5586:
5214:
5581:{\displaystyle {\begin{aligned}{\mathcal {A}}_{m}&=m(\mathbf {u} +{\frac {1}{m}}\mathbf {v} )\otimes (\mathbf {u} +{\frac {1}{m}}\mathbf {v} )\otimes (\mathbf {u} +{\frac {1}{m}}\mathbf {v} )-m\mathbf {u} \otimes \mathbf {u} \otimes \mathbf {u} \\&=\mathbf {u} \otimes \mathbf {u} \otimes \mathbf {v} +\mathbf {u} \otimes \mathbf {v} \otimes \mathbf {u} +\mathbf {v} \otimes \mathbf {u} \otimes \mathbf {u} +{\frac {1}{m}}(\mathbf {u} \otimes \mathbf {v} \otimes \mathbf {v} +\mathbf {v} \otimes \mathbf {u} \otimes \mathbf {v} +\mathbf {v} \otimes \mathbf {v} \otimes \mathbf {u} )+{\frac {1}{m^{2}}}\mathbf {v} \otimes \mathbf {v} \otimes \mathbf {v} \end{aligned}}}
5203:
9481:
1554:
6854:
10165:
5042:
9320:
9644:
6557:
1352:
1783:
932:
9948:
5758:
9221:
5198:{\displaystyle {\mathcal {A}}=\mathbf {u} \otimes \mathbf {u} \otimes \mathbf {v} +\mathbf {u} \otimes \mathbf {v} \otimes \mathbf {u} +\mathbf {v} \otimes \mathbf {u} \otimes \mathbf {u} ,\quad {\text{with }}\|\mathbf {u} \|=\|\mathbf {v} \|=1{\text{ and }}\langle \mathbf {u} ,\mathbf {v} \rangle \neq 1.}
9476:{\displaystyle {\mathcal {A}}=\mathbf {u} \otimes \mathbf {u} \otimes \mathbf {v} +\mathbf {u} \otimes \mathbf {v} \otimes \mathbf {u} +\mathbf {v} \otimes \mathbf {u} \otimes \mathbf {u} ,\quad {\text{with }}\|\mathbf {u} \|=\|\mathbf {v} \|=1{\text{ and }}\langle \mathbf {u} ,\mathbf {v} \rangle \neq 1}
10203:
A common partial solution to the ill-posedness problem consists of imposing an additional inequality constraint that bounds the norm of the individual rank-1 terms by some constant. Other constraints that result in a closed set, and, thus, well-posed optimization problem, include imposing positivity
10294:
In machine learning, the CP-decomposition is the central ingredient in learning probabilistic latent variables models via the technique of moment-matching. For example, consider the multi-view model which is a probabilistic latent variable model. In this model, the generation of samples are posited
9492:
2368:
will be a rank-1 tensor with probability zero, a rank-2 tensor with positive probability, and rank-3 with positive probability. On the other hand, a randomly sampled complex tensor of the same size will be a rank-1 tensor with probability zero, a rank-2 tensor with probability one, and a rank-3
1549:{\displaystyle {\mathcal {A}}=\mathbf {x} _{1}\otimes \mathbf {x} _{2}\otimes \mathbf {x} _{3}+\mathbf {x} _{1}\otimes \mathbf {y} _{2}\otimes \mathbf {y} _{3}-\mathbf {y} _{1}\otimes \mathbf {x} _{2}\otimes \mathbf {y} _{3}+\mathbf {y} _{1}\otimes \mathbf {y} _{2}\otimes \mathbf {x} _{3},}
9832:
1345:
The rank of a tensor depends on the field over which the tensor is decomposed. It is known that some real tensors may admit a complex decomposition whose rank is strictly less than the rank of a real decomposition of the same tensor. As an example, consider the following real tensor
6849:{\displaystyle {\mathcal {A}}=\sum _{i=1}^{r}\mathbf {a} _{i}\otimes \mathbf {b} _{i}=\sum _{i=1}^{r}\mathbf {a} _{i}\mathbf {b} _{i}^{T}=AB^{T}=(AX^{-1})(BX^{T})^{T}=\sum _{i=1}^{r}\mathbf {c} _{i}\mathbf {d} _{i}^{T}=\sum _{i=1}^{r}\mathbf {c} _{i}\otimes \mathbf {d} _{i},}
10200:. It was, in addition, shown that a random low-rank tensor over the reals may not admit a rank-2 approximation with positive probability, leading to the understanding that the ill-posedness problem is an important consideration when employing the tensor rank decomposition.
1628:
765:
4624:
10160:{\displaystyle \|\mathbf {a} _{i,n}^{1}\otimes \mathbf {a} _{i,n}^{2}\otimes \cdots \otimes \mathbf {a} _{i,n}^{M}\|_{F}\to \infty {\text{ and }}\|\mathbf {a} _{j,n}^{1}\otimes \mathbf {a} _{j,n}^{2}\otimes \cdots \otimes \mathbf {a} _{j,n}^{M}\|_{F}\to \infty }
5637:
9055:
10673:, this can be interpreted as the co-occurrence of words in a document. Then the coefficients in the decomposition of this empirical moment tensor can be interpreted as the probability of choosing a specific topic and each column of the factor matrix
449:
3118:
9270:. A solution to aforementioned problem may sometimes not exist because the set over which one optimizes is not closed. As such, a minimizer may not exist, even though an infimum would exist. In particular, it is known that certain so-called
77:
in 1927 and later rediscovered several times, notably in psychometrics. The CP decomposition is referred to as CANDECOMP, PARAFAC, or CANDECOMP/PARAFAC (CP). Note that the PARAFAC2 rank decomposition is a variation of the CP decomposition.
2688:
When the first factor is very large with respect to the other factors in the tensor product, then the tensor space essentially behaves as a matrix space. The generic rank of tensors living in an unbalanced tensor spaces is known to equal
7589:
705:
2996:
6086:
2206:
only forms an open set of positive measure in the
Euclidean topology. There may exist Euclidean-open sets of tensors of rank strictly higher than the generic rank. All ranks appearing on open sets in the Euclidean topology are called
9639:{\displaystyle {\mathcal {A}}_{n}=n(\mathbf {u} +{\frac {1}{n}}\mathbf {v} )\otimes (\mathbf {u} +{\frac {1}{n}}\mathbf {v} )\otimes (\mathbf {u} +{\frac {1}{n}}\mathbf {v} )-n\mathbf {u} \otimes \mathbf {u} \otimes \mathbf {u} }
2805:
6187:
3931:
3800:
2426:
2334:
2275:
3704:
9701:
7369:
2671:
8509:
3182:
8634:
1274:
7261:
is a closed set in the
Zariski topology, the decomposition on the right-hand side is a sum of a different set of rank-1 tensors than the decomposition on the left-hand side, entailing that order-2 tensors of rank
6526:
3496:
1616:
617:
7951:
1848:
8051:
6303:
3606:
10254:
4278:
that can be admitted by any of the tensors in a tensor space is unknown in general; even a conjecture about this maximum rank is missing. Presently, the best general upper bound states that the maximum rank
2870:
5895:
5841:
1778:{\displaystyle {\mathcal {A}}={\frac {1}{2}}({\bar {\mathbf {z} }}_{1}\otimes \mathbf {z} _{2}\otimes {\bar {\mathbf {z} }}_{3}+\mathbf {z} _{1}\otimes {\bar {\mathbf {z} }}_{2}\otimes \mathbf {z} _{3}),}
8878:
8202:
1335:
1031:
10664:
7493:
927:{\displaystyle {\mathcal {A}}=\sum _{r=1}^{R}\lambda _{r}\mathbf {a} _{0,r}\otimes \mathbf {a} _{1,r}\otimes \mathbf {a} _{2,r}\dots \otimes \mathbf {a} _{c,r}\otimes \cdots \otimes \mathbf {a} _{C,r},}
4746:
4401:
2489:
2047:
531:
4468:
1205:
7722:
7841:
4460:
4264:
2548:
10226:
9876:
7239:
5219:
4341:
3387:
4787:
8812:
8715:
8401:
8319:
8107:
5753:{\displaystyle {\mathcal {A}}=\mathbf {a} _{1}\otimes \mathbf {a} _{2}\otimes \cdots \otimes \mathbf {a} _{M}=\mathbf {b} _{1}\otimes \mathbf {b} _{2}\otimes \cdots \otimes \mathbf {b} _{M}}
4177:
5952:
972:
3976:
9216:{\displaystyle \min _{\mathbf {a} _{i}^{m}\in F^{I_{m}}}\|{\mathcal {A}}-\sum _{i=1}^{r}\mathbf {a} _{i}^{1}\otimes \mathbf {a} _{i}^{2}\otimes \cdots \otimes \mathbf {a} _{i}^{M}\|_{F},}
8949:
4843:
3330:
3268:
6901:
4682:
2211:. The smallest typical rank is called the generic rank; this definition applies to both complex and real tensors. The generic rank of tensor spaces was initially studied in 1983 by
2204:
2102:
1943:
4093:
4015:
232:
9940:
9695:
tensors that converges to a tensor of strictly higher rank needs to admit at least two individual rank-1 terms whose norms become unbounded. Stated formally, whenever a sequence
7117:
6381:
6218:
10350:
7186:
2366:
9257:
9021:
6405:
6350:
5976:
4997:
4949:
4895:
338:
7639:
1063:
363:
264:
186:
11291:
11269:
10194:
9902:
9673:
7045:
6986:
5785:
5615:
1877:
311:
289:
7428:
10220:
3010:
6927:
11832:
Domanov, Ignat; De
Lathauwer, Lieven (January 2017). "Canonical polyadic decomposition of third-order tensors: Relaxed uniqueness conditions and algebraic algorithm".
8747:
8228:
4119:
9047:
8908:
8263:
7395:
7286:
6552:
5024:
4925:
7752:
7669:
7498:
630:
11779:
Domanov, Ignat; Lathauwer, Lieven De (January 2014). "Canonical
Polyadic Decomposition of Third-Order Tensors: Reduction to Generalized Eigenvalue Decomposition".
2910:
8535:
8427:
8345:
8133:
6431:
6001:
11958:
Lorber, Avraham. (October 1985). "Features of quantifying chemical composition from two-dimensional data array by the rank annihilation factor analysis method".
