2607:
1233:
2195:
842:
2602:{\displaystyle {\begin{array}{rcl}{\mathcal {A}}&=&{\mathcal {A}}\times ({\bf {U}}_{1}{\bf {U}}_{1}^{H},{\bf {U}}_{2}{\bf {U}}_{2}^{H},\ldots ,{\bf {U}}_{M}{\bf {U}}_{M}^{H})\\&=&\left({\mathcal {A}}\times ({\bf {U}}_{1}^{H},{\bf {U}}_{2}^{H},\ldots ,{\bf {U}}_{M}^{H})\right)\times ({\bf {U}}_{1},{\bf {U}}_{2},\ldots ,{\bf {U}}_{M})\\&=&{\mathcal {S}}\times ({\bf {U}}_{1},{\bf {U}}_{2},\ldots ,{\bf {U}}_{M}),\end{array}}}
1228:{\displaystyle {\begin{array}{rcl}{\mathcal {A}}&=&{\mathcal {A}}\times ({\bf {I}},{\bf {I}},\ldots ,{\bf {I}})\\&=&{\mathcal {A}}\times ({\bf {U}}_{1}{\bf {U}}_{1}^{H},{\bf {U}}_{2}{\bf {U}}_{2}^{H},\ldots ,{\bf {U}}_{M}{\bf {U}}_{M}^{H})\\&=&\left({\mathcal {A}}\times ({\bf {U}}_{1}^{H},{\bf {U}}_{2}^{H},\ldots ,{\bf {U}}_{M}^{H})\right)\times ({\bf {U}}_{1},{\bf {U}}_{2},\ldots ,{\bf {U}}_{M}),\end{array}}}
5980:
3814:
3457:
5739:
4348:, and by Vasilescu and Terzopulous. The term HOSVD was coined by Lieven De Lathauwer, but the algorithm typically referred to in the literature as HOSVD was introduced by Vasilescu and Terzopoulos with the name M-mode SVD. It is a parallel computation that employs the matrix SVD to compute the orthonormal mode matrices.
4779:
3563:
3206:
2130:
6877:
Starting in the early 2000s, Vasilescu addressed causal questions by reframing the data analysis, recognition and synthesis problems as multilinear tensor problems. The power of the tensor framework was showcased by decomposing and representing an image in terms of its causal factors of data
7633:"Multilinear Analysis of Image Ensembles: TensorFaces," Proc. 7th European Conference on Computer Vision (ECCV'02), Copenhagen, Denmark, May, 2002, in Computer Vision -- ECCV 2002, Lecture Notes in Computer Science, Vol. 2350, A. Heyden et al. (Eds.), Springer-Verlag, Berlin, 2002, 447–460.
4062:
6863:
5975:{\displaystyle \min _{{\mathcal {\bar {A}}}\in \mathbb {C} ^{I_{1}\times I_{2}\times \cdots \times I_{M}}}{\frac {1}{2}}\|{\mathcal {A}}-{\mathcal {\bar {A}}}\|_{F}^{2}\quad {\text{s.t.}}\quad \mathrm {rank-} ({\bar {R}}_{1},{\bar {R}}_{2},\ldots ,{\bar {R}}_{M}),}
1413:
5390:
222:
5707:
409:
7896:"Tensor GSVD of Patient- and Platform-Matched Tumor and Normal DNA Copy-Number Profiles Uncovers Chromosome Arm-Wide Patterns of Tumor-Exclusive Platform-Consistent Alterations Encoding for Cell Transformation and Predicting Ovarian Cancer Survival"
85:
The term higher order singular value decomposition (HOSVD) was coined be DeLathauwer, but the algorithm referred to commonly in the literature as the HOSVD and attributed to either Tucker or DeLathauwer was developed by
Vasilescu and Terzopoulos.
1953:
1537:
4628:
5289:
via the Fused In-place
Sequentially Truncated Higher Order Singular Value Decomposition (FIST-HOSVD) algorithm by overwriting the original tensor by the HOSVD core tensor, significantly reducing the memory consumption of computing HOSVD.
4218:
3201:
6087:
7645:"TensorTextures: Multilinear Image-Based Rendering", M. A. O. Vasilescu and D. Terzopoulos, Proc. ACM SIGGRAPH 2004 Conference Los Angeles, CA, August, 2004, in Computer Graphics Proceedings, Annual Conference Series, 2004, 336–342.
3809:{\displaystyle {\mathcal {A}}=\sum _{r_{1}=1}^{R_{1}}\sum _{r_{2}=1}^{R_{2}}\cdots \sum _{r_{M}=1}^{R_{M}}s_{r_{1},r_{2},\ldots ,r_{M}}\mathbf {u} _{r_{1}}\otimes \mathbf {u} _{r_{2}}\otimes \cdots \otimes \mathbf {u} _{r_{M}},}
3452:{\displaystyle {\mathcal {S}}=\sum _{r_{1}=1}^{R_{1}}\sum _{r_{2}=1}^{R_{2}}\cdots \sum _{r_{M}=1}^{R_{M}}s_{r_{1},r_{2},\ldots ,r_{M}}\mathbf {e} _{r_{1}}\otimes \mathbf {e} _{r_{2}}\otimes \cdots \otimes \mathbf {e} _{r_{M}},}
4535:
5202:
5515:
6906:. This extension led to the definition of the HOSVD-based canonical form of tensor product functions and Linear Parameter Varying system models and to convex hull manipulation based control optimization theory, see
6750:
6560:
3923:
3918:
6317:
5060:
1622:
649:
4594:
1303:
2690:
5272:
2890:
6667:. Unfortunately, truncation does not result in an optimal solution for the best low multilinear rank optimization problem,. However, both the classically and interleaved truncated HOSVD result in a
6665:
6422:
6155:
6885:
The HOSVD has been successfully applied to signal processing and big data, e.g., in genomic signal processing. These applications also inspired a higher-order GSVD (HO GSVD) and a tensor GSVD.
4872:
1442:
5120:
3021:
4283:
3097:
2791:
7606:"Human Motion Signatures: Analysis, Synthesis, Recognition," Proceedings of International Conference on Pattern Recognition (ICPR 2002), Vol. 3, Quebec City, Canada, Aug, 2002, 456–460.
6745:
6707:
2190:
2136:(in the Hermitian inner product) and the last equality is due to the properties of multilinear multiplication. As flattenings are bijective maps and the above formula is valid for all
8158:"Identification of candidate drugs using tensor-decomposition-based unsupervised feature extraction in integrated analysis of gene expression between diseases and DrugMatrix datasets"
4971:
5301:
3558:
133:
7135:
5546:
2965:
6865:
in practice this means that if there exists an optimal solution with a small error, then a truncated HOSVD will for many intended purposes also yield a sufficiently good solution.
5575:
3850:
3493:
278:
82:
in their
Multilinear SVD work that employs the power method, or advocated by Vasilescu and Terzopoulos that developed M-mode SVD a parallel algorithm that employs the matrix SVD.
4442:
1948:
1880:
1783:
1663:
795:
690:
559:
478:
2919:
2820:
723:
3026:
The compact HOSVD is a rank-revealing decomposition in the sense that the dimensions of its core tensor correspond with the components of the multilinear rank of the tensor.
6596:
6474:
6353:
6242:
5570:
5414:
4916:
4827:
4623:
4130:
4086:
3121:
2726:
2631:
1687:
1437:
583:
522:
441:
124:
4396:
1911:
1814:
1718:
1295:
829:
754:
4774:{\displaystyle {\mathcal {S}}={\mathcal {A}}\times _{1}{\bf {U}}_{1}^{H}\times _{2}{\bf {U}}_{2}^{H}\ldots \times _{m}{\bf {U}}_{m}^{H}\ldots \times _{M}{\bf {U}}_{M}^{H}}
4327:
4305:
270:
248:
1260:
6188:
5734:
4143:
3520:
3126:
1842:
1745:
6888:
A combination of HOSVD and SVD also has been applied for real-time event detection from complex data streams (multivariate data with space and time dimensions) in
2125:{\displaystyle {\mathcal {A}}_{}={\bf {U}}_{m}{\bf {U}}_{m}^{H}{\mathcal {A}}_{}={\bigl (}{\mathcal {A}}\times _{m}({\bf {U}}_{m}{\bf {U}}_{m}^{H}){\bigr )}_{},}
5985:
4106:
498:
6197:
A simple idea for trying to solve this optimization problem is to truncate the (compact) SVD in step 2 of either the classic or the interlaced computation. A
4453:
7131:
M. A. O. Vasilescu, D. Terzopoulos (2002) with the name M-mode SVD. The M-mode SVD is suitable for parallel computation and employs the matrix SVD
1624:
is a matrix with unitary columns containing a basis of the left singular vectors corresponding to the nonzero singular values of the standard factor-
7281:
Markopoulos, Panos P.; Chachlakis, Dimitris G.; Papalexakis, Evangelos (April 2018). "The Exact
Solution to Rank-1 L1-Norm TUCKER2 Decomposition".
7337:
Markopoulos, Panos P.; Chachlakis, Dimitris G.; Prater-Bennette, Ashley (21 February 2019). "L1-Norm Higher-Order
Singular-Value Decomposition".
5128:
5419:
6479:
6879:
4057:{\displaystyle B=\{\mathbf {u} _{r_{1}}\otimes \mathbf {u} _{r_{2}}\otimes \cdots \otimes \mathbf {u} _{r_{M}}\}_{r_{1},r_{2},\ldots ,r_{M}}}
7770:"Tensor Decomposition Reveals Concurrent Evolutionary Convergences and Divergences and Correlations with Structural Motifs in Ribosomal RNA"
6913:
HOSVD was proposed to be applied to multi-view data analysis and was successfully applied to in silico drug discovery from gene expression.
7620:
M. A. O. Vasilescu, D. Terzopoulos, Proc. Computer Vision and
Pattern Recognition Conf. (CVPR '03), Vol.2, Madison, WI, June, 2003, 93–99.
7961:
Hadi Fanaee-T; João Gama (May 2015). "EigenEvent: An algorithm for event detection from complex data streams in
Syndromic surveillance".
7132:
6907:
3855:
6247:
8055:
P. Baranyi; D. Tikk; Y. Yam; R. J. Patton (2003). "From
Differential Equations to PDC Controller Design via Numerical Transformation".
4979:
1561:
588:
7175:, "Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR’05), San Diego, CA, June 2005, vol.1, 547–553."
4540:
1551:
of a matrix, where the rows and columns corresponding to vanishing singular values are dropped, it is also possible to consider a
6858:{\displaystyle \|{\mathcal {A}}-{\mathcal {\bar {A}}}_{t}\|_{F}\leq {\sqrt {M}}\|{\mathcal {A}}-{\mathcal {\bar {A}}}^{*}\|_{F};}
2636:
5207:
2825:
7494:
7354:
6601:
6358:
6092:
412:
7658:"A Tensor Higher-Order Singular Value Decomposition for Integrative Analysis of DNA Microarray Data From Different Studies"
7833:"A Higher-Order Generalized Singular Value Decomposition for Comparison of Global mRNA Expression from Multiple Organisms"
7058:
In N. Frederiksen and H. Gulliksen (Eds.), Contributions to
Mathematical Psychology. New York: Holt, Rinehart and Winston
6896:
4836:
1548:
8099:"Tensor decomposition-based unsupervised feature extraction applied to matrix products for multi-view data processing"
7544:
1408:{\displaystyle {\mathcal {S}}:={\mathcal {A}}\times ({\bf {U}}_{1}^{H},{\bf {U}}_{2}^{H},\ldots ,{\bf {U}}_{M}^{H}).}
7155:, "Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR’03), Madison, WI, June, 2003"
5065:
4223:
3037:
2970:
2731:
6712:
6674:
5385:{\displaystyle {\mathcal {A}}\in \mathbb {C} ^{I_{1}\times I_{2}\times \cdots \times I_{m}\cdots \times I_{M}}}
2139:
217:{\displaystyle {\mathcal {A}}\in \mathbb {C} ^{I_{1}\times I_{2}\cdots \times \cdots I_{m}\cdots \times I_{M}}}
4064:
is an orthonormal set of tensors. This means that the HOSVD can be interpreted as a way to express the tensor
5702:{\displaystyle \mathrm {rank-} ({\bar {R}}_{1},{\bar {R}}_{2},\ldots ,{\bar {R}}_{m},\ldots ,{\bar {R}}_{M})}
4928:
404:{\displaystyle {\mathcal {A}}_{}\in \mathbb {C} ^{I_{m}\times (I_{1}I_{2}\cdots I_{m-1}I_{m+1}\cdots I_{M})}}
3525:
8218:
7605:
5520:
4448:
2924:
40:
7078:
De
Lathauwer, L.; De Moor, B.; Vandewalle, J. (2000-01-01). "A Multilinear Singular Value Decomposition".
