and Applications, 2019, 40 (2): 739- 773. |
216 | Ibrahim S , Fu X , Li X G . On recoverability of randomly compressed tensors with low CP rank[J]. IEEE Signal Processing Letters, 2020, 27, 1125- 1129. |
217 | Breiding P , Vannieuwenhoven N . Convergence analysis of Riemannian Gauss-Newton methods and its connection with the geometric condition number[J]. Applied Mathematics Letters, 2018, 78, 42- 50. |
218 | Dewaele N , Vannieuwenhoven N . Which constraints of a numerical problem cause ill-conditioning?[J]. Numerische Mathematik, 2024, 156 (4): 1427- 1453. |
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