6326:
10714:
10452:
10390:
10370:
9693:
9312:
9292:
8997:
8977:
7609:
7259:
6238:
6110:
5996:
4969:
4871:
4139:
3202:
2890:
2149:
2125:
1987:
1963:
1103:
1083:
757:
737:
358:
151:
131:
10196:. This phenomenon is often encountered when attempting to approximate a tensor using numerical optimization algorithms. It is sometimes called the problem of
12036:
Sands, Richard; Young, Forrest W. (March 1980). "Component models for three-way data: An alternating least squares algorithm with optimal scaling features".
2695:
6115:
11236:
11091:
6243:
3808:
3710:
2372:
2280:
2221:
1618:. The rank of this tensor over the reals is known to be 3, while its complex rank is only 2 because it is the sum of a complex rank-1 tensor with its
12185:
Anandkumar, Animashree; Ge, Rong; Hsu, Daniel; Kakade, Sham M; Telgarsky, Matus (2014). "Tensor decompositions for learning latent variable models".
11462:
Chiantini, L.; Ottaviani, G.; Vannieuwenhoven, N. (2014-01-01). "An
Algorithm For Generic and Low-Rank Specific Identifiability of Complex Tensors".
11153:
10980:
12132:
Bernardi, Alessandra; Daleo, Noah S.; Hauenstein, Jonathan D.; Mourrain, Bernard (December 2017). "Tensor decomposition and homotopy continuation".
11240:
11095:
11012:
9827:{\displaystyle {\mathcal {A}}_{n}=\sum _{i=1}^{r}\mathbf {a} _{i,n}^{1}\otimes \mathbf {a} _{i,n}^{2}\otimes \cdots \otimes \mathbf {a} _{i,n}^{M}}
3978:
is defective (13 and not the expected 14), the generic rank in that space is still the expected one, 4. Similarly, the set of tensors of rank 5 in
10283:
4121:. In 2011, a major breakthrough was established by Catalisano, Geramita, and Gimigliano who proved that the expected dimension of the set of rank
11677:
Hauenstein, J. D.; Oeding, L.; Ottaviani, G.; Sommese, A. J. (2016). "Homotopy techniques for tensor decomposition and perfect identifiability".
11332:
11244:
11099:
10823:
3614:
7294:
2560:
11144:
10928:
8435:
3126:
5617:. Therefore, its border rank is 2, which is strictly less than its rank. When the two vectors are orthogonal, this example is also known as a
11198:
11148:
11003:
1229:
8543:
6444:
3395:
11336:
1562:
536:
2336:
is 2. Practically, this means that a randomly sampled real tensor (from a continuous probability measure on the space of tensors) of size
11394:
7849:
1791:
11568:
Chiantini, L.; Ottaviani, G.; Vannieuwenhoven, N. (2017-01-01). "Effective
Criteria for Specific Identifiability of Tensors and Forms".
11390:
10827:
3508:
11885:
Faber, Nicolaas (Klaas) M.; Ferré, Joan; Boqué, Ricard (January 2001). "Iteratively reweighted generalized rank annihilation method".
7956:
4179:
is the expected one except for rank 3 tensors in the 4 factor case, yet the expected rank in that case is still 4. As a consequence,
11386:
10745:
1220:
82:
2816:
12672:
11140:
11007:
10932:
5846:
5790:
8817:
8141:
1279:
977:
12079:
Bernardi, A.; Brachat, J.; Comon, P.; Mourrain, B. (May 2013). "General tensor decomposition, moment matrices and applications".
10457:
7433:
4619:{\displaystyle r_{\mbox{max}}(I_{1},\ldots ,I_{M})\leq \min \left\{\prod _{m=2}^{M}I_{m},2\cdot r(I_{1},\ldots ,I_{M})\right\},}
4691:
4346:
2434:
1992:
12821:
454:
11443:
1143:
7677:
1219:
that the tensor represents. A simple polynomial-time algorithm exists for certifying that a tensor is of rank 1, namely the
11399:
7765:
4406:
4182:
2494:
2431:
The generic rank of tensor spaces depends on the distinction between balanced and unbalanced tensor spaces. A tensor space
11515:
Bocci, Cristiano; Chiantini, Luca; Ottaviani, Giorgio (2014-12-01). "Refined methods for the identifiability of tensors".
10878:
9840:
7191:
11207:
11108:
11066:
4282:
3339:
188:. Multiple indices that one might encounter when referring to the multiple modes of a tensor are conveniently denoted by
12856:
12535:
10793:
8951:
that is not identifiability-unbalanced is expected to be identifiable (modulo the exceptional cases in small spaces).
4751:
12315:
8764:
8353:
8271:
8059:
4144:
8639:
5904:
940:
12737:
3936:
2127:: every tensor in the aforementioned space is either of rank less than the generic rank, or it is the limit in the
17:
10299:
random variables known as the different "views" of the hidden variable. For example, assume there are three views
5030:
4792:
4020:
The AOP conjecture has been proved completely in a number of special cases. Lickteig showed already in 1985 that
12273:
10295:
as follows: there exists a hidden random variable that is not observed directly, given which, there are several
8913:
3273:
3211:
11993:
Sanchez, Eugenio; Kowalski, Bruce R. (January 1990). "Tensorial resolution: A direct trilinear decomposition".
6859:
10879:"Foundations of the PARAFAC procedure: Models and conditions for an "explanatory" multi-modal factor analysis"
12588:
12520:
10730:
4748:. It is well-known that the foregoing inequality may be strict. For instance, the generic rank of tensors in
6352:
can be obtained by permuting the order of the summands. Observe that in a tensor rank decomposition all the
4632:
2154:
2052:
1893:
12959:
12613:
10735:
10266:
10248:
4023:
3981:
191:
12851:
9907:
12278:
7050:
6355:
6192:
12662:
12482:
10302:
7430:. For simplicity in notation, assume without loss of generality that the factors are ordered such that
7122:
4017:
is defective (44 and not the expected 45), but the generic rank in that space is still the expected 6.
2339:
9266:
It was shown in a 2008 paper by de Silva and Lim that the above standard approximation problem may be
444:{\displaystyle {\mathcal {A}}\in \mathbb {C} ^{I_{1}\times I_{2}\times \dots I_{m}\times \dots I_{M}}}
12334:
10296:
12816:
12233:
11621:
Chiantini, L.; Ottaviani, G. (2012-01-01). "On
Generic Identifiability of 3-Tensors of Small Rank".
9229:
9002:
6386:
6331:
5957:
4978:
4930:
4876:
319:
12918:
12836:
12790:
12497:
7618:
3113:{\displaystyle \Pi =\prod _{m=1}^{M}I_{m}\quad {\text{and}}\quad \Sigma =\sum _{m=1}^{M}(I_{m}-1).}
1036:
237:
159:
11274:
11252:
10173:
9881:
9652:
6991:
6932:
5763:
5594:
3389:
because it is only conjecturally correct. It is known that the true generic rank always satisfies
1860:
294:
272:
12888:
12575:
12492:
12462:
10260:
7612:
7400:
12846:
12702:
12657:
12287:
12228:
10911:
6906:
74:
57:
is a variant of the tensor rank decomposition, in which the tensor is approximated as a sum of
7584:{\displaystyle S_{r}\subset F^{I_{1}}\otimes \cdots F^{I_{m}}\otimes \cdots \otimes F^{I_{M}}}
3270:
is the least rank that is expected to occur on a set of positive
Euclidean measure. The value
1121:. If the number of terms is not minimal, then the above decomposition is often referred to as
700:{\displaystyle {\mathcal {A}}\in {\mathbb {F} }^{I_{0}\times I_{1}\times \ldots \times I_{C}}}
12928:
12883:
12363:
12308:
11723:
11341:
10725:
8732:
8207:
4098:
2991:{\displaystyle r_{E}(I_{1},\ldots ,I_{M})=\left\lceil {\frac {\Pi }{\Sigma +1}}\right\rceil }
2218:
As an illustration of the above concepts, it is known that both 2 and 3 are typical ranks of
10771:
9026:
8887:
8233:
7374:
7265:
6531:
6081:{\displaystyle {\mathcal {A}}\in F^{I_{1}}\otimes F^{I_{2}}\otimes \cdots \otimes F^{I_{M}}}
5006:
4904:
12903:
12831:
12717:
12583:
12545:
12477:
12220:
10750:
10740:
9294:, even though the limit of the sequence converges to a tensor of rank strictly higher than
7730:
7647:
8:
12780:
12603:
12593:
12442:
12427:
12383:
10874:
10208:
strictly less than unity between the rank-1 terms appearing in the sought decomposition.
8514:
8406:
8324:
8112:
6410:
28:
12224:
11010:(2008). "Tensor Rank and the Ill-Posedness of the Best Low-Rank Approximation Problem".
6308:
12913:
12770:
12623:
12437:
12373:
12246:
12167:
12141:
12114:
12088:
12061:
12018:
11867:
11841:
11814:
11788:
11761:
11735:
11704:
11686:
11656:
11630:
11603:
11577:
11550:
11524:
11497:
11471:
11368:
11350:
11180:
11162:
11039:
11021:
10962:
10944:
10853:
10676:
10395:
10375:
10355:
9678:
9297:
9277:
8982:
8962:
7594:
7244:
6223:
6095:
5981:
4954:
4856:
4124:
3187:
2875:
2134:
2128:
2110:
1972:
1948:
1088:
1068:
742:
722:
343:
136:
116:
11898:
11122:
11103:
4971:
is the least value for which such a convergent sequence exists, then it is called the
2800:{\displaystyle r(I_{1},\ldots ,I_{M})=\min \left\{I_{1},\prod _{m=2}^{M}I_{m}\right\}}
2369:
tensor with probability zero. It is even known that the generic rank-3 real tensor in
12908:
12677:
12652:
12467:
12378:
12358:
12171:
12159:
12106:
12065:
12053:
12022:
12010:
11975:
11940:
11902:
11871:
11859:
11806:
11765:
11753:
11648:
11595:
11554:
11542:
11489:
11439:
11221:
11202:
11076:
11061:
3001:
2810:
1619:
1208:
98:
90:
12250:
12118:
11818:
11749:
11708:
11660:
11607:
11501:
11372:
11184:
10857:
10716:
corresponds to probabilities of words in the vocabulary in the corresponding topic.