5298:
In applications, such as those mentioned below, a common problem consists of approximating a given tensor
3819:
3462:
7880:
7817:
7644:
7041:
Tucker, L. R. (1963). "Implications of factor analysis of three-way matrices for measurement of change".
2133:
7754:
6952:
Hitchcock, Frank L (1928-04-01). "Multiple Invariants and Generalized Rank of a M-Way Array or Tensor".
1532:{\displaystyle {\mathcal {A}}={\mathcal {S}}\times ({\bf {U}}_{1},{\bf {U}}_{2},\ldots ,{\bf {U}}_{M}).}
4410:
1916:
1848:
1750:
1631:
836:
763:
658:
527:
446:
2895:
2796:
699:
7717:"Global Effects of DNA Replication and DNA Replication Origin Activity on Eukaryotic Gene Expression"
7943:
7574:
7092:
6874:
The HOSVD is most commonly applied to the extraction of relevant information from multi-way arrays.
6565:
6443:
6322:
6211:
5551:
5395:
4880:
4792:
4604:
4111:
4067:
3102:
2707:
2612:
1668:
1418:
564:
503:
422:
105:
4363:
1885:
1788:
1692:
1269:
803:
728:
7632:
4310:
4288:
253:
231:
7715:
L. Omberg; J. R. Meyerson; K. Kobayashi; L. S. Drury; J. F. X. Diffley; O. Alter (October 2009).
6903:
4337:
The strategy for computing the Multilinear SVD and the M-mode SVD was introduced in the 1960s by
4829:
consists of interlacing the computation of the core tensor and the factor matrices, as follows:
1238:
8086:. 3rd International Conference on Mechatronics (ICM 2006). Budapest, Hungary. pp. 660–665.
7569:
7185:
Godfarb, Donald; Zhiwei, Qin (2014). "Robust low-rank tensor recovery: Models and algorithms".
7087:
6160:
5712:
4213:{\displaystyle {\mathcal {A}}\in {\mathbb {C} }^{I_{1}\times I_{2}\times \cdots \times I_{M}}}
3196:{\displaystyle {\mathcal {S}}\in {\mathbb {C} }^{R_{1}\times R_{2}\times \cdots \times R_{M}}}
7142:, Proc. 7th European Conference on Computer Vision (ECCV'02), Copenhagen, Denmark, May, 2002
8017:
P. Baranyi (April 2004). "TP model transformation as a way to LMI based controller design".
7235:
Chachlakis, Dimitris G.; Prater-Bennette, Ashley; Markopoulos, Panos P. (22 November 2019).
6878:
formation, in the context of Human Motion Signatures for gait recognition, face recognition—
8169:
8110:
7980:
7907:
7844:
7781:
7669:
7527:
FIST-HOSVD: Fused in-Place Sequentially Truncated Higher Order Singular Value Decomposition
7432:
7300:
6930:
6889:
3498:
1820:
1723:
72:
56:
36:
6987:
Tucker, Ledyard R. (1966-09-01). "Some mathematical notes on three-mode factor analysis".
6082:{\displaystyle ({\bar {R}}_{1},{\bar {R}}_{2},\ldots ,{\bar {R}}_{M})\in \mathbb {N} ^{M}}
127:
8:
7152:
6747:
denotes the optimal solution to the best low multilinear rank approximation problem, then
4342:
3034:
The following geometric interpretation is valid for both the full and compact HOSVD. Let
1263:
76:
20:
8173:
8114:
7984:
7911:
7848:
7785:
7673:
7617:
7436:
7304:
7056:
Tucker, L. R. (1964). "The extension of factor analysis to three-dimensional matrices".
8190:
8157:
8133:
8098:
8034:
7996:
7970:
7930:
7895:
7867:
7832:
7804:
7769:
7741:
7716:
7692:
7657:
7500:
7456:
7360:
7316:
7290:
7248:
7212:
7194:
7020:
4091:
483:
8068:
7947:
2200:
847:
8195:
8138:
7935:
7872:
7809:
7746:
7697:
7587:
7540:
7504:
7490:
7448:
7420:
7350:
7105:
7043:
In C. W. Harris (Ed.), Problems in Measuring Change. Madison, Wis.: Univ. Wis. Press.
7012:
7004:
6969:
6926:
87:
60:
48:
8000:
7893:
7830:
7560:
Grasedyck, L. (2010-01-01). "Hierarchical Singular Value Decomposition of Tensors".
7460:
7364:
7024:
8223:
8185:
8177:
8128:
8118:
8064:
8038:
8026:
7988:
7925:
7915:
7862:
7852:
7799:
7789:
7736:
7728:
7687:
7677:
7579:
7530:
7482:
7440:
7394:
7346:
7342:
7320:
7308:
7258:
7216:
7204:
7097:
6996:
6961:
4530:{\displaystyle {\mathcal {A}}_{}={\bf {U}}_{m}{\bf {\Sigma }}_{m}{\bf {V}}_{m}^{T}}
52:
8123:
7920:
7857:
7794:
7172:
7139:
64:
44:
7263:
7236:
8181:
7399:
7382:
6829:
6775:
6724:
6686:
6191:
5848:
5755:
5531:
652:
7714:
7486:
7101:
6902:
The concept of HOSVD was carried over to functions by Baranyi and Yam via the
8212:
7767:
7591:
7452:
7421:"A New Truncation Strategy for the Higher-Order Singular Value Decomposition"
7312:
7109:
7008:
6973:
5392:
by one with a reduced multilinear rank. Formally, if the multilinear rank of
8030:
7831:
S. P. Ponnapalli; M. A. Saunders; C. F. Van Loan; O. Alter (December 2011).
7682:
7535:
7339:
2018 IEEE Global Conference on Signal and Information Processing (GlobalSIP)
8199:
8142:
7939:
7876:
7813:
7750:
7701:
5197:{\displaystyle {\mathcal {A}}^{m}=U_{m}^{H}\times _{m}{\mathcal {A}}^{m-1}}
4338:
68:
7525:
Cobb, Benjamin; Kolla, Hemanth; Phipps, Eric; Çatalyürek, Ümit V. (2022).
7016:
7894:
P. Sankaranarayanan; T. E. Schomay; K. A. Aiello; O. Alter (April 2015).
6965:
5510:{\displaystyle \mathrm {rank-} (R_{1},R_{2},\ldots ,R_{m},\ldots ,R_{M})}
831:
does not depend on the particular on the specific definition of the mode
8084:
Definition of the HOSVD-based canonical form of polytopic dynamic models
7732:
6555:{\displaystyle {\mathcal {A}}_{}^{m-1}\approx U_{m}\Sigma _{m}V_{m}^{T}}
7992:
7000:
7583:
7444:
7208:
126:
is assumed to be given in coordinates with respect to some basis as a
7336:
7234:
6933:. L1-HOSVD is the analogous of HOSVD for the solution to L1-Tucker.
8081:
8054:
7295:
7253:
6922:
6436:) is obtained by replacing step 2 in the interlaced computation by
3913:{\displaystyle {\bf {U}}_{m}\in {\mathbb {C} }^{I_{m}\times R_{m}}}
91:
7975:
7768:
C. Muralidhara; A. M. Gross; R. R. Gutell; O. Alter (April 2011).
7418:
7199:
6312:{\displaystyle {\mathcal {A}}_{}\approx U_{m}\Sigma _{m}V_{m}^{T}}
7419:
Vannieuwenhoven, N.; Vandebril, R.; Meerbergen, K. (2012-01-01).
5055:{\displaystyle {\mathcal {A}}_{}^{m-1}=U_{m}\Sigma _{m}V_{m}^{T}}
1617:{\displaystyle {\bf {U}}_{m}\in \mathbb {C} ^{I_{m}\times R_{m}}}
644:{\displaystyle {\bf {U}}_{m}\in \mathbb {C} ^{I_{m}\times I_{m}}}
39:. It may be regarded as one type of generalization of the matrix
7280:
3560:. By definition of the multilinear multiplication, it holds that
32:
7481:. Springer Series in Computational Mathematics. Vol. 42.
6204:
is obtained by replacing step 2 in the classic computation by
7077:
4589:{\displaystyle {\bf {U}}\in \mathbb {C} ^{I_{m}\times R_{m}}}
250:
is the complex numbers and it includes both the real numbers
8082:
P. Baranyi; L. Szeidl; P. Várlaki; Y. Yam (July 3–5, 2006).
7655:
6709:
denotes the classically or sequentially truncated HOSVD and
2921:
are multilinear ranks. The multilinear ranks are bounded by
1539:
The above construction shows that every tensor has a HOSVD.
2685:{\displaystyle R_{1}\times R_{2}\times \cdots \times R_{M}}
7479:
Tensor Spaces and Numerical Tensor Calculus | SpringerLink
5267:{\displaystyle {\mathcal {A}}_{}^{m}=\Sigma _{m}V_{m}^{T}}
4108:
with the coefficients given as the multidimensional array
7524:
4789:
A strategy that is significantly faster when some or all
2885:{\displaystyle R_{m}:=\mathrm {rank} ({\mathcal {A}}_{})}
4088:
with respect to a specifically chosen orthonormal basis
3203:
is a multidimensional array, we can expand it as follows
6660:{\displaystyle U_{m}\in F^{I_{m}\times {\bar {R}}_{m}}}
6417:{\displaystyle U_{m}\in F^{I_{m}\times {\bar {R}}_{m}}}
6150:{\displaystyle 1\leq {\bar {R}}_{m}<R_{m}\leq I_{m}}
655:
containing a basis of the left singular vectors of the
7960:
7133:"Multilinear Analysis of Image Ensembles: TensorFaces"
2973:
6753:
6715:
6677:
6604:
6568:
6482:
6446:
6361:
6325:
6250:
6214:
6163:
6095:
5988:
5742:
5715:
5578:
5554:
5523:
5422:
5398:
5304:
5210:
5131:
5068:
4982:
4931:
4883:
4839:
4795:
4631:
4607:
4543:
4456:
4413:
4366:
4313:
4291:
4226:
4146:
4114:
4094:
4070:
3926:
3858:
3822:
3566:
3528:
3501:
3465:
3209:
3129:
3105:
3040:
2927:
2898:
2828:
2799:
2734:
2710:
2639:
2615:
2198:
2142:
2132:
where the first equality is due to the properties of
1956:
1919:
1888:
1851:
1823:
1791:
1753:
1726:
1695:
1671:
1634:
1564:
1445:
1421:
1306:
1272:
1241:
845:
806:
766:
731:
702:
661:
591:
567:
530:
506:
486:
449:
425:
281:
256:
234:
136:
108:
102:
For the purpose of this article, the abstract tensor
7529:. Platform for Advanced Scientific Computing(PASC).
228:
is the number of modes and the order of the tensor.
7618:"Multilinear Subspace Analysis for Image Ensembles,
4976:Compute the (compact) singular value decomposition
7656:L. Omberg; G. H. Golub; O. Alter (November 2007).