6182:{\displaystyle \{{\mathcal {A}}_{1},{\mathcal {A}}_{2},\ldots ,{\mathcal {A}}_{r}\}}
5208:
It can be approximated arbitrarily well by the following sequence of rank-2 tensors
12964:
12923:
12598:
12565:
12550:
12432:
12301:
12238:
12151:
12098:
12045:
12002:
11967:
11932:
11923:; Ross, R. T.; Abel, R. B. (October 1993). "A Decomposition for Three-Way Arrays".
11894:
11851:
11798:
11745:
11696:
11640:
11587:
11534:
11481:
11431:
11408:
11360:
11312:
11216:
11172:
11117:
11071:
11043:
11031:
10966:
10954:
10845:
10802:
8761:
expected to be at least two in case 8 with exception of the two identifiable cases
3205:
1966:
11317:
11248:
11176:
2892:
is some indeterminate closed set in the
Zariski topology, equals the above value.
12893:
12841:
12785:
12765:
12667:
12555:
12422:
12393:
12208:
12155:
11057:
10788:
9486:
can be approximated arbitrarily well by the following sequence of rank-2 tensors
3926:{\displaystyle r(I_{1},\ldots ,I_{m},\ldots ,I_{M})=r_{E}(I_{1},\ldots ,I_{M})+1}
3795:{\displaystyle F^{(m+1)\times (m+1)\times 2\times 2}{\text{ with }}m=2,3,\ldots }
2212:
1854:
156:
Indices are denoted by a combination of lowercase and upper case italic letters,
94:
9675:. This example neatly illustrates the general principle that a sequence of rank-
2421:{\displaystyle \mathbb {R} ^{2}\otimes \mathbb {R} ^{2}\otimes \mathbb {R} ^{2}}
2329:{\displaystyle \mathbb {C} ^{2}\otimes \mathbb {C} ^{2}\otimes \mathbb {C} ^{2}}
2270:{\displaystyle \mathbb {R} ^{2}\otimes \mathbb {R} ^{2}\otimes \mathbb {R} ^{2}}
12933:
12898:
12795:
12628:
12618:
12608:
12530:
12502:
12487:
12472:
12388:
10893:
10830:(1970). "Analysis of individual differences in multidimensional scaling via an
10670:
9260:
12878:
12102:
11855:
11538:
11435:
11364:
12953:
12870:
12775:
12687:
12560:
12163:
12110:
12057:
12014:
11979:
11944:
11906:
11863:
11810:
11757:
11700:
11652:
11599:
11546:
11493:
11397:(1980). "Approximate solutions for the bilinear form computational problem".
10836:
10205:
9274:
may be approximated arbitrarily well by a sequence of tensor of rank at most
1216:
102:
7758:
in this case), unless either one of the following exceptional cases holds:
1226:
The rank of the tensor of zeros is zero by convention. The rank of a tensor
12938:
12742:
12727:
12692:
12540:
12525:
12006:
11920:
3699:{\displaystyle F^{(2m+1)\times (2m+1)\times 3}{\text{ with }}m=1,2,\ldots }
86:
70:
10806:
7364:{\displaystyle F^{I_{1}}\otimes F^{I_{2}}\otimes \cdots \otimes F^{I_{M}}}
5026:
they may differ. Border tensors were first studied in the context of fast
2666:{\displaystyle I_{1}>1+\prod _{m=2}^{M}I_{m}-\sum _{m=2}^{M}(I_{m}-1),}
12826:
12800:
12722:
12411:
12350:
10791:(1927). "The expression of a tensor or a polyadic as a sum of products".
8504:{\displaystyle F^{2}\otimes F^{2}\otimes F^{2}\otimes F^{2}\otimes F^{2}}
6092:
if every of its tensor rank decompositions is the sum of the same set of
3177:{\displaystyle \mathbb {C} ^{I_{1}\times \cdots \times I_{M}}\setminus Z}
2813:. More precisely, the rank of every tensor in an unbalanced tensor space
1212:
66:
11971:
3004:
for complex tensors and on a
Euclidean-open set for real tensors, where
12707:
12049:
11591:
10849:
8629:{\textstyle r_{E}(I_{1},I_{2},\ldots ,I_{M})={\frac {\Pi }{\Sigma +1}}}
1879:: real matrix rank and complex matrix rank coincide for real matrices.
1269:{\displaystyle \mathbf {a} _{1}\otimes \cdots \otimes \mathbf {a} _{M}}
12242:
11802:
11644:
11485:
11104:"Ranks of tensors, secant varieties of Segre varieties and fat points"
11035:
8954:
6521:{\displaystyle F^{I_{1}}\otimes F^{I_{2}}\simeq F^{I_{1}\times I_{2}}}
3491:{\displaystyle r(I_{1},\ldots ,I_{M})\geq r_{E}(I_{1},\ldots ,I_{M}).}
2904:
generic rank of tensors living in a balanced tensor space is equal to
12682:
12633:
11167:
11026:
8979:
decomposition closest (in the usual Euclidean topology) to some rank-
8721:
In these exceptional cases, the generic (and also minimum) number of
2105:
1611:{\displaystyle \mathbf {x} _{i},\mathbf {y} _{j}\in \mathbb {R} ^{2}}
612:{\displaystyle {\mathcal {A}}_{i_{1},i_{2},\dots ,i_{m},\dots i_{M}}}
133:
and an upper bound scalar is denoted by an upper case italic letter,
12211:; Bader, Brett W. (2009). "Tensor Decompositions and Applications".
11936:
11412:
10958:
7946:{\textstyle I_{1}>\prod _{m=2}^{M}i_{m}-\sum _{m=2}^{M}(I_{m}-1)}
3805:
In each of these exceptional cases, the generic rank is known to be
1136:
Contrary to the case of matrices, computing the rank of a tensor is
12712:
12697:
12146:
11846:
11691:
11582:
1853:
In contrast, the rank of real matrices will never decrease under a
1843:{\displaystyle \mathbf {z} _{k}=\mathbf {x} _{k}+i\mathbf {y} _{k}}
12093:
11793:
11740:
11635:
11529:
11476:
11355:
10949:
2151:. In the case of real tensors, the set of tensors of rank at most
12406:
12368:
11567:
11461:
8046:{\textstyle r\geq \prod _{m=2}^{M}I_{m}-\sum _{m=2}^{M}(I_{m}-1)}
7611:. Then, the following statement was proved to be correct using a
6298:{\displaystyle {\mathcal {A}}=\sum _{i=1}^{r}{\mathcal {A}}_{i},}
5618:
3601:{\displaystyle r(I_{1},\ldots ,I_{M})=r_{E}(I_{1},\ldots ,I_{M})}
1137:
85:
computes orthonormal mode matrices and has found applications in
12286:(free exploratory multivariate data analysis software linked to
11676:
10392:. Then the empirical third moment of this latent variable model
12732:
12324:
12131:
10272:
1140:. The only notable well-understood case consists of tensors in
1109:
of the tensor, and the decomposition is often referred to as a
10772:"Automatic Unsupervised Tensor Mining with Quality Assessment"
2865:{\displaystyle F^{I_{1}\times \cdots \times I_{M}}\setminus Z}
707:
is a collection of multivariate observations organized into a
81:
Another popular generalization of the matrix SVD known as the
12283:
11151:(2009). "Induction for secant varieties of Segre varieties".
7291:
The situation changes completely for higher-order tensors in
5890:{\displaystyle \mathbf {a} _{m}=\lambda _{m}\mathbf {b} _{m}}
5836:{\displaystyle \lambda _{1}\lambda _{2}\cdots \lambda _{M}=1}
51:
is minimal. Computing this decomposition is an open problem.
12078:
11235:
11090:
8873:{\displaystyle F^{3}\otimes F^{2}\otimes F^{2}\otimes F^{2}}
8197:{\displaystyle F^{n}\otimes F^{n}\otimes F^{2}\otimes F^{2}}
4999:. For order-2 tensors, i.e., matrices, rank and border rank
1330:{\displaystyle \mathbf {a} _{m}\in F^{I_{m}}\setminus \{0\}}
1026:{\displaystyle \mathbf {a} _{m,r}\in {\mathbb {F} }^{I_{m}}}
10659:{\displaystyle E=\sum _{i=1}^{k}Pr(h=i)E\otimes E\otimes E}
7488:{\displaystyle I_{1}\geq I_{2}\geq \cdots \geq I_{M}\geq 2}
113:
A scalar variable is denoted by lower case italic letters,
12293:
5036:
A classic example of a border tensor is the rank-3 tensor
3208:, is expected to equal the above value. For real tensors,
2049:. In the case of complex tensors, tensors of rank at most
719:+1. Every tensor may be represented with a suitably large
4741:{\displaystyle F^{I_{1}}\otimes \cdots \otimes F^{I_{M}}}
4396:{\displaystyle F^{I_{1}}\otimes \cdots \otimes F^{I_{M}}}
2484:{\displaystyle F^{I_{1}}\otimes \cdots \otimes F^{I_{M}}}
2042:{\displaystyle F^{I_{1}}\otimes \cdots \otimes F^{I_{M}}}
316:
A higher order tensor is denoted by calligraphic letters,
526:{\displaystyle a_{i_{1},i_{2},\dots ,i_{m},\dots i_{M}}}
11514:
1200:{\displaystyle F^{I_{m}}\otimes F^{I_{n}}\otimes F^{2}}
10834:-way generalization of 'Eckart–Young' decomposition".
8916:
8642:
8546:
7959:
7852:
7717:{\displaystyle {\mathcal {A}}\in S_{r}\setminus Z_{r}}
5978:
are called identifiable or essentially unique. A rank-
4901:
if there exists a sequence of tensors of rank at most
4801:
4477:
4291:
269:
A vector is denoted by a lower case bold Times Roman,
65:. The CP decomposition has found some applications in
12184:
11277:
11255:
10679:
10460:
10398:
10378:
10358:
10305:
10176:
9951:
9910:
9884:
9843:
9704:
9681:
9655:
9495:
9323:
9300:
9280:
9232:
9058:
9029:
9005:
8985:
8965:
8890:
8820:
8767:
8735:
8517:
8438:
8409:
8356:
8327:
8274:
8236:
8210:
8144:
8115:
8062:
7836:{\displaystyle r>r_{E}(I_{1},I_{2},\ldots ,I_{M})}
7768:
7733:
7680:
7650:
7621:
7597:
7501:
7436:
7403:
7377:
7297:
7268:
7247:
7194:
7125:
7053:
6994:
6935:
6909:
6862:
6560:
6534:
6447:
6413:
6389:
6358:
6334:
6311:
6246:
6226:
6195:
6118:
6098:
6004:
5984:
5960:
5907:
5849:
5793:
5766:
5640:
5634:
It follows from the definition of a pure tensor that
5597:
5217:
5045:
5009:
4981:
4957:
4933:
4907:
4879:
4859:
4795:
4754:
4694:
4635:
4471:
4455:{\displaystyle I_{1}\geq I_{2}\geq \cdots \geq I_{M}}
4409:
4349:
4285:
4259:{\displaystyle r(2,2,\ldots ,2)=r_{E}(2,2,\ldots ,2)}
4185:
4147:
4127:
4101:
4026:
3984:
3939:
3811:
3713:
3617:
3511:
3398:
3342:
3276:
3214:
3190:
3129:
3013:
2913:
2878:
2819:
2698:
2563:
2543:{\displaystyle I_{1}\geq I_{2}\geq \cdots \geq I_{M}}
2497:
2437:
2375:
2342:
2283:
2224:
2157:
2137:
2113:
2055:
1995:
1975:
1951:
1896:
1863:
1794:
1631:
1565:
1355:
1282:
1232:
1146:
1091:
1071:
1039:
980:
943:
768:
745:
725:
633:
539:
457:
366:
346:
322:
297:
275:
240:
194:
162:
139:
119:
12178:
11139:
9871:{\displaystyle {\mathcal {A}}_{n}\to {\mathcal {A}}}
7234:{\displaystyle X\in \mathrm {GL} _{n}(F)\setminus Z}
4845:, while it is known that the maximum rank equals 3.
11831:
8955:
Ill-posedness of the standard approximation problem
6240:thus has only one essentially unique decomposition
4336:{\displaystyle r_{\mbox{max}}(I_{1},\ldots ,I_{M})}
3382:{\displaystyle F^{I_{1}\times \cdots \times I_{M}}}
291:and a matrix is denoted by bold upper case letters
11620:
11285:
11263:
10708:
10658:
10446:
10384:
10364:
10344:
10188:
10159:
9934:
9896:
9870:
9826:
9687:
9667:
9638:
9475:
9306:
9286:
9251:
9215:
9041:
9015:
8991:
8971:
8943:
8902:
8872:
8806:
8741:
8709:
8628:
8529:
8503:
8421:
8395:
8339:
8313:
8257:
8222:
8196:
8127:
8101:
8045:
7945:
7835:
7746:
7716:
7663:
7633:
7603:
7583:
7487:
7422:
7389:
7363:
7280:
7253:
7233:
7180:
7111:
7039:
6980:
6921:
6895:
6848:
6546:
6520:
6425:
6399:
6375:
6344:
6320:
6297:
6232:
6212:
6181:
6104:
6080:
5990:
5970:
5946:
5889:
5835:
5779:
5752:
5609:
5580:
5197:
5018:
4991:
4963:
4943:
4919:
4889:
4865:
4837:
4781:
4740:
4676:
4618:
4454:
4395:
4335:
4258:
4171:
4133:
4113:
4087:
4009:
3970:
3933:. Note that while the set of tensors of rank 3 in
3925:
3794:
3698:
3600:
3490:
3381:
3324:
3262:
3196:
3176:
3112:
2990:
2884:
2864:
2799:
2665:
2542:
2483:
2420:
2360:
2328:
2269:
2198:
2143:
2119:
2096:
2041:
1981:
1957:
1937:
1871:
1842:
1777:
1610:
1548:
1329:
1268:
1199:
1097:
1077:
1057:
1025:
966:
926:
751:
731:
699:
611:
525:
443:
352:
332:
305:
283:
258:
226:
180:
145:
125:
11724:"On the identifiability of binary Segre products"
11339:(2015). "On maximum, typical and generic ranks".
11331:
11154:Transactions of the American Mathematical Society
11062:"Rank and optimal computation of generic tensors"
8959:The rank approximation problem asks for the rank-
12951:
11925:SIAM Journal on Matrix Analysis and Applications
11919:
11781:SIAM Journal on Matrix Analysis and Applications
11778:
11623:SIAM Journal on Matrix Analysis and Applications
11570:SIAM Journal on Matrix Analysis and Applications
11464:SIAM Journal on Matrix Analysis and Applications
11430:. Graduate Texts in Mathematics. Vol. 133.
11013:SIAM Journal on Matrix Analysis and Applications
9060:
7641:, and it is conjectured to be valid in general:
6554:. This follows essentially from the observation
4782:{\displaystyle \mathbb {R} ^{2\times 2\times 2}}
4522:
2740:
11992:
11887:Chemometrics and Intelligent Laboratory Systems
11884:
11385:
10787:
8807:{\displaystyle F^{5}\otimes F^{4}\otimes F^{3}}
8710:{\textstyle r=r_{E}(I_{1},I_{2},\ldots ,I_{M})}
8396:{\displaystyle F^{6}\otimes F^{6}\otimes F^{3}}
8314:{\displaystyle F^{4}\otimes F^{4}\otimes F^{4}}
8102:{\displaystyle F^{4}\otimes F^{4}\otimes F^{3}}
7846:The space is identifiability-unbalanced, i.e.,
4172:{\displaystyle 2\times 2\times \cdots \times 2}
11721:
10279:General polynomial system solving algorithms:
5947:{\displaystyle \{\mathbf {a} _{m}\}_{m=1}^{M}}
967:{\displaystyle \lambda _{r}\in {\mathbb {R} }}
12309:
10454:is a rank 3 tensor and can be decomposed as:
7591:denote the set of tensors of rank bounded by
3971:{\displaystyle F^{2\times 2\times 2\times 2}}
43:is the decomposition of a tensor as a sum of
10935:(2013). "Most tensor problems are NP-Hard".
10912:"Aptera: Automatic PARAFAC2 Tensor Analysis"
10822:
10255:simultaneous generalized Schur decomposition
10142:
10057:
10037:
9952:
9464:
9448:
9434:
9426:
9420:
9412:
9240:
9233:
9201:
9103:
6176:
6119:
5924:
5908:
5186:
5170:
5156:
5148:
5142:
5134:
3123:More precisely, the rank of every tensor in
1324:
1318:
11002:
8944:{\textstyle r<{\frac {\Pi }{\Sigma +1}}}
6528:, i.e., matrices, are not identifiable for
6383:'s are distinct, for otherwise the rank of
4838:{\displaystyle r_{\mbox{max}}(2,2,2)\leq 4}
2683:
12316:
12302:
12207:
12134:Differential Geometry and Its Applications
12035:
11722:Bocci, Cristiano; Chiantini, Luca (2013).
11135:
11133:
6436:
3325:{\displaystyle r_{E}(I_{1},\ldots ,I_{M})}
3263:{\displaystyle r_{E}(I_{1},\ldots ,I_{M})}
12257:
12232:
12145:
12092:
11845:
11792:
11739:
11690:
11634:
11581:
11528:
11475:
11354:
11316:
11279:
11257:
11220:
11166:
11121:
11075:
11025:
10998:
10996:
10994:
10979:
10948:
10927:
10746:Higher-order singular value decomposition
6896:{\displaystyle X\in \mathrm {GL} _{r}(F)}
4757:
3132:
2895:
2408:
2393:
2378:
2316:
2301:
2286:
2257:
2242:
2227:
1865:
1598:
1221:higher-order singular value decomposition
1085:is minimal in the above expression, then
1005:
959:
647:
379:
83:higher-order singular value decomposition
12673:Covariance and contravariance of vectors
12187:The Journal of Machine Learning Research
11197:
11056:
10873:
10818:
10816:
8884:In summary, the generic tensor of order
5003:coincide, however, for tensors of order
3608:, with the following exceptional cases:
3505:states that equality is expected, i.e.,
3204:is some indeterminate closed set in the
11130:
10869:
10867:
10769:
6220:'s are of rank 1. An identifiable rank-
4789:is two, so that the above bound yields
14:
12952:
11957:
11517:Annali di Matematica Pura ed Applicata
11425:
10991:
10227:alternating slice-wise diagonalisation
10211:
4677:{\displaystyle r(I_{1},\ldots ,I_{M})}
2199:{\displaystyle r(I_{1},\ldots ,I_{M})}
2097:{\displaystyle r(I_{1},\ldots ,I_{M})}
1969:of the set of tensors of rank at most
1938:{\displaystyle r(I_{1},\ldots ,I_{M})}
1207:, whose rank can be obtained from the
1119:Canonical Polyadic Decomposition (CPD)
55:Canonical polyadic decomposition (CPD)
12297:
11672:
11670:
11457:
11455:
10813:
4088:{\displaystyle r(n,n,n)=r_{E}(n,n,n)}
4010:{\displaystyle F^{4\times 4\times 3}}
227:{\displaystyle 1\leq i_{m}\leq I_{m}}
11400:SIAM Journal on Scientific Computing
10864:
5033:by Bini, Lotti, and Romani in 1980.
2428:will be of complex rank equal to 2.
61:rank-1 tensors for a user-specified
11834:Linear Algebra and Its Applications
11208:Linear Algebra and Its Applications
11109:Linear Algebra and Its Applications
11067:Linear Algebra and Its Applications
10372:-state categorical hidden variable
9935:{\displaystyle 1\leq i\neq j\leq r}
9904:, then there should exist at least
1340:
24:
12536:Tensors in curvilinear coordinates
12279:Parallel Factor Analysis (PARAFAC)
12260:Tensors: Geometry and Applications
12201:
11667:
11452:
10985:Tensors: Geometry and Applications
10909:
10794:Journal of Mathematics and Physics
10183:
10154:
10049:
9891:
9863:
9847:
9708:
9662:
9499:
9326:
9108:
9008:
8929:
8925:
8736:
8614:
8610:
7683:
7671:in the Zariski topology such that
7622:
7288:are generically not identifiable.