7153:"Multilinear Subspace Analysis of Image Ensembles"
6857:
6739:
6701:
6659:
6590:
6554:
6468:
6416:
6347:
6311:
6236:
6182:
6149:
6081:
5974:
5728:
5701:
5564:
5540:
5509:
5408:
5384:
5266:
5196:
5114:
5054:
4965:
4910:
4866:
4821:
4773:
4617:
4588:
4529:
4436:
4390:
4321:
4299:
4277:
4212:
4124:
4100:
4080:
4056:
3912:
3844:
3808:
3552:
3514:
3487:
3451:
3195:
3115:
3091:
3015:
2959:
2913:
2884:
2814:
2785:
2720:
2684:
2625:
2601:
2184:
2124:
1942:
1905:
1874:
1836:
1808:
1777:
1739:
1712:
1681:
1657:
1616:
1531:
1431:
1407:
1289:
1254:
1227:
823:
789:
748:
717:
684:
643:
577:
553:
516:
492:
472:
435:
403:
264:
242:
216:
118:
94:-based variants of HOSVD have also been proposed.
71:who developed for third-order tensors the general
4867:{\displaystyle {\mathcal {A}}^{0}={\mathcal {A}}}
8210:
7562:SIAM Journal on Matrix Analysis and Applications
7187:SIAM Journal on Matrix Analysis and Applications
7167:
7165:
7163:
7161:
7127:
7125:
7123:
7121:
7119:
7080:SIAM Journal on Matrix Analysis and Applications
5744:
7625:
7598:
7637:
7610:
7520:
7518:
7516:
7514:
5115:{\displaystyle U_{m}\in F^{I_{m}\times R_{m}}}
8012:
8010:
7380:
7184:
7158:
7116:
2102:
2038:
8155:
8096:
7332:
7330:
7276:
7274:
7173:"Multilinear Independent Component Analysis"
7145:
7036:
7034:
6843:
6808:
6789:
6754:
6171:
6164:
5856:
5828:
4006:
3933:
3016:{\textstyle R_{m}\leq \prod _{i\neq m}R_{i}}
63:. Some aspects can be traced as far back as
8050:
8048:
8019:IEEE Transactions on Industrial Electronics
7511:
7230:
7228:
7226:
6908:TP model transformation in control theories
4278:{\displaystyle (R_{1},R_{2},\ldots ,R_{M})}
3092:{\displaystyle (R_{1},R_{2},\ldots ,R_{M})}
2786:{\displaystyle (R_{1},R_{2},\ldots ,R_{M})}
8016:
8007:
7381:Carlini, Enrico; Kleppe, Johannes (2011).
7178:
7171:M. A. O. Vasilescu, D. Terzopoulos (2005)
7151:M. A. O. Vasilescu, D. Terzopoulos (2003)
4784:
8189:
8132:
8122:
7974:
7929:
7919:
7866:
7856:
7803:
7793:
7740:
7691:
7681:
7573:
7559:
7534:
7476:
7398:
7327:
7294:
7271:
7262:
7252:
7198:
7091:
7049:
7031:
6951:
6916:
6740:{\displaystyle {\mathcal {\bar {A}}}^{*}}
6702:{\displaystyle {\mathcal {\bar {A}}}_{t}}
6069:
5767:
5317:
4556:
4315:
4293:
4160:
3879:
3532:
3143:
2901:
2802:
2185:{\displaystyle m=1,2,\ldots ,m,\ldots ,M}
1584:
611:
307:
258:
236:
149:
25:higher-order singular value decomposition
8045:
7643:M.A.O. Vasilescu, D. Terzopoulos (2004)
7631:M.A.O. Vasilescu, D. Terzopoulos (2002)
7616:M.A.O. Vasilescu, D. Terzopoulos (2003)
7223:
1555:, which is very useful in applications.
1297:'s are unitary matrices. Define now the
8075:
5280:
4966:{\displaystyle {\mathcal {A}}_{}^{m-1}}
8211:
7055:
7040:
6986:
6882:and computer graphics—TensorTextures.
5062:, and store the left singular vectors
4537:, and store the left singular vectors
4332:
3553:{\displaystyle {\mathbb {C} }^{I_{m}}}
3099:be the multilinear rank of the tensor
2793:. The multilinear rank is a tuple in
7472:
7470:
7414:
7412:
7410:
7383:"Ranks derived from multilinear maps"
7376:
7374:
7237:"L1-norm Tucker Tensor Decomposition"
6089:is the reduced multilinear rank with
5541:{\displaystyle {\mathcal {\bar {A}}}}
2960:{\displaystyle 1\leq R_{m}\leq I_{m}}
1845:th largest nonzero singular value of
1266:. The second equality is because the
835:flattening. By the properties of the
7425:SIAM Journal on Scientific Computing
7073:
7071:
7069:
7067:
6947:
6945:
3845:{\displaystyle \mathbf {u} _{r_{m}}}
3488:{\displaystyle \mathbf {e} _{r_{m}}}
1549:compact singular value decomposition
561:corresponds to all other indices of
7387:Journal of Pure and Applied Algebra
6897:tensor product model transformation
4625:via the multilinear multiplication
2695:
75:in the 1960s, further advocated by
13:
7467:
7407:
7371:
6954:Journal of Mathematics and Physics
6826:
6813:
6772:
6759:
6721:
6683:
6528:
6486:
6285:
6254:
5887:
5884:
5881:
5878:
5845:
5833:
5752:
5589:
5586:
5583:
5580:
5557:
5528:
5433:
5430:
5427:
5424:
5401:
5307:
5240:
5214:
5177:
5135:
5028:
4986:
4935:
4859:
4843:
4644:
4634:
4610:
4460:
4417:
4149:
4117:
4073:
3569:
3212:
3132:
3108:
2862:
2852:
2849:
2846:
2843:
2713:
2618:
2524:
2361:
2219:
2205:
2045:
2016:
1960:
1923:
1855:
1674:
1638:
1458:
1448:
1424:
1319:
1309:
1067:
925:
866:
852:
770:
665:
570:
534:
509:
453:
428:
285:
139:
111:
14:
8235:
7064:
6980:
6942:
4437:{\displaystyle {\mathcal {A}}_{}}
3029:
1943:{\displaystyle {\mathcal {A}}_{}}
1875:{\displaystyle {\mathcal {A}}_{}}
1778:{\displaystyle {\bf {u}}_{r_{m}}}
1658:{\displaystyle {\mathcal {A}}_{}}
790:{\displaystyle {\mathcal {A}}_{}}
685:{\displaystyle {\mathcal {A}}_{}}
554:{\displaystyle {\mathcal {A}}_{}}
473:{\displaystyle {\mathcal {A}}_{}}
5293:
4755:
4723:
4691:
4662:
4546:
4511:
4497:
4483:
3988:
3960:
3938:
3862:
3825:
3786:
3758:
3736:
3468:
3429:
3401:
3379:
2914:{\displaystyle \mathbb {N} ^{M}}
2815:{\displaystyle \mathbb {N} ^{M}}
2578:
2555:
2538:
2498:
2475:
2458:
2425:
2397:
2375:
2325:
2311:
2283:
2269:
2247:
2233:
2080:
2066:
1997:
1983:
1892:
1795:
1757:
1699:
1568:
1542:
1512:
1489:
1472:
1383:
1355:
1333:
1276:
1204:
1181:
1164:
1131:
1103:
1081:
1031:
1017:
989:
975:
953:
939:
905:
889:
879:
810:
735:
718:{\displaystyle \mathbf {u} _{j}}
705:
595:
272:and the pure imaginary numbers.
8149:
8090:
7954:
7887:
7824:
7761:
7708:
7649:
7553:
6869:
5876:
5870:
7944:AAAS EurekAlert! Press Release
7347:10.1109/GlobalSIP.2018.8646385
7283:IEEE Signal Processing Letters
6643:
6591:{\displaystyle {\bar {R}}_{m}}
6576:
6498:
6492:
6469:{\displaystyle {\bar {R}}_{m}}
6454:
6400:
6348:{\displaystyle {\bar {R}}_{m}}
6333:
6266:
6260:
6237:{\displaystyle {\bar {R}}_{m}}
6222:
6109:
6061:
6049:
6021:
5999:
5989:
5966:
5954:
5926:
5904:
5894:
5696:
5684:
5656:
5628:
5606:
5596:
5565:{\displaystyle {\mathcal {A}}}
5504:
5440:
5409:{\displaystyle {\mathcal {A}}}
5226:
5220:
4998:
4992:
4947:
4941:
4911:{\displaystyle m=1,2\ldots ,M}
4822:{\displaystyle R_{m}\ll I_{m}}
4618:{\displaystyle {\mathcal {S}}}
4472:
4466:
4429:
4423:
4272:
4227:
4135:
4125:{\displaystyle {\mathcal {S}}}
4081:{\displaystyle {\mathcal {A}}}
3116:{\displaystyle {\mathcal {A}}}
3086:
3041:
2967:and it satisfy the constraint
2879:
2874:
2868:
2856:
2780:
2735:
2721:{\displaystyle {\mathcal {A}}}
2626:{\displaystyle {\mathcal {S}}}
2589:
2532:
2509:
2452:
2441:
2369:
2341:
2227:
2114:
2108:
2096:
2060:
2028:
2022:
1972:
1966:
1935:
1929:
1913:form a basis for the image of
1867:
1861:
1682:{\displaystyle {\mathcal {A}}}
1650:
1644:
1523:
1466:
1432:{\displaystyle {\mathcal {A}}}
1399:
1327:
1215:
1158:
1147:
1075:
1047:
933:
910:
874:
782:
776:
677:
671:
578:{\displaystyle {\mathcal {A}}}
546:
540:
517:{\displaystyle {\mathcal {A}}}
465:
459:
436:{\displaystyle {\mathcal {A}}}
396:
325:
297:
291:
119:{\displaystyle {\mathcal {A}}}
1:
8156:Y-h. Taguchi (October 2017).
8069:10.1016/s0166-3615(03)00058-7
6936:
5517:, then computing the optimal
4391:{\displaystyle m=1,\ldots ,M}
4351:
1906:{\displaystyle {\bf {U}}_{m}}
1809:{\displaystyle {\bf {U}}_{m}}
1713:{\displaystyle {\bf {U}}_{m}}
1290:{\displaystyle {\bf {U}}_{m}}
824:{\displaystyle {\bf {U}}_{m}}
760:th largest singular value of
749:{\displaystyle {\bf {U}}_{m}}
97:
8124:10.1371/journal.pone.0183933
8097:Y-h. Taguchi (August 2017).
7921:10.1371/journal.pone.0121396
7858:10.1371/journal.pone.0028072
7795:10.1371/journal.pone.0018768
7477:Hackbusch, Wolfgang (2012).
6434:successively truncated HOSVD
6430:sequentially truncated HOSVD
4921:Construct the standard mode-
4449:singular value decomposition
4322:{\displaystyle \mathbb {R} }
4300:{\displaystyle \mathbb {C} }
3920:. It is easy to verify that
3522:th standard basis vector of
443:, so that the left index of
265:{\displaystyle \mathbb {R} }
243:{\displaystyle \mathbb {C} }
41:singular value decomposition
7:
7264:10.1109/ACCESS.2019.2955134
10:
8240:
8182:10.1038/s41598-017-13003-0
7604:M. A. O. Vasilescu (2002)
7400:10.1016/j.jpaa.2010.11.010
6899:-based controller design.
5709:is a nonlinear non-convex
5285:The HOSVD can be computed
1255:{\displaystyle \cdot ^{H}}
837:multilinear multiplication
7963:Intelligent Data Analysis
7721:Molecular Systems Biology
7487:10.1007/978-3-642-28027-6
7102:10.1137/s0895479896305696
6183:{\displaystyle \|.\|_{F}}
5729:{\displaystyle \ell _{2}}
43:. It has applications in
35:is a specific orthogonal
7313:10.1109/LSP.2018.2790901
4601:Compute the core tensor
4220:be a tensor with a rank-
2192:, we find as before that
1720:be sorted such that the
8031:10.1109/tie.2003.822037
7683:10.1073/pnas.0709146104
7536:10.1145/3539781.3539798
6904:TP model transformation
4918:perform the following:
4785:Interlacing computation
4341:, further advocated by
1882:. Since the columns of
524:and the right index of
7341:. pp. 1353–1357.