7206:
7203:
7112:{\displaystyle AX^{-1}=_{i=1}^{r}}
6874:
6871:
6563:
6392:
6376:{\displaystyle {\mathcal {A}}_{i}}
6362:
6337:
6281:
6249:
6213:{\displaystyle {\mathcal {A}}_{i}}
6199:
6165:
6142:
6125:
6007:
5963:
5901:. For this reason, the parameters
5643:
5629:
5604:
5225:
5048:
4984:
4936:
4882:
3058:
3014:
2972:
2968:
1634:
1358:
771:
636:
543:
369:
325:
25:
12976:
12267:
10345:{\displaystyle x_{1},x_{2},x_{3}}
10244:General optimization algorithms:
7701:
7225:
7188:. It can be shown that for every
7181:{\displaystyle BX^{T}=_{i=1}^{r}}
3503:Abo–Ottaviani–Peterson conjecture
3168:
2856:
2361:{\displaystyle 2\times 2\times 2}
1315:
10892:: 84. No. 10,085. Archived from
10886:UCLA Working Papers in Phonetics
10120:
10088:
10062:
10015:
9983:
9957:
9803:
9771:
9745:
9632:
9624:
9616:
9602:
9584:
9570:
9552:
9538:
9520:
9460:
9452:
9430:
9416:
9399:
9391:
9383:
9375:
9367:
9359:
9351:
9343:
9335:
9185:
9159:
9139:
9066:
8138:The space is the defective case
8056:The space is the defective case
7147:
7078:
7006:
6947:
6833:
6818:
6777:
6765:
6657:
6645:
6609:
6594:
5913:
5877:
5852:
5740:
5719:
5704:
5689:
5668:
5653:
5570:
5562:
5554:
5526:
5518:
5510:
5502:
5494:
5486:
5478:
5470:
5462:
5441:
5433:
5425:
5417:
5409:
5401:
5393:
5385:
5377:
5362:
5354:
5346:
5332:
5314:
5300:
5282:
5268:
5250:
5182:
5174:
5152:
5138:
5121:
5113:
5105:
5097:
5089:
5081:
5073:
5065:
5057:
5031:matrix multiplication algorithms
1830:
1812:
1797:
1759:
1738:
1720:
1699:
1681:
1660:
1583:
1568:
1533:
1518:
1503:
1488:
1473:
1458:
1443:
1428:
1413:
1398:
1383:
1368:
1285:
1256:
1235:
983:
905:
878:
854:
833:
812:
299:
277:
12125:
12081:Journal of Symbolic Computation
12072:
12029:
11986:
11951:
11913:
11878:
11825:
11772:
11750:10.1090/s1056-3911-2011-00592-4
11715:
11614:
11561:
11508:
11428:Algebraic Geometry SpringerLink
11419:
11379:
11325:
11229:
11191:
11084:
10289:
9878:(in the Euclidean topology) as
9406:
8758:proved to be two in case 7; and
8636:is an integer, and the rank is
5128:
4269:
3057:
3051:
1882:
11297:-times) are not defective for
11050:
10973:
10921:
10903:
10781:
10763:
10703:
10690:
10683:
10653:
10640:
10626:
10617:
10604:
10590:
10581:
10568:
10554:
10548:
10536:
10503:
10464:
10441:
10402:
10180:
10151:
10046:
9888:
9858:
9659:
9606:
9580:
9574:
9548:
9542:
9516:
9252:{\displaystyle \|\cdot \|_{F}}
9049:. That is, one seeks to solve
9016:{\displaystyle {\mathcal {A}}}
8704:
8659:
8602:
8557:
8040:
8021:
7940:
7921:
7830:
7785:
7222:
7216:
7158:
7142:
7089:
7073:
7017:
7001:
6958:
6942:
6890:
6884:
6730:
6713:
6710:
6691:
6400:{\displaystyle {\mathcal {A}}}
6345:{\displaystyle {\mathcal {A}}}
6328:tensor rank decompositions of
5971:{\displaystyle {\mathcal {A}}}
5601:
5530:
5458:
5336:
5310:
5304:
5278:
5272:
5246:
4992:{\displaystyle {\mathcal {A}}}
4944:{\displaystyle {\mathcal {A}}}
4890:{\displaystyle {\mathcal {A}}}
4848:
4826:
4808:
4671:
4639:
4605:
4573:
4516:
4484:
4330:
4298:
4253:
4229:
4213:
4189:
4082:
4064:
4048:
4030:
3914:
3882:
3866:
3815:
3749:
3737:
3731:
3719:
3659:
3644:
3638:
3623:
3595:
3563:
3547:
3515:
3482:
3450:
3434:
3402:
3319:
3287:
3257:
3225:
3104:
3085:
2956:
2924:
2734:
2702:
2657:
2638:
2193:
2161:
2131:of a sequence of tensors from
2091:
2059:
1932:
1900:
1769:
1742:
1703:
1664:
1652:
1131:
333:{\displaystyle {\mathcal {A}}}
13:
1:
12589:Exterior covariant derivative
12521:Tensor (intrinsic definition)
12258:Landsberg, Joseph M. (2012).
11899:10.1016/s0169-7439(00)00117-9
11728:Journal of Algebraic Geometry
11318:10.1090/s1056-3911-10-00537-0
11305:Journal of Algebraic Geometry
11177:10.1090/s0002-9947-08-04725-9
11123:10.1016/s0024-3795(02)00352-x
10756:
10731:Multilinear subspace learning
8755:expected to be six in case 6;
7953:, and the rank is too large:
7634:{\displaystyle \Pi <15000}
5624:
1965:such that the closure in the
1945:is defined as the least rank
1058:{\displaystyle 1\leq m\leq M}
622:
259:{\displaystyle 1\leq m\leq M}
181:{\displaystyle 1\leq i\leq I}
12614:Raising and lowering indices
12156:10.1016/j.difgeo.2017.07.009
11286:{\displaystyle \mathbb {P} }
11264:{\displaystyle \mathbb {P} }
11222:10.1016/0024-3795(85)90070-9
11077:10.1016/0024-3795(83)80041-x
10736:Singular value decomposition
10267:nonlinear conjugate gradient
10249:simultaneous diagonalization
10189:{\displaystyle n\to \infty }
9897:{\displaystyle n\to \infty }
9668:{\displaystyle n\to \infty }
8540:The space is perfect, i.e.,
7615:for all spaces of dimension
7040:{\displaystyle B=_{i=1}^{r}}
6981:{\displaystyle A=_{i=1}^{r}}
5780:{\displaystyle \lambda _{k}}
5610:{\displaystyle m\to \infty }
3332:is often referred to as the
1872:{\displaystyle \mathbb {C} }
306:{\displaystyle \mathbf {A} }
284:{\displaystyle \mathbf {a} }
7:
12852:Gluon field strength tensor
12323:
10719:
8752:proved to be two in case 5;
7423:{\displaystyle I_{m}\geq 2}
5760:if and only if there exist
1111:(tensor) rank decomposition
1065:. When the number of terms
739:as a linear combination of
108:
10:
12981:
12663:Cartan formalism (physics)
12483:Penrose graphical notation
7644:There exists a closed set
2277:while the generic rank of
1215:normal form of the linear
12869:
12809:
12758:
12751:
12643:
12574:
12511:
12455:
12402:
12349:
12342:
12335:Glossary of tensor theory
12331:
12103:10.1016/j.jsc.2012.05.012
11856:10.1016/j.laa.2016.10.019
11539:10.1007/s10231-013-0352-8
11436:10.1007/978-1-4757-2189-8
11365:10.1007/s00208-014-1150-3
10297:conditionally independent
10221:alternating least squares
6922:{\displaystyle r\times r}
33:tensor rank decomposition
12919:Gregorio Ricci-Curbastro
12791:Riemann curvature tensor
12498:Van der Waerden notation
11701:10.1515/crelle-2016-0067
11203:"Typical tensorial rank"
10770:Papalexakis, Evangelos.
10669:In applications such as
10216:Alternating algorithms:
7756:generically identifiable
4266:for all binary tensors.
2684:Unbalanced tensor spaces
1127:Polyadic decomposition'.
1115:minimal CP decomposition
12889:Elwin Bruno Christoffel
12822:Angular momentum tensor
12493:Tetrad (index notation)
12463:Abstract index notation
11995:Journal of Chemometrics
10240:moment-based algorithms
10237:pencil-based algorithms
8742:{\displaystyle \infty }
8223:{\displaystyle n\geq 2}
7762:The rank is too large:
7613:computer-assisted proof
6437:Generic identifiability
4114:{\displaystyle n\neq 3}
73:. It was introduced by
12703:Levi-Civita connection
12007:10.1002/cem.1180040105
11287:
11265:
10710:
10660:
10529:
10448:
10386:
10366:
10346:
10190:
10161:
9936:
9898:
9872:
9837:has the property that
9828:
9742:
9689:
9669:
9640:
9477:
9308:
9288:
9253:
9217:
9136:
9043:
9042:{\displaystyle r<s}
9017:
8993:
8973:
8945:
8904:
8903:{\displaystyle M>2}
8874:
8808:
8743:
8711:
8630:
8531:
8505:
8423:
8397:
8341:
8315:
8259:
8258:{\displaystyle r=2n-1}
8224:
8198:
8129:
8103:
8047:
8020:
7986:
7947:
7920:
7886:
7837:
7748:
7718:
7665:
7635:
7605:
7585:
7489:
7424:
7391:
7390:{\displaystyle M>2}
7365:
7282:
7281:{\displaystyle r>1}
7255:
7235:
7182:
7113:
7041:
6982:
6923:
6897:
6850:
6815:
6762:
6642:
6591:
6548:
6547:{\displaystyle r>1}
6522:
6427:
6401:
6377:
6346:
6322:
6299:
6277:
6234:
6214:
6183:
6106:
6082:
5992:
5972:
5948:
5891:
5837:
5781:
5754:
5611:
5582:
5199:
5020:
5019:{\displaystyle \geq 3}
4993:
4965:
4945:
4921:
4920:{\displaystyle r<s}
4891:
4867:
4839:
4783:
4742:
4678:
4620:
4550:
4456:
4397:
4337:
4260:
4173:
4135:
4115:
4089:
4011:
3972:
3927:
3796:
3700:
3602:
3492:
3383:
3326:
3264:
3198:
3178:
3114:
3084:
3040:
2992:
2896:Balanced tensor spaces
2886:
2866:
2801:
2781:
2667:
2637:
2603:
2544:
2485:
2422:
2362:
2330:
2271:
2200:
2145:
2121:
2098:
2043:
1983:
1959:
1939:
1873:
1844:
1779:
1612:
1550:
1331:
1276:is one, provided that
1270:
1201:
1099:
1079:
1059:
1027:
968:
928:
799:
753:
733:
701:
613:
527:
445:
354:
334:
307:
285:
260:
228:
182:
147:
127:
75:Frank Lauren Hitchcock
47:rank-1 tensors, where
12929:Jan Arnoldus Schouten
12884:Augustin-Louis Cauchy
12364:Differential geometry
11342:Mathematische Annalen
11288:
11266:
11249:"Secant varieties of
10807:10.1002/sapm192761164
10726:Latent class analysis
10711:
10661:
10509:
10449:
10387:
10367:
10347:
10284:homotopy continuation
10191:
10162:
9937:
9899:
9873:
9829:
9722:
9690:
9670:
9641:
9478:
9309:
9289:
9254:
9218:
9116:
9044:
9018:
8994:
8974:
8946:
8905:
8875:
8809:
8749:in the first 4 cases;
8744:
8712:
8631:
8532:
8506:
8424:
8398:
8342:
8316:
8260:
8225:
8199:
8130:
8104:
8048:
8000:
7966:
7948:
7900:
7866:
7838:
7749:
7747:{\displaystyle S_{r}}
7719:
7666:
7664:{\displaystyle Z_{r}}
7636:
7606:
7586:
7490:
7425:
7392:
7366:
7283:
7256:
7236:
7183:
7114:
7042:
6983:
6924:
6898:
6851:
6795:
6742:
6622:
6571:
6549:
6523:
6428:
6402:
6378:
6347:
6323:
6300:
6257:
6235:
6215:
6184:
6107:
6083:
5993:
5973:
5949:
5892:
5838:
5782:
5755:
5612:
5583:
5200:
5021:
4994:
4966:
4946:
4922:
4892:
4868:
4840:
4784:
4743:
4679:
4621:
4530:
4457:
4398:
4338:
4261:
4174:
4136:
4116:
4090:
4012:
3973:
3928:
3797:
3701:
3603:
3493:
3384:
3334:expected generic rank
3327:
3265:
3199:
3179:
3115:
3064:
3020:
2993:
2887:
2867:
2802:
2761:
2668:
2617:
2583:
2545:
2486:
2423:
2363:
2331:
2272:
2201:
2146:
2122:
2099:
2044:
1984:
1960:
1940:
1874:
1845:
1780:
1613:
1551:
1332:
1271:
1202:
1100:
1080:
1060:
1028:
969:
929:
779:
754:
734:
702:
614:
528:
446:
355:
335:
308:
286:
261:
229:
183:
148:
128:
12904:Carl Friedrich Gauss
12837:stress–energy tensor
12832:Cauchy stress tensor
12584:Covariant derivative
12546:Antisymmetric tensor
12478:Multi-index notation
11960:Analytical Chemistry
11679:J. Reine Angew. Math
11426:Harris, Joe (1992).
11275:
11253:
10899:on October 10, 2004.
10875:Harshman, Richard A.
10751:Tensor decomposition
10741:Tucker decomposition
10677:
10458:
10396:
10376:
10356:
10303:
10198:diverging components
10174:
9949:
9908:
9882:
9841:
9702:
9679:
9653:
9493:
9321:
9314:. The rank-3 tensor
9298:
9278:
9230:
9056:
9027:
9003:
8983:
8963:
8914:
8888:
8818:
8765:
8733:
8640:
8544:
8515:
8436:
8407:
8354:
8325:
8272:
8234:
8208:
8142:
8113:
8060:
7957:
7850:
7766:
7731:
7678:
7648:
7619:
7595:
7499:
7434:
7401:
7375:
7295:
7266:
7245:
7192:
7123:
7051:
6992:
6933:
6907:
6860:
6558:
6532:
6445:
6411:
6387:
6356:
6332:
6309:
6244:
6224:
6193:
6116:
6096:
6002:
5982:
5958:
5905:
5847:
5791:
5764:
5638:
5595:
5215:
5043:
5007:
4979:
4955:
4931:
4905:
4877:
4857:
4793:
4752:
4692:
4633:
4469:
4407:
4347:
4283:
4183:
4145:
4125:
4099:
4024:
3982:
3937:
3809:
3711:
3615:
3509:
3396:
3340:
3336:of the tensor space
3274:
3212:
3188:
3127:
3011:
2911:
2876:
2817:
2696:
2561:
2495:
2435:
2373:
2340:
2281:
2222:
2155:
2135:
2111:
2053:
1993:
1989:is the entire space
1973:
1949:
1894:
1861:
1792:
1629:
1563:
1353:
1280:
1230:
1144:
1089:
1069:
1037:
978:
941:
766:
743:
723:
631:
537:
455:
364:
344:
320:
295:
273:
238:
192:
160:
137:
117:
12960:Multilinear algebra
12781:Nonmetricity tensor
12636:(2nd-order tensors)
12604:Hodge star operator
12594:Exterior derivative
12443:Transport phenomena
12428:Continuum mechanics
12384:Multilinear algebra
12225:2009SIAMR..51..455K
11972:10.1021/ac00289a052
10273:limited memory BFGS
10261:Levenberg–Marquardt
10233:Direct algorithms:
10212:Calculating the CPD
10140:
10108:
10082:
10035:
10003:
9977:
9823:
9791:
9765:
9199:
9173:
9153:
9080:
8725:decompositions is
8530:{\displaystyle r=5}
8422:{\displaystyle r=8}
8340:{\displaystyle r=6}
8128:{\displaystyle r=5}
7177:
7108:
7036:
6977:
6791:
6671:
6441:Order-2 tensors in
6426:{\displaystyle r-1}
5954:of a rank-1 tensor
5943:
340:. An element of an
29:multilinear algebra
12914:Tullio Levi-Civita
12857:Metric tensor (GR)
12771:Levi-Civita symbol
12624:Tensor contraction
12438:General relativity
12374:Euclidean geometry
12050:10.1007/bf02293598
11592:10.1137/16m1090132
11283:
11261:
11070:. 52/53: 645–685.
10937:Journal of the ACM
10850:10.1007/BF02310791
10706:
10656:
10444:
10382:
10362:
10342:
10186:
10157:
10118:
10086:
10060:
10013:
9981:
9955:
9932:
9894:
9868:
9824:
9801:
9769:
9743:
9685:
9665:
9636:
9473:
9304:
9284:
9249:
9213:
9183:
9157:
9137:
9102:
9064:
9039:
9013:
8989:
8969:
8941:
8900:
8870:
8804:
8739:
8707:
8626:
8527:
8501:
8419:
8393:
8337:
8311:
8255:
8230:, and the rank is
8220:
8194:
8125:
8099:
8043:
7943:
7833:
7744:
7714:
7661:
7631:
7601:
7581:
7485:
7420:
7387:
7361:
7278:
7251:
7231:
7178:
7157:
7109:
7088:
7037:
7016:
6978:
6957:
6919:
6893:
6846:
6775:
6655:
6544:
6518:
6423:
6397:
6373:
6342:
6321:{\displaystyle r!}
6318:
6295:
6230:
6210:
6179:
6102:
6078:
5988:
5968:
5944:
5923:
5887:
5833:
5777:
5750:
5607:
5578:
5576:
5195:
5016:
4989:
4961:
4941:
4917:
4887:
4863:
4835:
4805:
4779:
4738:
4674:
4616:
4481:
4452:
4393:
4333:
4295:
4256:
4169:
4141:tensors of format
4131:
4111:
4085:
4007:
3968:
3923:
3792:
3696:
3598:
3488:
3379:
3322:
3260:
3194:
3174:
3110:
2988:
2882:
2862:
2797:
2663:
2540:
2481:
2418:
2358:
2326:
2267:
2196:
2141:
2129:Euclidean topology
2117:
2094:
2039:
1979:
1955:
1935:
1869:
1840:
1775:
1608:
1546:
1327:
1266:
1197:
1095:
1075:
1055:
1023:
964:
924:
749:
729:
697:
609:
523:
441:
350:
330:
303:
281:
256:
224:
178:
143:
123:
12947:
12946:
12909:Hermann Grassmann
12865:
12864:
12817:Moment of inertia
12678:Differential form
12653:Affine connection
12468:Einstein notation
12451:
12450:
12379:Exterior calculus
12359:Coordinate system
12243:10.1137/07070111X
11966:(12): 2395–2397.
11803:10.1137/130916084
11645:10.1137/110829180
11486:10.1137/140961389
11445:978-1-4419-3099-6
11237:Catalisano, M. V.
11092:Catalisano, M. V.
11036:10.1137/06066518x
10709:{\displaystyle E}
10447:{\displaystyle E}
10385:{\displaystyle h}
10365:{\displaystyle k}
10055:
9688:{\displaystyle r}
9599:
9567:
9535:
9446:
9410:
9307:{\displaystyle r}
9287:{\displaystyle r}
9059:
8992:{\displaystyle s}
8972:{\displaystyle r}
8939:
8624:
7604:{\displaystyle r}
7254:{\displaystyle Z}
6903:is an invertible
6407:would be at most
6233:{\displaystyle r}
6112:distinct tensors
6105:{\displaystyle r}
5991:{\displaystyle r}
5551:
5456:
5329:
5297:
5265:
5168:
5132:
4964:{\displaystyle r}
4866:{\displaystyle s}
4804:
4480:
4294:
4134:{\displaystyle s}
3769:
3673:
3197:{\displaystyle Z}
3055:
3002:almost everywhere
2982:
2885:{\displaystyle Z}
2811:almost everywhere
2676:and it is called
2144:{\displaystyle S}
2120:{\displaystyle S}
1982:{\displaystyle r}
1958:{\displaystyle r}
1745:
1706:
1667:
1650:
1620:complex conjugate
1123:CANDECOMP/PARAFAC
1098:{\displaystyle R}
1078:{\displaystyle R}
752:{\displaystyle r}
732:{\displaystyle R}
711:-way array where
353:{\displaystyle M}
146:{\displaystyle A}
126:{\displaystyle a}
99:computer graphics
91:signal processing
16:(Redirected from
12972:
12924:Bernhard Riemann
12756:
12755:
12599:Exterior product
12566:Two-point tensor
12551:Symmetric tensor
12433:Electromagnetism
12347:
12346:
12318:
12311:
12304:
12295:
12294:
12274:PARAFAC Tutorial
12263:
12254:
12236:
12209:Kolda, Tamara G.