6917:Robust L1-norm variant
6859:
6741:
6703:
6661:
6598:left singular vectors
6592:
6556:
6470:
6418:
6355:left singular vectors
6349:
6313:
6238:
6184:
6151:
6083:
5976:
5736:-optimization problem
5730:
5703:
5566:
5542:
5511:
5410:
5386:
5268:
5198:
5116:
5056:
4967:
4912:
4868:
4823:
4775:
4619:
4590:
4531:
4447:Compute the (compact)
4438:
4392:
4323:
4301:
4279:
4214:
4126:
4102:
4082:
4058:
3914:
3846:
3810:
3684:
3646:
3611:
3554:
3516:
3489:
3453:
3327:
3289:
3254:
3197:
3117:
3093:
3017:
2961:
2915:
2886:
2816:
2787:
2722:
2686:
2627:
2609:where the core tensor
2603:
2186:
2134:orthogonal projections
2126:
1944:
1907:
1876:
1838:
1810:
1779:
1741:
1714:
1683:
1659:
1618:
1547:As in the case of the
1533:
1433:
1409:
1291:
1256:
1229:
825:
791:
750:
719:
686:
645:
579:
555:
518:
494:
474:
437:
405:
266:
244:
218:
120:
8057:Computers in Industry
6860:
6742:
6704:
6662:
6593:
6557:
6471:
6419:
6350:
6314:
6239:
6185:
6152:
6084:
5977:
5731:
5704:
5567:
5543:
5512:
5411:
5387:
5269:
5199:
5117:
5057:
4968:
4913:
4869:
4824:
4776:
4620:
4591:
4532:
4439:
4393:
4324:
4302:
4280:
4215:
4127:
4103:
4083:
4059:
3915:
3847:
3811:
3650:
3612:
3577:
3555:
3517:
3515:{\displaystyle r_{m}}
3490:
3454:
3293:
3255:
3220:
3198:
3118:
3094:
3018:
2962:
2916:
2887:
2817:
2788:
2728:is denoted with rank-
2723:
2687:
2628:
2604:
2187:
2127:
1945:
1908:
1877:
1839:
1837:{\displaystyle r_{m}}
1811:
1780:
1742:
1740:{\displaystyle r_{m}}
1715:
1689:. Let the columns of
1684:
1660:
1619:
1534:
1434:
1410:
1292:
1257:
1230:
826:
792:
751:
720:
687:
646:
580:
556:
519:
495:
475:
438:
406:
267:
245:
219:
121:
6966:10.1002/sapm19287139
6931:Tucker decomposition
6890:disease surveillance
6751:
6713:
6675:
6602:
6566:
6562:, and store the top
6480:
6444:
6359:
6323:
6319:, and store the top
6248:
6212:
6161:
6093:
5986:
5740:
5713:
5576:
5572:for a given reduced
5552:
5521:
5420:
5396:
5302:
5281:In-place computation
5208:
5204:, or, equivalently,
5129:
5066:
4980:
4929:
4881:
4837:
4793:
4629:
4605:
4541:
4454:
4411:
4364:
4311:
4289:
4224:
4144:
4112:
4092:
4068:
3924:
3856:
3820:
3564:
3526:
3499:
3463:
3207:
3127:
3103:
3038:
2971:
2925:
2896:
2892:. Not all tuples in
2826:
2797:
2732:
2708:
2637:
2613:
2196:
2140:
1954:
1917:
1886:
1849:
1821:
1789:
1751:
1724:
1693:
1669:
1632:
1562:
1443:
1439:is the decomposition
1419:
1304:
1270:
1239:
843:
804:
764:
729:
700:
659:
589:
565:
528:
504:
484:
447:
423:
279:
254:
232:
134:
106:
73:Tucker decomposition
67:in 1928, but it was
57:scientific computing
37:Tucker decomposition
16:Tensor decomposition
8219:Multilinear algebra
8174:2017NatSR...713733T
8115:2017PLoSO..1283933T
7985:2014arXiv1406.3496F
7948:NAE Podcast Feature
7912:2015PLoSO..1021396S
7849:2011PLoSO...628072P
7786:2011PLoSO...618768M
7733:10.1038/msb.2009.70
7674:2007PNAS..10418371O
7668:(47): 18371–18376.
7437:2012SJSC...34A1027V
7305:2018ISPL...25..511M
6895:It is also used in
6551:
6513:
6308:
5869:
5263:
5235:
5163:
5051:
5013:
4962:
4770:
4738:
4706:
4677:
4526:
4403:Construct the mode-
4398:, do the following:
4333:Classic computation
4307:contains the reals
3852:are the columns of
2440:
2412:
2390:
2340:
2298:
2262:
2095:
2012:
1816:corresponds to the
1415:Then, the HOSVD of
1398:
1370:
1348:
1264:conjugate transpose
1146:
1118:
1096:
1046:
1004:
968:
797:. Observe that the
756:corresponds to the
480:corresponds to the
21:multilinear algebra
8162:Scientific Reports
7993:10.3233/IDA-150734
7431:(2): A1027–A1052.
7138:2022-12-29 at the
7001:10.1007/bf02289464
6855:
6737:
6699:
6657:
6588:
6552:
6537:
6483:
6466:
6414:
6345:
6309:
6294:
6234:
6180:
6147:
6079:
5972:
5855:
5817:
5726:
5699:
5562:
5548:that approximates
5538:
5507:
5406:
5382:
5264:
5249:
5211:
5194:
5149:
5112:
5052:
5037:
4983:
4963:
4932:
4908:
4864:
4819:
4771:
4752:
4720:
4688:
4659:
4615:
4586:
4527:
4508:
4434:
4388:
4319:
4297:
4275:
4210:
4122:
4098:
4078:
4054:
3910:
3842:
3806:
3550:
3512:
3485:
3449:
3193:
3113:
3089:
3013:
3002:
2957:
2911:
2882:
2812:
2783:
2718:
2682:
2623:
2599:
2597:
2422:
2394:
2372:
2322:
2280:
2244:
2182:
2122:
2077:
1994:
1940:
1903:
1872:
1834:
1806:
1775:
1737:
1710:
1679:
1655:
1614:
1529:
1429:
1405:
1380:
1352:
1330:
1287:
1252:
1225:
1223:
1128:
1100:
1078:
1028:
986:
950:
821:
799:mode/factor matrix
787:
746:
715:
682:
641:
575:
551:
514:
490:
470:
433:
401:
262:
240:
214:
130:, also denoted by
116:
7584:10.1137/090764189
7496:978-3-642-28026-9
7445:10.1137/110836067
7356:978-1-7281-1295-4
7247:: 178454–178465.
7209:10.1137/130905010
6921:L1-Tucker is the
6832:
6806:
6778:
6727:
6689:
6646:
6579:
6457:
6403:
6336:
6225:
6112:
6052:
6024:
6002:
5957:
5929:
5907:
5874:
5851:
5826:
5758:
5743:
5687:
5659:
5631:
5609:
5534:
4101:{\displaystyle B}
2987:
493:{\displaystyle m}
61:signal processing
49:computer graphics
8231:
8204:
8203:
8193:
8153:
8147:
8146:
8136:
8126:
8094:
8088:
8087:
8079:
8073:
8072:
8052:
8043:
8042:
8014:
8005:
8004:
7978:
7958:
7952:
7951:
7933:
7923:
7891:
7885:
7884:
7870:
7860:
7828:
7822:
7821:
7807:
7797:
7765:
7759:
7758:
7744:
7712:
7706:
7705:
7695:
7685:
7653:
7647:
7641:
7635:
7629:
7623:
7614:
7608:
7602:
7596:
7595:
7577:
7568:(4): 2029–2054.
7557:
7551:
7550:
7538:
7522:
7509:
7508:
7474:
7465:
7464:
7416:
7405:
7404:
7402:
7393:(8): 1999–2004.
7378:
7369:
7368:
7334:
7325:
7324:
7298:
7278:
7269:
7268:
7266:
7256:
7232:
7221:
7220:
7202:
7182:
7176:
7169:
7156:
7149:
7143:
7129:
7114:
7113:
7095:
7086:(4): 1253–1278.
7075:
7062:
7061:
7053:
7047:
7046:
7038:
7029:
7028:
6984:
6978:
6977:
6949:
6864:
6862:
6861:
6856:
6851:
6850:
6841:
6840:
6835:
6834:
6833:
6825:
6817:
6816:
6807:
6802:
6797:
6796:
6787:
6786:
6781:
6780:
6779:
6771:
6763:
6762:
6746:
6744:
6743:
6738:
6736:
6735:
6730:
6729:
6728:
6720:
6708:
6706:
6705:
6700:
6698:
6697:
6692:
6691:
6690:
6682:
6666:
6664:
6663:
6658:
6656:
6655:
6654:
6653:
6648:
6647:
6639:
6632:
6631:
6614:
6613:
6597:
6595:
6594:
6589:
6587:
6586:
6581:
6580:
6572:
6561:
6559:
6558:
6553:
6550:
6545:
6536:
6535:
6526:
6525:
6512:
6501:
6490:
6489:
6475:
6473:
6472:
6467:
6465:
6464:
6459:
6458:
6450:
6423:
6421:
6420:
6415:
6413:
6412:
6411:
6410:
6405:
6404:
6396:
6389:
6388:
6371:
6370:
6354:
6352:
6351:
6346:
6344:
6343:
6338:
6337:
6329:
6318:
6316:
6315:
6310:
6307:
6302:
6293:
6292:
6283:
6282:
6270:
6269:
6258:
6257:
6243:
6241:
6240:
6235:
6233:
6232:
6227:
6226:
6218:
6189:
6187:
6186:
6181:
6179:
6178:
6156:
6154:
6153:
6148:
6146:
6145:
6133:
6132:
6120:
6119:
6114:
6113:
6105:
6088:
6086:
6085:
6080:
6078:
6077:
6072:
6060:
6059:
6054:
6053:
6045:
6032:
6031:
6026:
6025:
6017:
6010:
6009:
6004:
6003:
5995:
5981:
5979:
5978:
5973:
5965:
5964:
5959:
5958:
5950:
5937:
5936:
5931:
5930:
5922:
5915:
5914:
5909:
5908:
5900:
5893:
5875:
5872:
5868:
5863:
5854:
5853:
5852:
5844:
5837:
5836:
5827:
5819:
5816:
5815:
5814:
5813:
5812:
5794:
5793:
5781:
5780:
5770:
5761:
5760:
5759:
5751:
5735:
5733:
5732:
5727:
5725:
5724:
5708:
5706:
5705:
5700:
5695:
5694:
5689:
5688:
5680:
5667:
5666:
5661:
5660:
5652:
5639:
5638:
5633:
5632:
5624:
5617:
5616:
5611:
5610:
5602:
5595:
5571:
5569:
5568:
5563:
5561:
5560:
5547:
5545:
5544:
5539:
5537:
5536:
5535:
5527:
5516:
5514:
5513:
5508:
5503:
5502:
5484:
5483:
5465:
5464:
5452:
5451:
5439:
5415:
5413:
5412:
5407:
5405:
5404:
5391:
5389:
5388:
5383:
5381:
5380:
5379:
5378:
5363:
5362:
5344:
5343:
5331:
5330:
5320:
5311:
5310:
5273:
5271:
5270:
5265:
5262:
5257:
5248:
5247:
5234:
5229:
5218:
5217:
5203:
5201:
5200:
5195:
5193:
5192:
5181:
5180:
5173:
5172:
5162:
5157:
5145:
5144:
5139:
5138:
5121:
5119:
5118:
5113:
5111:
5110:
5109:
5108:
5096:
5095:
5078:
5077:
5061:
5059:
5058:
5053:
5050:
5045:
5036:
5035:
5026:
5025:
5012:
5001:
4990:
4989:
4972:
4970:
4969:
4964:
4961:
4950:
4939:
4938:
4917:
4915:
4914:
4909:
4873:
4871:
4870:
4865:
4863:
4862:
4853:
4852:
4847:
4846:
4828:
4826:
4825:
4820:
4818:
4817:
4805:
4804:
4780:
4778:
4777:
4772:
4769:
4764:
4759:
4758:
4751:
4750:
4737:
4732:
4727:
4726:
4719:
4718:
4705:
4700:
4695:
4694:
4687:
4686:
4676:
4671:
4666:
4665:
4658:
4657:
4648:
4647:
4638:
4637:
4624:
4622:
4621:
4616:
4614:
4613:
4595:
4593:
4592:
4587:
4585:
4584:
4583:
4582:
4570:
4569:
4559:
4550:
4549:
4536:
4534:
4533:
4528:
4525:
4520:
4515:
4514:
4507:
4506:
4501:
4500:
4493:
4492:
4487:
4486:
4476:
4475:
4464:
4463:
4443:
4441:
4440:
4435:
4433:
4432:
4421:
4420:
4397:
4395:
4394:
4389:
4328:
4326:
4325:
4320:
4318:
4306:
4304:
4303:
4298:
4296:
4284:
4282:
4281:
4276:
4271:
4270:
4252:
4251:
4239:
4238:
4219:
4217:
4216:
4211:
4209:
4208:
4207:
4206:
4188:
4187:
4175:
4174:
4164:
4163:
4153:
4152:
4131:
4129:
4128:
4123:
4121:
4120:
4107:
4105:
4104:
4099:
4087:
4085:
4084:
4079:
4077:
4076:
4063:
4061:
4060:
4055:
4053:
4052:
4051:
4050:
4032:
4031:
4019:
4018:
4004:
4003:
4002:
4001:
3991:
3976:
3975:
3974:
3973:
3963:
3954:
3953:
3952:
3951:
3941:
3919:
3917:
3916:
3911:
3909:
3908:
3907:
3906:
3894:
3893:
3883:
3882:
3872:
3871:
3866:
3865:
3851:
3849:
3848:
3843:
3841:
3840:
3839:
3838:
3828:
3815:
3813:
3812:
3807:
3802:
3801:
3800:
3799:
3789:
3774:
3773:
3772:
3771:
3761:
3752:
3751:
3750:
3749:
3739:
3733:
3732:
3731:
3730:
3712:
3711:
3699:
3698:
3683:
3682:
3681:
3671:
3664:
3663:
3645:
3644:
3643:
3633:
3626:
3625:
3610:
3609:
3608:
3598:
3591:
3590:
3573:
3572:
3559:
3557:
3556:
3551:
3549:
3548:
3547:
3546:
3536:
3535:
3521:
3519:
3518:
3513:
3511:
3510:
3494:
3492:
3491:
3486:
3484:
3483:
3482:
3481:
3471:
3458:
3456:
3455:
3450:
3445:
3444:
3443:
3442:
3432:
3417:
3416:
3415:
3414:
3404:
3395:
3394:
3393:
3392:
3382:
3376:
3375:
3374:
3373:
3355:
3354:
3342:
3341:
3326:
3325:
3324:
3314:
3307:
3306:
3288:
3287:
3286:
3276:
3269:
3268:
3253:
3252:
3251:
3241:
3234:
3233:
3216:
3215:
3202:
3200:
3199:
3194:
3192:
3191:
3190:
3189:
3171:
3170:
3158:
3157:
3147:
3146:
3136:
3135:
3122:
3120:
3119:
3114:
3112:
3111:
3098:
3096:
3095:
3090:
3085:
3084:
3066:
3065:
3053:
3052:
3022:
3020:
3019:
3014:
3012:
3011:
3001:
2983:
2982:
2966:
2964:
2963:
2958:
2956:
2955:
2943:
2942:
2920:
2918:
2917:
2912:
2910:
2909:
2904:
2891:
2889:
2888:
2883:
2878:
2877:
2866:
2865:
2855:
2838:
2837:
2821:
2819:
2818:
2813:
2811:
2810:
2805:
2792:
2790:
2789:
2784:
2779:
2778:
2760:
2759:
2747:
2746:
2727:
2725:
2724:
2719:
2717:
2716:
2702:multilinear rank
2696:Multilinear rank
2691:
2689:
2688:
2683:
2681:
2680:
2662:
2661:
2649:
2648:
2632:
2630:
2629:
2624:
2622:
2621:
2608:
2606:
2605:
2600:
2598:
2588:
2587:
2582:
2581:
2565:
2564:
2559:
2558:
2548:
2547:
2542:
2541:
2528:
2527:
2515:
2508:
2507:
2502:
2501:
2485:
2484:
2479:
2478:
2468:
2467:
2462:
2461:
2448:
2444:
2439:
2434:
2429:
2428:
2411:
2406:
2401:
2400:
2389:
2384:
2379:
2378:
2365:
2364:
2347:
2339:
2334:
2329:
2328:
2321:
2320:
2315:
2314:
2297:
2292:
2287:
2286:
2279:
2278:
2273:
2272:
2261:
2256:
2251:
2250:
2243:
2242:
2237:
2236:
2223:
2222:
2209:
2208:
2191:
2189:
2188:
2183:
2131:
2129:
2128:
2123:
2118:
2117:
2106:
2105:
2094:
2089:
2084:
2083:
2076:
2075:
2070:
2069:
2059:
2058:
2049:
2048:
2042:
2041:
2032:
2031:
2020:
2019:
2011:
2006:
2001:
2000:
1993:
1992:
1987:
1986:
1976:
1975:
1964:
1963:
1949:
1947:
1946:
1941:
1939:
1938:
1927:
1926:
1912:
1910:
1909:
1904:
1902:
1901:
1896:
1895:
1881:
1879:
1878:
1873:
1871:
1870:
1859:
1858:
1843:
1841:
1840:
1835:
1833:
1832:
1815:
1813:
1812:
1807:
1805:
1804:
1799:
1798:
1784:
1782:
1781:
1776:
1774:
1773:
1772:
1771:
1761:
1760:
1746:
1744:
1743:
1738:
1736:
1735:
1719:
1717:
1716:
1711:
1709:
1708:
1703:
1702:
1688:
1686:
1685:
1680:
1678:
1677:
1664:
1662:
1661:
1656:
1654:
1653:
1642:
1641:
1623:
1621:
1620:
1615:
1613:
1612:
1611:
1610:
1598:
1597:
1587:
1578:
1577:
1572:
1571:
1538:
1536:
1535:
1530:
1522:
1521:
1516:
1515:
1499:
1498:
1493:
1492:
1482:
1481:
1476:
1475:
1462:
1461:
1452:
1451:
1438:
1436:
1435:
1430:
1428:
1427:
1414:
1412:
1411:
1406:
1397:
1392:
1387:
1386:
1369:
1364:
1359:
1358:
1347:
1342:
1337:
1336:
1323:
1322:
1313:
1312:
1296:
1294:
1293:
1288:
1286:
1285:
1280:
1279:
1261:
1259:
1258:
1253:
1251:
1250:
1234:
1232:
1231:
1226:
1224:
1214:
1213:
1208:
1207:
1191:
1190:
1185:
1184:
1174:
1173:
1168:
1167:
1154:
1150:
1145:
1140:
1135:
1134:
1117:
1112:
1107:
1106:
1095:
1090:
1085:
1084:
1071:
1070:
1053:
1045:
1040:
1035:
1034:
1027:
1026:
1021:
1020:
1003:
998:
993:
992:
985:
984:
979:
978:
967:
962:
957:
956:
949:
948:
943:
942:
929:
928:
916:
909:
908:
893:
892:
883:
882:
870:
869:
856:
855:
830:
828:
827:
822:
820:
819:
814:
813:
796:
794:
793:
788:
786:
785:
774:
773:
755:
753:
752:
747:
745:
744:
739:
738:
724:
722:
721:
716:
714:
713:
708:
691:
689:
688:
683:
681:
680:
669:
668:
650:
648:
647:
642:
640:
639:
638:
637:
625:
624:
614:
605:
604:
599:
598:
584:
582:
581:
576:
574:
573:
560:
558:
557:
552:
550:
549:
538:
537:
523:
521:
520:
515:
513:
512:
499:
497:
496:
491:
479:
477:
476:
471:
469:
468:
457:
456:
442:
440:
439:
434:
432:
431:
410:
408:
407:
402:
400:
399:
395:
394:
382:
381:
366:
365:
347:
346:
337:
336:
321:
320:
310:
301:
300:
289:
288:
271:
269:
268:
263:
261:
249:
247:
246:
241:
239:
223:
221:
220:
215:
213:
212:
211:
210:
195:
194:
176:
175:
163:
162:
152:
143:
142:
125:
123:
122:
117:
115:
114:
53:machine learning
8239:
8238:
8234:
8233:
8232:
8230:
8229:
8228:
8209:
8208:
8207:
8154:
8150:
8109:(8): e0183933.
8095:
8091:
8080:
8076:
8053:
8046:
8015:
8008:
7959:
7955:
7906:(4): e0121396.
7892:
7888:
7829:
7825:
7766:
7762:
7713:
7709:
7654:
7650:
7642:
7638:
7630:
7626:
7615:
7611:
7603:
7599:
7575:10.1.1.660.8333
7558:
7554:
7547:
7523:
7512:
7497:
7475:
7468:
7417:
7408:
7379:
7372:
7357:
7335:
7328:
7279:
7272:
7233:
7224:
7183:
7179:
7170:
7159:
7150:
7146:
7140:Wayback Machine
7130:
7117:
7093:10.1.1.102.9135
7076:
7065:
7054:
7050:
7039:
7032:
6985:
6981:
6950:
6943:
6939:
6919:
6872:
6846:
6842:
6836:
6824:
6823:
6822:
6821:
6812:
6811:
6801:
6792:
6788:
6782:
6770:
6769:
6768:
6767:
6758:
6757:
6752:
6749:
6748:
6731:
6719:
6718:
6717:
6716:
6714:
6711:
6710:
6693:
6681:
6680:
6679:
6678:
6676:
6673:
6672:
6649:
6638:
6637:
6636:
6627:
6623:
6622:
6618:
6609:
6605:
6603:
6600:
6599:
6582:
6571:
6570:
6569:
6567:
6564:
6563:
6546:
6541:
6531:
6527:
6521:
6517:
6502:
6491:
6485:
6484:
6481:
6478:
6477:
6460:
6449:
6448:
6447:
6445:
6442:
6441:
6440:Compute a rank-
6406:
6395:
6394:
6393:
6384:
6380:
6379:
6375:
6366:
6362:
6360:
6357:
6356:
6339:
6328:
6327:
6326:
6324:
6321:
6320:
6303:
6298:
6288:
6284:
6278:
6274:
6259:
6253:
6252:
6251:
6249:
6246:
6245:
6228:
6217:
6216:
6215:
6213:
6210:
6209:
6208:Compute a rank-
6202:truncated HOSVD
6174:
6170:
6162:
6159:
6158:
6157:, and the norm
6141:
6137:
6128:
6124:
6115:
6104:
6103:
6102:
6094:
6091:
6090:
6073:
6068:
6067:
6055:
6044:
6043:
6042:
6027:
6016:
6015:
6014:
6005:
5994:
5993:
5992:
5987:
5984:
5983:
5960:
5949:
5948:
5947:
5932:
5921:
5920:
5919:
5910:
5899:
5898:
5897:
5877:
5871:
5864:
5859:
5843:
5842:
5841:
5832:
5831:
5818:
5808:
5804:
5789:
5785:
5776:
5772:
5771:
5766:
5765:
5750:
5749:
5748:
5747:
5741:
5738:
5737:
5720:
5716:
5714:
5711:
5710:
5690:
5679:
5678:
5677:
5662:
5651:
5650:
5649:
5634:
5623:
5622:
5621:
5612:
5601:
5600:
5599:
5579:
5577:
5574:
5573:
5556:
5555:
5553:
5550:
5549:
5526:
5525:
5524:
5522:
5519:
5518:
5498:
5494:
5479:
5475:
5460:
5456:
5447:
5443:
5423:
5421:
5418:
5417:
5400:
5399:
5397:
5394:
5393:
5374:
5370:
5358:
5354:
5339:
5335:
5326:
5322:
5321:
5316:
5315:
5306:
5305:
5303:
5300:
5299:
5296:
5283:
5258:
5253:
5243:
5239:
5230:
5219:
5213:
5212:
5209:
5206:
5205:
5182:
5176:
5175:
5174:
5168:
5164:
5158:
5153:
5140:
5134:
5133:
5132:
5130:
5127:
5126:
5104:
5100:
5091:
5087:
5086:
5082:
5073:
5069:
5067:
5064:
5063:
5046:
5041:
5031:
5027:
5021:
5017:
5002:
4991:
4985:
4984:
4981:
4978:
4977:
4951:
4940:
4934:
4933:
4930:
4927:
4926:
4882:
4879:
4878:
4858:
4857:
4848:
4842:
4841:
4840:
4838:
4835:
4834:
4813:
4809:
4800:
4796:
4794:
4791:
4790:
4787:
4765:
4760:
4754:
4753:
4746:
4742:
4733:
4728:
4722:
4721:
4714:
4710:
4701:
4696:
4690:
4689:
4682:
4678:
4672:
4667:
4661:
4660:
4653:
4649:
4643:
4642:
4633:
4632:
4630:
4627:
4626:
4609:
4608:
4606:
4603:
4602:
4578:
4574:
4565:
4561:
4560:
4555:
4554:
4545:
4544:
4542:
4539:
4538:
4521:
4516:
4510:
4509:
4502:
4496:
4495:
4494:
4488:
4482:
4481:
4480:
4465:
4459:
4458:
4457:
4455:
4452:
4451:
4422:
4416:
4415:
4414:
4412:
4409:
4408:
4365:
4362:
4361:
4354:
4343:L. De Lathauwer
4335:
4314:
4312:
4309:
4308:
4292:
4290:
4287:
4286:
4266:
4262:
4247:
4243:
4234:
4230:
4225:
4222:
4221:
4202:
4198:
4183:
4179:
4170:
4166:
4165:
4159:
4158:
4157:
4148:
4147:
4145:
4142:
4141:
4138:
4116:
4115:
4113:
4110:
4109:
4093:
4090:
4089:
4072:
4071:
4069:
4066:
4065:
4046:
4042:
4027:
4023:
4014:
4010:
4009:
4005:
3997:
3993:
3992:
3987:
3986:
3969:
3965:
3964:
3959:
3958:
3947:
3943:
3942:
3937:
3936:
3925:
3922:
3921:
3902:
3898:
3889:
3885:
3884:
3878:
3877:
3876:
3867:
3861:
3860:
3859:
3857:
3854:
3853:
3834:
3830:
3829:
3824:
3823:
3821:
3818:
3817:
3795:
3791:
3790:
3785:
3784:
3767:
3763:
3762:
3757:
3756:
3745:
3741:
3740:
3735:
3734:
3726:
3722:
3707:
3703:
3694:
3690:
3689:
3685:
3677:
3673:
3672:
3659:
3655:
3654:
3639:
3635:
3634:
3621:
3617:
3616:
3604:
3600:
3599:
3586:
3582:
3581:
3568:
3567:
3565:
3562:
3561:
3542:
3538:
3537:
3531:
3530:
3529:
3527:
3524:
3523:
3506:
3502:
3500:
3497:
3496:
3477:
3473:
3472:
3467:
3466:
3464:
3461:
3460:
3438:
3434:
3433:
3428:
3427:
3410:
3406:
3405:
3400:
3399:
3388:
3384:
3383:
3378:
3377:
3369:
3365:
3350:
3346:
3337:
3333:
3332:
3328:
3320:
3316:
3315:
3302:
3298:
3297:
3282:
3278:
3277:
3264:
3260:
3259:
3247:
3243:
3242:
3229:
3225:
3224:
3211:
3210:
3208:
3205:
3204:
3185:
3181:
3166:
3162:
3153:
3149:
3148:
3142:
3141:
3140:
3131:
3130:
3128:
3125:
3124:
3107:
3106:
3104:
3101:
3100:
3080:
3076:
3061:
3057:
3048:
3044:
3039:
3036:
3035:
3032:
3007:
3003:
2991:
2978:
2974:
2972:
2969:
2968:
2951:
2947:
2938:
2934:
2926:
2923:
2922:
2905:
2900:
2899:
2897:
2894:
2893:
2867:
2861:
2860:
2859:
2842:
2833:
2829:
2827:
2824:
2823:
2806:
2801:
2800:
2798:
2795:
2794:
2774:
2770:
2755:
2751:
2742:
2738:
2733:
2730:
2729:
2712:
2711:
2709:
2706:
2705:
2698:
2676:
2672:
2657:
2653:
2644:
2640:
2638:
2635:
2634:
2633:is now of size
2617:
2616:
2614:
2611:
2610:
2596:
2595:
2583:
2577:
2576:
2575:
2560:
2554:
2553:
2552:
2543:
2537:
2536:
2535:
2523:
2522:
2520:
2513:
2512:
2503:
2497:
2496:
2495:
2480:
2474:
2473:
2472:
2463:
2457:
2456:
2455:
2435:
2430:
2424:
2423:
2407:
2402:
2396:
2395:
2385:
2380:
2374:
2373:
2360:
2359:
2358:
2354:
2352:
2345:
2344:
2335:
2330:
2324:
2323:
2316:
2310:
2309:
2308:
2293:
2288:
2282:
2281:
2274:
2268:
2267:
2266:
2257:
2252:
2246:
2245:
2238:
2232:
2231:
2230:
2218:
2217:
2215:
2210:
2204:
2203:
2199:
2197:
2194:
2193:
2141:
2138:
2137:
2107:
2101:
2100:
2099:
2090:
2085:
2079:
2078:
2071:
2065:
2064:
2063:
2054:
2050:
2044:
2043:
2037:
2036:
2021:
2015:
2014:
2013:
2007:
2002:
1996:
1995:
1988:
1982:
1981:
1980:
1965:
1959:
1958:
1957:
1955:
1952:
1951:
1928:
1922:
1921:
1920:
1918:
1915:
1914:
1897:
1891:
1890:
1889:
1887:
1884:
1883:
1860:
1854:
1853:
1852:
1850:
1847:
1846:
1828:
1824:
1822:
1819:
1818:
1800:
1794:
1793:
1792:
1790:
1787:
1786:
1767:
1763:
1762:
1756:
1755:
1754:
1752:
1749:
1748:
1731:
1727:
1725:
1722:
1721:
1704:
1698:
1697:
1696:
1694:
1691:
1690:
1673:
1672:
1670:
1667:
1666:
1643:
1637:
1636:
1635:
1633:
1630:
1629:
1606:
1602:
1593:
1589:
1588:
1583:
1582:
1573:
1567:
1566:
1565:
1563:
1560:
1559:
1545:
1517:
1511:
1510:
1509:
1494:
1488:
1487:
1486:
1477:
1471:
1470:
1469:
1457:
1456:
1447:
1446:
1444:
1441:
1440:
1423:
1422:
1420:
1417:
1416:
1393:
1388:
1382:
1381:
1365:
1360:
1354:
1353:
1343:
1338:
1332:
1331:
1318:
1317:
1308:
1307:
1305:
1302:
1301:
1281:
1275:
1274:
1273:
1271:
1268:
1267:
1246:
1242:
1240:
1237:
1236:
1222:
1221:
1209:
1203:
1202:
1201:
1186:
1180:
1179:
1178:
1169:
1163:
1162:
1161:
1141:
1136:
1130:
1129:
1113:
1108:
1102:
1101:
1091:
1086:
1080:
1079:
1066:
1065:
1064:
1060:
1058:
1051:
1050:
1041:
1036:
1030:
1029:
1022:
1016:
1015:
1014:
999:
994:
988:
987:
980:
974:
973:
972:
963:
958:
952:
951:
944:
938:
937:
936:
924:
923:
921:
914:
913:
904:
903:
888:
887:
878:
877:
865:
864:
862:
857:
851:
850:
846:
844:
841:
840:
815:
809:
808:
807:
805:
802:
801:
775:
769:
768:
767:
765:
762:
761:
740:
734:
733:
732:
730:
727:
726:
709:
704:
703:
701:
698:
697:
670:
664:
663:
662:
660:
657:
656:
633:
629:
620:
616:
615:
610:
609:
600:
594:
593:
592:
590:
587:
586:
569:
568:
566:
563:
562:
539:
533:
532:
531:
529:
526:
525:
508:
507:
505:
502:
501:
485:
482:
481:
458:
452:
451:
450:
448:
445:
444:
427:
426:
424:
421:
420:
390:
386:
371:
367:
355:
351:
342:
338:
332:
328:
316:
312:
311:
306:
305:
290:
284:
283:
282:
280:
277:
276:
257:
255:
252:
251:
235:
233:
230:
229:
206:
202:
190:
186:
171:
167:
158:
154:
153:
148:
147:
138:
137:
135:
132:
131:
110:
109:
107:
104:
103:
100:
77:L. De Lathauwer
65:F. L. Hitchcock
45:computer vision
17:
12:
11:
5:
8237:
8227:
8226:
8221:
8206:
8205:
8148:
8089:
8074:
8063:(3): 281–297.
8044:
8025:(2): 387–400.
8006:
7969:(3): 597–616.
7953:
7886:
7843:(12): e28072.
7823:
7760:
7707:
7648:
7636:
7624:
7609:
7597:
7552:
7545:
7510:
7495:
7466:
7406:
7370:
7355:
7326:
7289:(4): 511–515.
7270:
7222:
7193:(1): 225–253.
7177:
7157:
7144:
7115:
7063:
7048:
7030:
6995:(3): 279–311.
6979:
6960:(1–4): 39–79.
6940:
6938:
6935:
6918:
6915:
6871:
6868:
6867:
6866:
6854:
6849:
6845:
6839:
6831:
6828:
6820:
6815:
6810:
6805:
6800:
6795:
6791:
6785:
6777:
6774:
6766:
6761:
6756:
6734:
6726:
6723:
6696:
6688:
6685:
6652:
6645:
6642:
6635:
6630:
6626:
6621:
6617:
6612:
6608:
6585:
6578:
6575:
6549:
6544:
6540:
6534:
6530:
6524:
6520:
6516:
6511:
6508:
6505:
6500:
6497:
6494:
6488:
6476:truncated SVD
6463:
6456:
6453:
6426:
6425:
6409:
6402:
6399:
6392:
6387:
6383:
6378:
6374:
6369:
6365:
6342:
6335:
6332:
6306:
6301:
6297:
6291:
6287:
6281:
6277:
6273:
6268:
6265:
6262:
6256:
6244:truncated SVD
6231:
6224:
6221:
6192:Frobenius norm
6177:
6173:
6169:
6166:
6144:
6140:
6136:
6131:
6127:
6123:
6118:
6111:
6108:
6101:
6098:
6076:
6071:
6066:
6063:
6058:
6051:
6048:
6041:
6038:
6035:
6030:
6023:
6020:
6013:
6008:
6001:
5998:
5991:
5971:
5968:
5963:
5956:
5953:
5946:
5943:
5940:
5935:
5928:
5925:
5918:
5913:
5906:
5903:
5896:
5892:
5889:
5886:
5883:
5880:
5867:
5862:
5858:
5850:
5847:
5840:
5835:
5830:
5825:
5822:
5811:
5807:
5803:
5800:
5797:
5792:
5788:
5784:
5779:
5775:
5769:
5764:
5757:
5754:
5746:
5723:
5719:
5698:
5693:
5686:
5683:
5676:
5673:
5670:
5665:
5658:
5655:
5648:
5645:
5642:
5637:
5630:
5627:
5620:
5615:
5608:
5605:
5598:
5594:
5591:
5588:
5585:
5582:
5559:
5533:
5530:
5506:
5501:
5497:
5493:
5490:
5487:
5482:
5478:
5474:
5471:
5468:
5463:
5459:
5455:
5450:
5446:
5442:
5438:
5435:
5432:
5429:
5426:
5416:is denoted by
5403:
5377:
5373:
5369:
5366:
5361:
5357:
5353:
5350:
5347:
5342:
5338:
5334:
5329:
5325:
5319:
5314:
5309:
5295:
5292:
5282:
5279:
5278:
5277:
5276:
5275:
5261:
5256:
5252:
5246:
5242:
5238:
5233:
5228:
5225:
5222:
5216:
5191:
5188:
5185:
5179:
5171:
5167:
5161:
5156:
5152:
5148:
5143:
5137:
5123:
5107:
5103:
5099:
5094:
5090:
5085:
5081:
5076:
5072:
5049:
5044:
5040:
5034:
5030:
5024:
5020:
5016:
5011:
5008:
5005:
5000:
4997:
4994:
4988:
4974:
4960:
4957:
4954:
4949:
4946:
4943:
4937:
4907:
4904:
4901:
4898:
4895:
4892:
4889:
4886:
4875:
4861:
4856:
4851:
4845:
4816:
4812:
4808:
4803:
4799:
4786:
4783:
4782:
4781:
4768:
4763:
4757:
4749:
4745:
4741:
4736:
4731:
4725:
4717:
4713:
4709:
4704:
4699:
4693:
4685:
4681:
4675:
4670:
4664:
4656:
4652:
4646:
4641:
4636:
4612:
4598:
4597:
4581:
4577:
4573:
4568:
4564:
4558:
4553:
4548:
4524:
4519:
4513:
4505:
4499:
4491:
4485:
4479:
4474:
4471:
4468:
4462:
4445:
4431:
4428:
4425:
4419:
4400:
4399:
4387:
4384:
4381:
4378:
4375:
4372:
4369:
4353:
4350:
4334:
4331:
4317:
4295:
4274:
4269:
4265:
4261:
4258:
4255:
4250:
4246:
4242:
4237:
4233:
4229:
4205:
4201:
4197:
4194:
4191:
4186:
4182:
4178:
4173:
4169:
4162:
4156:
4151:
4137:
4134:
4119:
4097:
4075:
4049:
4045:
4041:
4038:
4035:
4030:
4026:
4022:
4017:
4013:
4008:
4000:
3996:
3990:
3985:
3982:
3979:
3972:
3968:
3962:
3957:
3950:
3946:
3940:
3935:
3932:
3929:
3905:
3901:
3897:
3892:
3888:
3881:
3875:
3870:
3864:
3837:
3833:
3827:
3805:
3798:
3794:
3788:
3783:
3780:
3777:
3770:
3766:
3760:
3755:
3748:
3744:
3738:
3729:
3725:
3721:
3718:
3715:
3710:
3706:
3702:
3697:
3693:
3688:
3680:
3676:
3670:
3667:
3662:
3658:
3653:
3649:
3642:
3638:
3632:
3629:
3624:
3620:
3615:
3607:
3603:
3597:
3594:
3589:
3585:
3580:
3576:
3571:
3545:
3541:
3534:
3509:
3505:
3480:
3476:
3470:
3448:
3441:
3437:
3431:
3426:
3423:
3420:
3413:
3409:
3403:
3398:
3391:
3387:
3381:
3372:
3368:
3364:
3361:
3358:
3353:
3349:
3345:
3340:
3336:
3331:
3323:
3319:
3313:
3310:
3305:
3301:
3296:
3292:
3285:
3281:
3275:
3272:
3267:
3263:
3258:
3250:
3246:
3240:
3237:
3232:
3228:
3223:
3219:
3214:
3188:
3184:
3180:
3177:
3174:
3169:
3165:
3161:
3156:
3152:
3145:
3139:
3134:
3110:
3088:
3083:
3079:
3075:
3072:
3069:
3064:
3060:
3056:
3051:
3047:
3043:
3031:
3030:Interpretation
3028:
3010:
3006:
3000:
2997:
2994:
2990:
2986:
2981:
2977:
2954:
2950:
2946:
2941:
2937:
2933:
2930:
2908:
2903:
2881:
2876:
2873:
2870:
2864:
2858:
2854:
2851:
2848:
2845:
2841:
2836:
2832:
2809:
2804:
2782:
2777:
2773:
2769:
2766:
2763:
2758:
2754:
2750:
2745:
2741:
2737:
2715:
2697:
2694:
2679:
2675:
2671:
2668:
2665:
2660:
2656:
2652:
2647:
2643:
2620:
2594:
2591:
2586:
2580:
2574:
2571:
2568:
2563:
2557:
2551:
2546:
2540:
2534:
2531:
2526:
2521:
2519:
2516:
2514:
2511:
2506:
2500:
2494:
2491:
2488:
2483:
2477:
2471:
2466:
2460:
2454:
2451:
2447:
2443:
2438:
2433:
2427:
2421:
2418:
2415:
2410:
2405:
2399:
2393:
2388:
2383:
2377:
2371:
2368:
2363:
2357:
2353:
2351:
2348:
2346:
2343:
2338:
2333:
2327:
2319:
2313:
2307:
2304:
2301:
2296:
2291:
2285:
2277:
2271:
2265:
2260:
2255:
2249:
2241:
2235:
2229:
2226:
2221:
2216:
2214:
2211:
2207:
2202:
2201:
2181:
2178:
2175:
2172:
2169:
2166:
2163:
2160:
2157:
2154:
2151:
2148:
2145:
2121:
2116:
2113:
2110:
2104:
2098:
2093:
2088:
2082:
2074:
2068:
2062:
2057:
2053:
2047:
2040:
2035:
2030:
2027:
2024:
2018:
2010:
2005:
1999:
1991:
1985:
1979:
1974:
1971:
1968:
1962:
1937:
1934:
1931:
1925:
1900:
1894:
1869:
1866:
1863:
1857:
1831:
1827:
1803:
1797:
1770:
1766:
1759:
1734:
1730:
1707:
1701:
1676:
1652:
1649:
1646:
1640:
1609:
1605:
1601:
1596:
1592:
1586:
1581:
1576:
1570:
1544:
1541:
1528:
1525:
1520:
1514:
1508:
1505:
1502:
1497:
1491:
1485:
1480:
1474:
1468:
1465:
1460:
1455:
1450:
1426:
1404:
1401:
1396:
1391:
1385:
1379:
1376:
1373:
1368:
1363:
1357:
1351:
1346:
1341:
1335:
1329:
1326:
1321:
1316:
1311:
1284:
1278:
1249:
1245:
1220:
1217:
1212:
1206:
1200:
1197:
1194:
1189:
1183:
1177:
1172:
1166:
1160:
1157:
1153:
1149:
1144:
1139:
1133:
1127:
1124:
1121:
1116:
1111:
1105:
1099:
1094:
1089:
1083:
1077:
1074:
1069:
1063:
1059:
1057:
1054:
1052:
1049:
1044:
1039:
1033:
1025:
1019:
1013:
1010:
1007:
1002:
997:
991:
983:
977:
971:
966:
961:
955:
947:
941:
935:
932:
927:
922:
920:
917:
915:
912:
907:
902:
899:
896:
891:
886:
881:
876:
873:
868:
863:
861:
858:
854:
849:
848:
818:
812:
784:
781:
778:
772:
743:
737:
712:
707:
692:such that the
679:
676:
673:
667:
653:unitary matrix
636:
632:
628:
623:
619:
613:
608:
603:
597:
585:combined. Let
572:
548:
545:
542:
536:
511:
489:
467:
464:
461:
455:
430:
413:standard mode-
398:
393:
389:
385:
380:
377:
374:
370:
364:
361:
358:
354:
350:
345:
341:
335:
331:
327:
324:
319:
315:
309:
304:
299:
296:
293:
287:
260:
238:
209:
205:
201:
198:
193:
189:
185:
182:
179:
174:
170:
166:
161:
157:
151:
146:
141:
113:
99:
96:
15:
9:
6:
4:
3:
2:
8236:
8225:
8222:
8220:
8217:
8216:
8214:
8201:
8197:
8192:
8187:
8183:
8179:
8175:
8171:
8167:
8163:
8159:
8152:
8144:
8140:
8135:
8130:
8125:
8120:
8116:
8112:
8108:
8104:
8100:
8093:
8085:
8078:
8070:
8066:
8062:
8058:
8051:
8049:
8040:
8036:
8032:
8028:
8024:
8020:
8013:
8011:
8002:
7998:
7994:
7990:
7986:
7982:
7977:
7972:
7968:
7964:
7957:
7949:
7945:
7941:
7937:
7932:
7927:
7922:
7917:
7913:
7909:
7905:
7901:
7897:
7890:
7882:
7878:
7874:
7869:
7864:
7859:
7854:
7850:
7846:
7842:
7838:
7834:
7827:
7819:
7815:
7811:
7806:
7801:
7796:
7791:
7787:
7783:
7780:(4): e18768.
7779:
7775:
7771:
7764:
7756:
7752:
7748:
7743:
7738:
7734:
7730:
7726:
7722:
7718:
7711:
7703:
7699:
7694:
7689:
7684:
7679:
7675:
7671:
7667:
7663:
7659:
7652:
7646:
7640:
7634:
7628:
7622:
7621:
7613:
7607:
7601:
7593:
7589:
7585:
7581:
7576:
7571:
7567:
7563:
7556:
7548:
7546:9781450394109
7542:
7537:
7532:
7528:
7521:
7519:
7517:
7515:
7506:
7502:
7498:
7492:
7488:
7484:
7480:
7473:
7471:
7462:
7458:
7454:
7450:
7446:
7442:
7438:
7434:
7430:
7426:
7422:
7415:
7413:
7411:
7401:
7396:
7392:
7388:
7384:
7377:
7375:
7366:
7362:
7358:
7352:
7348:
7344:
7340:
7333:
7331:
7322:
7318:
7314:
7310:
7306:
7302:
7297:
7292:
7288:
7284:
7277:
7275:
7265:
7260:
7255:
7250:
7246:
7242:
7238:
7231:
7229:
7227:
7218:
7214:
7210:
7206:
7201:
7196:
7192:
7188:
7181:
7174:
7168:
7166:
7164:
7162:
7154:
7148:
7141:
7137:
7134:
7128:
7126:
7124:
7122:
7120:
7111:
7107:
7103:
7099:
7094:
7089:
7085:
7081:
7074:
7072:
7070:
7068:
7059:
7052:
7044:
7037:
7035:
7026:
7022:
7018:
7014:
7010:
7006:
7002:
6998:
6994:
6990:
6989:Psychometrika
6983:
6975:
6971:
6967:
6963:
6959:
6955:
6948:
6946:
6941:
6934:
6932:
6928:
6924:
6914:
6911:
6909:
6905:
6900:
6898:
6893:
6891:
6886:
6883:
6881:
6875:
6852:
6847:
6837:
6818:
6803:
6798:
6793:
6783:
6764:
6732:
6694:
6671:solution: if
6670:
6669:quasi-optimal
6650:
6640:
6633:
6628:
6624:
6619:
6615:
6610:
6606:
6583:
6573:
6547:
6542:
6538:
6532:
6522:
6518:
6514:
6509:
6506:
6503:
6495:
6461:
6451:
6439:
6438:
6437:
6435:
6431:
6407:
6397:
6390:
6385:
6381:
6376:
6372:
6367:
6363:
6340:
6330:
6304:
6299:
6295:
6289:
6279:
6275:
6271:
6263:
6229:
6219:
6207:
6206:
6205:
6203:
6200:
6195:
6193:
6175:
6167:
6142:
6138:
6134:
6129:
6125:
6121:
6116:
6106:
6099:
6096:
6074:
6064:
6056:
6046:
6039:
6036:
6033:
6028:
6018:
6011:
6006:
5996:
5969:
5961:
5951:
5944:
5941:
5938:
5933:
5923:
5916:
5911:
5901:
5890:
5865:
5860:
5838:
5823:
5820:
5809:
5805:
5801:
5798:
5795:
5790:
5786:
5782:
5777:
5773:
5762:
5721:
5717:
5691:
5681:
5674:
5671:
5668:
5663:
5653:
5646:
5643:
5640:
5635:
5625:
5618:
5613:
5603:
5592:
5499:
5495:
5491:
5488:
5485:
5480:
5476:
5472:
5469:
5466:
5461:
5457:
5453:
5448:
5444:
5436:
5375:
5371:
5367:
5364:
5359:
5355:
5351:
5348:
5345:
5340:
5336:
5332:
5327:
5323:
5312:
5294:Approximation
5291:
5288:
5259:
5254:
5250:
5244:
5236:
5231:
5223:
5189:
5186:
5183:
5169:
5165:
5159:
5154:
5150:
5146:
5141:
5124:
5105:
5101:
5097:
5092:
5088:
5083:
5079:
5074:
5070:
5047:
5042:
5038:
5032:
5022:
5018:
5014:
5009:
5006:
5003:
4995:
4975:
4958:
4955:
4952:
4944:
4924:
4920:
4919:
4905:
4902:
4899:
4896:
4893:
4890:
4887:
4884:
4876:
4854:
4849:
4832:
4831:
4830:
4814:
4810:
4806:
4801:
4797:
4766:
4761:
4747:
4743:
4739:
4734:
4729:
4715:
4711:
4707:
4702:
4697:
4683:
4679:
4673:
4668:
4654:
4650:
4639:
4600:
4599:
4579:
4575:
4571:
4566:
4562:
4551:
4522:
4517:
4503:
4489:
4477:
4469:
4450:
4446:
4426:
4406:
4402:
4401:
4385:
4382:
4379:
4376:
4373:
4370:
4367:
4359:
4358:
4357:
4349:
4347:
4344:
4340:
4330:
4329:as a subset.
4267:
4263:
4259:
4256:
4253:
4248:
4244:
4240:
4235:
4231:
4203:
4199:
4195:
4192:
4189:
4184:
4180:
4176:
4171:
4167:
4154:
4133:
4095:
4047:
4043:
4039:
4036:
4033:
4028:
4024:
4020:
4015:
4011:
3998:
3994:
3983:
3980:
3977:
3970:
3966:
3955:
3948:
3944:
3930:
3927:
3903:
3899:
3895:
3890:
3886:
3873:
3868:
3835:
3831:
3803:
3796:
3792:
3781:
3778:
3775:
3768:
3764:
3753:
3746:
3742:
3727:
3723:
3719:
3716:
3713:
3708:
3704:
3700:
3695:
3691:
3686:
3678:
3674:
3668:
3665:
3660:
3656:
3651:
3647:
3640:
3636:
3630:
3627:
3622:
3618:
3613:
3605:
3601:
3595:
3592:
3587:
3583:
3578:
3574:
3543:
3539:
3507:
3503:
3478:
3474:
3446:
3439:
3435:
3424:
3421:
3418:
3411:
3407:
3396:
3389:
3385:
3370:
3366:
3362:
3359:
3356:
3351:
3347:
3343:
3338:
3334:
3329:
3321:
3317:
3311:
3308:
3303:
3299:
3294:
3290:
3283:
3279:
3273:
3270:
3265:
3261:
3256:
3248:
3244:
3238:
3235:
3230:
3226:
3221:
3217:
3186:
3182:
3178:
3175:
3172:
3167:
3163:
3159:
3154:
3150:
3137:
3081:
3077:
3073:
3070:
3067:
3062:
3058:
3054:
3049:
3045:
3027:
3024:
3008:
3004:
2998:
2995:
2992:
2988:
2984:
2979:
2975:
2952:
2948:
2944:
2939:
2935:
2931:
2928:
2906:
2871:
2839:
2834:
2830:
2807:
2775:
2771:
2767:
2764:
2761:
2756:
2752:
2748:
2743:
2739:
2703:
2693:
2677:
2673:
2669:
2666:
2663:
2658:
2654:
2650:
2645:
2641:
2592:
2584:
2572:
2569:
2566:
2561:
2549:
2544:
2529:
2517:
2504:
2492:
2489:
2486:
2481:
2469:
2464:
2449:
2445:
2436:
2431:
2419:
2416:
2413:
2408:
2403:
2391:
2386:
2381:
2366:
2355:
2349:
2336:
2331:
2317:
2305:
2302:
2299:
2294:
2289:
2275:
2263:
2258:
2253:
2239:
2224:
2212:
2179:
2176:
2173:
2170:
2167:
2164:
2161:
2158:
2155:
2152:
2149:
2146:
2143:
2135:
2119:
2111:
2091:
2086:
2072:
2055:
2051:
2033:
2025:
2008:
2003:
1989:
1977:
1969:
1932:
1898:
1864:
1844:
1829:
1825:
1801:
1768:
1764:
1732:
1728:
1705:
1647:
1627:
1607:
1603:
1599:
1594:
1590:
1579:
1574:
1556:
1554:
1553:compact HOSVD
1550:
1543:Compact HOSVD
1540:
1526:
1518:
1506:
1503:
1500:
1495:
1483:
1478:
1463:
1453:
1402:
1394:
1389:
1377:
1374:
1371:
1366:
1361:
1349:
1344:
1339:
1324:
1314:
1300:
1282:
1265:
1247:
1243:
1218:
1210:
1198:
1195:
1192:
1187:
1175:
1170:
1155:
1151:
1142:
1137:
1125:
1122:
1119:
1114:
1109:
1097:
1092:
1087:
1072:
1061:
1055:
1042:
1037:
1023:
1011:
1008:
1005:
1000:
995:
981:
969:
964:
959:
945:
930:
918:
900:
897:
894:
884:
871:
859:
838:
834:
816:
800:
779:
759:
741:
710:
695:
674:
654:
634:
630:
626:
621:
617:
606:
601:
543:
487:
462:
418:
416:
391:
387:
383:
378:
375:
372:
368:
362:
359:
356:
352:
348:
343:
339:
333:
329:
322:
317:
313:
302:
294:
273:
227:
207:
203:
199:
196:
191:
187:
183:
180:
177:
172:
168:
164:
159:
155:
144:
129:
95:
93:
89:
83:
81:
78:
74:
70:
66:
62:
58:
54:
50:
46:
42:
38:
34:
30:
26:
22:
8168:(1): 13733.
8165:
8161:
8151:
8106:
8102:
8092:
8083:
8077:
8060:
8056:
8022:
8018:
7966:
7962:
7956:
7903:
7899:
7889:
7840:
7836:
7826:
7777:
7773:
7763:
7724:
7720:
7710:
7665:
7661:
7651:
7639:
7627:
7619:
7612:
7600:
7565:
7561:
7555:
7526:
7478:
7428:
7424:
7390:
7386:
7338:
7286:
7282:
7244:
7240:
7190:
7186:
7180:
7147:
7083:
7079:
7057:
7051:
7042:
6992:
6988:
6982:
6957:
6953:
6920:
6912:
6901:
6894:
6887:
6884:
6876:
6873:
6870:Applications
6668:
6433:
6429:
6427:
6201:
6198:
6196:
5297:
5286:
5284:
4922:
4788:
4404:
4355:
4345:
4339:L. R. Tucker
4336:
4139:
3033:
3025:
2701:
2699:
1817:
1625:
1558:Assume that
1557:
1552:
1546:
1298:
1262:denotes the
832:
798:
757:
693:
414:
274:
225:
101:
84:
79:
69:L. R. Tucker
28:
24:
18:
7241:IEEE Access
6929:variant of
6880:TensorFaces
6199:classically
4925:flattening
4407:flattening
4136:Computation
3023:must hold.
1628:flattening
1299:core tensor
411:denote the
128:M-way array
8213:Categories
7296:1710.11306
7254:1904.06455
7060:: 109–127.
7045:: 122–137.
6937:References
4352:M-mode SVD
3816:where the
1747:th column
696:th column
500:'th index
417:flattening
98:Definition
7976:1406.3496
7881:Highlight
7818:Highlight
7755:Highlight
7592:0895-4798
7570:CiteSeerX
7505:117253621
7453:1064-8275
7200:1311.6182
7110:0895-4798
7088:CiteSeerX
7009:0033-3123
6974:1467-9590
6844:‖
6838:∗
6830:¯
6819:−
6809:‖
6799:≤
6790:‖
6776:¯
6765:−
6755:‖
6733:∗
6725:¯
6687:¯
6644:¯
6634:×
6616:∈
6577:¯
6529:Σ
6515:≈
6507:−
6455:¯
6401:¯
6391:×
6373:∈
6334:¯
6286:Σ
6272:≈
6223:¯
6172:‖
6165:‖
6135:≤
6110:¯
6100:≤
6065:∈
6050:¯
6037:…
6022:¯
6000:¯
5955:¯
5942:…
5927:¯
5905:¯
5891:−
5857:‖
5849:¯
5839:−
5829:‖
5802:×
5799:⋯
5796:×
5783:×
5763:∈
5756:¯
5718:ℓ
5685:¯
5672:…
5657:¯
5644:…
5629:¯
5607:¯
5593:−
5532:¯
5489:…
5470:…
5437:−
5368:×
5365:⋯
5352:×
5349:⋯
5346:×
5333:×
5313:∈
5241:Σ
5187:−
5166:×
5098:×
5080:∈
5029:Σ
5007:−
4956:−
4900:…
4807:≪
4744:×
4740:…
4712:×
4708:…
4680:×
4651:×
4572:×
4552:∈
4498:Σ
4380:…
4356:Sources:
4257:…
4196:×
4193:⋯
4190:×
4177:×
4155:∈
4037:…
3984:⊗
3981:⋯
3978:⊗
3956:⊗
3896:×
3874:∈
3782:⊗
3779:⋯
3776:⊗
3754:⊗
3717:…
3652:∑
3648:⋯
3614:∑
3579:∑
3425:⊗
3422:⋯
3419:⊗
3397:⊗
3360:…
3295:∑
3291:⋯
3257:∑
3222:∑
3179:×
3176:⋯
3173:×
3160:×
3138:∈
3071:…
2996:≠
2989:∏
2985:≤
2945:≤
2932:≤
2765:…
2670:×
2667:⋯
2664:×
2651:×
2570:…
2530:×
2490:…
2450:×
2417:…
2367:×
2303:…
2225:×
2174:…
2162:…
2052:×
1950:, we have
1600:×
1580:∈
1504:…
1464:×
1375:…
1325:×
1244:⋅
1196:…
1156:×
1123:…
1073:×
1009:…
931:×
898:…
872:×
839:, we have
627:×
607:∈
384:⋯
360:−
349:⋯
323:×
303:∈
200:×
197:⋯
184:⋯
181:×
178:⋯
165:×
145:∈
8200:29062063
8143:28841719
8103:PLOS ONE
8001:17966555
7940:25875127
7900:PLOS ONE
7877:22216090
7837:PLOS ONE
7814:21625625
7774:PLOS ONE
7751:19888207
7702:18003902
7461:15318433
7365:67874182
7136:Archived
7025:44301099
6925:-based,
6428:while a
5287:in-place
4285:, where
3123:. Since
224:, where
8224:Tensors
8191:5653784
8170:Bibcode
8134:5571984
8111:Bibcode
8039:7957799
7981:Bibcode
7931:4398562
7908:Bibcode
7868:3245232
7845:Bibcode
7805:3094155
7782:Bibcode
7742:2779084
7727:: 312.
7693:2147680
7670:Bibcode
7433:Bibcode
7321:3693326
7301:Bibcode
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59:, and
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23:, the
8035:S2CID
7997:S2CID
7971:arXiv
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7457:S2CID
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7317:S2CID
7291:arXiv
7249:arXiv
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651:be a
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7946:and
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7810:PMID
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7588:ISSN
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