12195:
12194:
12182:
12176:
12175:
12149:
12129:
12123:
12122:
12096:
12076:
12070:
12069:
12033:
12027:
12026:
11990:
11984:
11983:
11955:
11949:
11948:
11931:(4): 1064–1083.
11917:
11911:
11910:
11882:
11876:
11875:
11849:
11829:
11823:
11822:
11796:
11776:
11770:
11769:
11743:
11719:
11713:
11712:
11694:
11674:
11665:
11664:
11638:
11629:(3): 1018–1037.
11618:
11612:
11611:
11585:
11565:
11559:
11558:
11532:
11523:(6): 1691–1702.
11512:
11506:
11505:
11479:
11470:(4): 1265–1287.
11459:
11450:
11449:
11423:
11417:
11416:
11383:
11377:
11376:
11358:
11329:
11323:
11322:
11320:
11292:
11290:
11289:
11284:
11282:
11270:
11268:
11267:
11262:
11260:
11233:
11227:
11226:
11224:
11199:Lickteig, Thomas
11195:
11189:
11188:
11170:
11137:
11128:
11127:
11125:
11116:(1–3): 263–285.
11088:
11082:
11081:
11079:
11054:
11048:
11047:
11029:
11020:(3): 1084–1127.
11000:
10989:
10988:
10981:Landsberg, J. M.
10977:
10971:
10970:
10952:
10925:
10919:
10918:
10916:
10907:
10901:
10900:
10898:
10883:
10871:
10862:
10861:
10820:
10811:
10810:
10801:(1–4): 164–189.
10785:
10779:
10778:
10776:
10767:
10715:
10713:
10712:
10707:
10693:
10665:
10663:
10662:
10657:
10643:
10638:
10637:
10607:
10602:
10601:
10571:
10566:
10565:
10528:
10523:
10502:
10501:
10489:
10488:
10476:
10475:
10453:
10451:
10450:
10445:
10440:
10439:
10427:
10426:
10414:
10413:
10391:
10389:
10388:
10383:
10371:
10369:
10368:
10363:
10351:
10349:
10348:
10343:
10341:
10340:
10328:
10327:
10315:
10314:
10195:
10193:
10192:
10187:
10166:
10164:
10163:
10158:
10150:
10149:
10139:
10134:
10123:
10107:
10102:
10091:
10081:
10076:
10065:
10056:
10053:
10045:
10044:
10034:
10029:
10018:
10002:
9997:
9986:
9976:
9971:
9960:
9941:
9939:
9938:
9933:
9903:
9901:
9900:
9895:
9877:
9875:
9874:
9869:
9867:
9866:
9857:
9856:
9851:
9850:
9833:
9831:
9830:
9825:
9822:
9817:
9806:
9790:
9785:
9774:
9764:
9759:
9748:
9741:
9736:
9718:
9717:
9712:
9711:
9694:
9692:
9691:
9686:
9674:
9672:
9671:
9666:
9645:
9643:
9642:
9637:
9635:
9627:
9619:
9605:
9600:
9592:
9587:
9573:
9568:
9560:
9555:
9541:
9536:
9528:
9523:
9509:
9508:
9503:
9502:
9482:
9480:
9479:
9474:
9463:
9455:
9447:
9444:
9433:
9419:
9411:
9408:
9402:
9394:
9386:
9378:
9370:
9362:
9354:
9346:
9338:
9330:
9329:
9313:
9311:
9310:
9305:
9293:
9291:
9290:
9285:
9258:
9256:
9255:
9250:
9248:
9247:
9222:
9220:
9219:
9214:
9209:
9208:
9198:
9193:
9188:
9172:
9167:
9162:
9152:
9147:
9142:
9135:
9130:
9112:
9111:
9101:
9100:
9099:
9098:
9097:
9079:
9074:
9069:
9048:
9046:
9045:
9040:
9022:
9020:
9019:
9014:
9012:
9011:
8998:
8996:
8995:
8990:
8978:
8976:
8975:
8970:
8950:
8948:
8947:
8942:
8940:
8938:
8924:
8909:
8907:
8906:
8901:
8879:
8877:
8876:
8871:
8869:
8868:
8856:
8855:
8843:
8842:
8830:
8829:
8813:
8811:
8810:
8805:
8803:
8802:
8790:
8789:
8777:
8776:
8748:
8746:
8745:
8740:
8716:
8714:
8713:
8708:
8703:
8702:
8684:
8683:
8671:
8670:
8658:
8657:
8635:
8633:
8632:
8627:
8625:
8623:
8609:
8601:
8600:
8582:
8581:
8569:
8568:
8556:
8555:
8536:
8534:
8533:
8528:
8511:and the rank is
8510:
8508:
8507:
8502:
8500:
8499:
8487:
8486:
8474:
8473:
8461:
8460:
8448:
8447:
8428:
8426:
8425:
8420:
8403:and the rank is
8402:
8400:
8399:
8394:
8392:
8391:
8379:
8378:
8366:
8365:
8346:
8344:
8343:
8338:
8321:and the rank is
8320:
8318:
8317:
8312:
8310:
8309:
8297:
8296:
8284:
8283:
8264:
8262:
8261:
8256:
8229:
8227:
8226:
8221:
8203:
8201:
8200:
8195:
8193:
8192:
8180:
8179:
8167:
8166:
8154:
8153:
8134:
8132:
8131:
8126:
8109:and the rank is
8108:
8106:
8105:
8100:
8098:
8097:
8085:
8084:
8072:
8071:
8052:
8050:
8049:
8044:
8033:
8032:
8019:
8014:
7996:
7995:
7985:
7980:
7952:
7950:
7949:
7944:
7933:
7932:
7919:
7914:
7896:
7895:
7885:
7880:
7862:
7861:
7842:
7840:
7839:
7834:
7829:
7828:
7810:
7809:
7797:
7796:
7784:
7783:
7753:
7751:
7750:
7745:
7743:
7742:
7723:
7721:
7720:
7715:
7713:
7712:
7700:
7699:
7687:
7686:
7670:
7668:
7667:
7662:
7660:
7659:
7640:
7638:
7637:
7632:
7610:
7608:
7607:
7602:
7590:
7588:
7587:
7582:
7580:
7579:
7578:
7577:
7554:
7553:
7552:
7551:
7531:
7530:
7529:
7528:
7511:
7510:
7494:
7492:
7491:
7486:
7478:
7477:
7459:
7458:
7446:
7445:
7429:
7427:
7426:
7421:
7413:
7412:
7396:
7394:
7393:
7388:
7370:
7368:
7367:
7362:
7360:
7359:
7358:
7357:
7334:
7333:
7332:
7331:
7314:
7313:
7312:
7311:
7287:
7285:
7284:
7279:
7260:
7258:
7257:
7252:
7240:
7238:
7237:
7232:
7215:
7214:
7209:
7187:
7185:
7184:
7179:
7176:
7171:
7156:
7155:
7150:
7138:
7137:
7118:
7116:
7115:
7110:
7107:
7102:
7087:
7086:
7081:
7069:
7068:
7046:
7044:
7043:
7038:
7035:
7030:
7015:
7014:
7009:
6987:
6985:
6984:
6979:
6976:
6971:
6956:
6955:
6950:
6928:
6926:
6925:
6920:
6902:
6900:
6899:
6894:
6883:
6882:
6877:
6855:
6853:
6852:
6847:
6842:
6841:
6836:
6827:
6826:
6821:
6814:
6809:
6790:
6785:
6780:
6774:
6773:
6768:
6761:
6756:
6738:
6737:
6728:
6727:
6709:
6708:
6687:
6686:
6670:
6665:
6660:
6654:
6653:
6648:
6641:
6636:
6618:
6617:
6612:
6603:
6602:
6597:
6590:
6585:
6567:
6566:
6553:
6551:
6550:
6545:
6527:
6525:
6524:
6519:
6517:
6516:
6515:
6514:
6502:
6501:
6484:
6483:
6482:
6481:
6464:
6463:
6462:
6461:
6432:
6430:
6429:
6424:
6406:
6404:
6403:
6398:
6396:
6395:
6382:
6380:
6379:
6374:
6372:
6371:
6366:
6365:
6351:
6349:
6348:
6343:
6341:
6340:
6327:
6325:
6324:
6319:
6304:
6302:
6301:
6296:
6291:
6290:
6285:
6284:
6276:
6271:
6253:
6252:
6239:
6237:
6236:
6231:
6219:
6217:
6216:
6211:
6209:
6208:
6203:
6202:
6188:
6186:
6185:
6180:
6175:
6174:
6169:
6168:
6152:
6151:
6146:
6145:
6135:
6134:
6129:
6128:
6111:
6109:
6108:
6103:
6087:
6085:
6084:
6079:
6077:
6076:
6075:
6074:
6051:
6050:
6049:
6048:
6031:
6030:
6029:
6028:
6011:
6010:
5997:
5995:
5994:
5989:
5977:
5975:
5974:
5969:
5967:
5966:
5953:
5951:
5950:
5945:
5942:
5937:
5922:
5921:
5916:
5896:
5894:
5893:
5888:
5886:
5885:
5880:
5874:
5873:
5861:
5860:
5855:
5842:
5840:
5839:
5834:
5826:
5825:
5813:
5812:
5803:
5802:
5786:
5784:
5783:
5778:
5776:
5775:
5759:
5757:
5756:
5751:
5749:
5748:
5743:
5728:
5727:
5722:
5713:
5712:
5707:
5698:
5697:
5692:
5677:
5676:
5671:
5662:
5661:
5656:
5647:
5646:
5616:
5614:
5613:
5608:
5587:
5585:
5584:
5579:
5577:
5573:
5565:
5557:
5552:
5550:
5549:
5537:
5529:
5521:
5513:
5505:
5497:
5489:
5481:
5473:
5465:
5457:
5449:
5444:
5436:
5428:
5420:
5412:
5404:
5396:
5388:
5380:
5369:
5365:
5357:
5349:
5335:
5330:
5322:
5317:
5303:
5298:
5290:
5285:
5271:
5266:
5258:
5253:
5235:
5234:
5229:
5228:
5204:
5202:
5201:
5196:
5185:
5177:
5169:
5166:
5155:
5141:
5133:
5130:
5124:
5116:
5108:
5100:
5092:
5084:
5076:
5068:
5060:
5052:
5051:
5025:
5023:
5022:
5017:
4998:
4996:
4995:
4990:
4988:
4987:
4970:
4968:
4967:
4962:
4950:
4948:
4947:
4942:
4940:
4939:
4926:
4924:
4923:
4918:
4896:
4894:
4893:
4888:
4886:
4885:
4872:
4870:
4869:
4864:
4844:
4842:
4841:
4836:
4807:
4806:
4802:
4788:
4786:
4785:
4780:
4778:
4777:
4760:
4747:
4745:
4744:
4739:
4737:
4736:
4735:
4734:
4711:
4710:
4709:
4708:
4683:
4681:
4680:
4675:
4670:
4669:
4651:
4650:
4625:
4623:
4622:
4617:
4612:
4608:
4604:
4603:
4585:
4584:
4560:
4559:
4549:
4544:
4515:
4514:
4496:
4495:
4483:
4482:
4478:
4461:
4459:
4458:
4453:
4451:
4450:
4432:
4431:
4419:
4418:
4402:
4400:
4399:
4394:
4392:
4391:
4390:
4389:
4366:
4365:
4364:
4363:
4342:
4340:
4339:
4334:
4329:
4328:
4310:
4309:
4297:
4296:
4292:
4265:
4263:
4262:
4257:
4228:
4227:
4178:
4176:
4175:
4170:
4140:
4138:
4137:
4132:
4120:
4118:
4117:
4112:
4095:, provided that
4094:
4092:
4091:
4086:
4063:
4062:
4016:
4014:
4013:
4008:
4006:
4005:
3977:
3975:
3974:
3969:
3967:
3966:
3932:
3930:
3929:
3924:
3913:
3912:
3894:
3893:
3881:
3880:
3865:
3864:
3846:
3845:
3827:
3826:
3801:
3799:
3798:
3793:
3770:
3768: with
3767:
3765:
3764:
3705:
3703:
3702:
3697:
3674:
3672: with
3671:
3669:
3668:
3607:
3605:
3604:
3599:
3594:
3593:
3575:
3574:
3562:
3561:
3546:
3545:
3527:
3526:
3497:
3495:
3494:
3489:
3481:
3480:
3462:
3461:
3449:
3448:
3433:
3432:
3414:
3413:
3388:
3386:
3385:
3380:
3378:
3377:
3376:
3375:
3357:
3356:
3331:
3329:
3328:
3323:
3318:
3317:
3299:
3298:
3286:
3285:
3269:
3267:
3266:
3261:
3256:
3255:
3237:
3236:
3224:
3223:
3206:Zariski topology
3203:
3201:
3200:
3195:
3183:
3181:
3180:
3175:
3167:
3166:
3165:
3164:
3146:
3145:
3135:
3119:
3117:
3116:
3111:
3097:
3096:
3083:
3078:
3056:
3053:
3050:
3049:
3039:
3034:
2997:
2995:
2994:
2989:
2987:
2983:
2981:
2967:
2955:
2954:
2936:
2935:
2923:
2922:
2891:
2889:
2888:
2883:
2871:
2869:
2868:
2863:
2855:
2854:
2853:
2852:
2834:
2833:
2806:
2804:
2803:
2798:
2796:
2792:
2791:
2790:
2780:
2775:
2757:
2756:
2733:
2732:
2714:
2713:
2672:
2670:
2669:
2664:
2650:
2649:
2636:
2631:
2613:
2612:
2602:
2597:
2573:
2572:
2549:
2547:
2546:
2541:
2539:
2538:
2520:
2519:
2507:
2506:
2490:
2488:
2487:
2482:
2480:
2479:
2478:
2477:
2454:
2453:
2452:
2451:
2427:
2425:
2424:
2419:
2417:
2416:
2411:
2402:
2401:
2396:
2387:
2386:
2381:
2367:
2365:
2364:
2359:
2335:
2333:
2332:
2327:
2325:
2324:
2319:
2310:
2309:
2304:
2295:
2294:
2289:
2276:
2274:
2273:
2268:
2266:
2265:
2260:
2251:
2250:
2245:
2236:
2235:
2230:
2205:
2203:
2202:
2197:
2192:
2191:
2173:
2172:
2150:
2148:
2147:
2142:
2126:
2124:
2123:
2118:
2103:
2101:
2100:
2095:
2090:
2089:
2071:
2070:
2048:
2046:
2045:
2040:
2038:
2037:
2036:
2035:
2012:
2011:
2010:
2009:
1988:
1986:
1985:
1980:
1967:Zariski topology
1964:
1962:
1961:
1956:
1944:
1942:
1941:
1936:
1931:
1930:
1912:
1911:
1878:
1876:
1875:
1870:
1868:
1849:
1847:
1846:
1841:
1839:
1838:
1833:
1821:
1820:
1815:
1806:
1805:
1800:
1784:
1782:
1781:
1776:
1768:
1767:
1762:
1753:
1752:
1747:
1746:
1741:
1736:
1729:
1728:
1723:
1714:
1713:
1708:
1707:
1702:
1697:
1690:
1689:
1684:
1675:
1674:
1669:
1668:
1663:
1658:
1651:
1643:
1638:
1637:
1617:
1615:
1614:
1609:
1607:
1606:
1601:
1592:
1591:
1586:
1577:
1576:
1571:
1555:
1553:
1552:
1547:
1542:
1541:
1536:
1527:
1526:
1521:
1512:
1511:
1506:
1497:
1496:
1491:
1482:
1481:
1476:
1467:
1466:
1461:
1452:
1451:
1446:
1437:
1436:
1431:
1422:
1421:
1416:
1407:
1406:
1401:
1392:
1391:
1386:
1377:
1376:
1371:
1362:
1361:
1341:Field dependence
1336:
1334:
1333:
1328:
1314:
1313:
1312:
1311:
1294:
1293:
1288:
1275:
1273:
1272:
1267:
1265:
1264:
1259:
1244:
1243:
1238:
1206:
1204:
1203:
1198:
1196:
1195:
1183:
1182:
1181:
1180:
1163:
1162:
1161:
1160:
1104:
1102:
1101:
1096:
1084:
1082:
1081:
1076:
1064:
1062:
1061:
1056:
1032:
1030:
1029:
1024:
1022:
1021:
1020:
1019:
1009:
1008:
998:
997:
986:
973:
971:
970:
965:
963:
962:
953:
952:
933:
931:
930:
925:
920:
919:
908:
893:
892:
881:
869:
868:
857:
848:
847:
836:
827:
826:
815:
809:
808:
798:
793:
775:
774:
759:rank-1 tensors:
758:
756:
755:
750:
738:
736:
735:
730:
718:
714:
710:
706:
704:
703:
698:
696:
695:
694:
693:
675:
674:
662:
661:
651:
650:
640:
639:
618:
616:
615:
610:
608:
607:
606:
605:
590:
589:
571:
570:
558:
557:
547:
546:
532:
530:
529:
524:
522:
521:
520:
519:
504:
503:
485:
484:
472:
471:
450:
448:
447:
442:
440:
439:
438:
437:
422:
421:
406:
405:
393:
392:
382:
373:
372:
359:
357:
356:
351:
339:
337:
336:
331:
329:
328:
312:
310:
309:
304:
302:
290:
288:
287:
282:
280:
265:
263:
262:
257:
233:
231:
230:
225:
223:
222:
210:
209:
187:
185:
184:
179:
152:
150:
149:
144:
132:
130:
129:
124:
21:
18:CP decomposition
12980:
12979:
12975:
12974:
12973:
12971:
12970:
12969:
12950:
12949:
12948:
12943:
12894:Albert Einstein
12861:
12842:Einstein tensor
12805:
12786:Ricci curvature
12766:Kronecker delta
12752:Notable tensors
12747:
12668:Connection form
12645:
12639:
12570:
12556:Tensor operator
12513:
12507:
12447:
12423:Computer vision
12416:
12398:
12394:Tensor calculus
12338:
12327:
12322:
12270:
12234:10.1.1.153.2059
12204:
12202:Further reading
12199:
12198:
12193:(1): 2773–2832.
12183:
12179:
12130:
12126:
12077:
12073:
12034:
12030:
11991:
11987:
11956:
11952:
11937:10.1137/0614071
11921:Leurgans, S. E.
11918:
11914:
11883:
11879:
11830:
11826:
11777:
11773:
11720:
11716:
11675:
11668:
11619:
11615:
11566:
11562:
11513:
11509:
11460:
11453:
11446:
11424:
11420:
11413:10.1137/0209053
11384:
11380:
11330:
11326:
11278:
11276:
11273:
11272:
11256:
11254:
11251:
11250:
11241:Geramita, A. V.
11234:
11230:
11196:
11192:
11138:
11131:
11096:Geramita, A. V.
11089:
11085:
11055:
11051:
11001:
10992:
10978:
10974:
10959:10.1145/2512329
10926:
10922:
10914:
10908:
10904:
10896:
10881:
10872:
10865:
10821:
10814:
10789:F. L. Hitchcock
10786:
10782:
10774:
10768:
10764:
10759:
10722:
10689:
10678:
10675:
10674:
10639:
10633:
10629:
10603:
10597:
10593:
10567:
10561:
10557:
10524:
10513:
10497:
10493:
10484:
10480:
10471:
10467:
10459:
10456:
10455:
10435:
10431:
10422:
10418:
10409:
10405:
10397:
10394:
10393:
10377:
10374:
10373:
10357:
10354:
10353:
10336:
10332:
10323:
10319:
10310:
10306:
10304:
10301:
10300:
10292:
10214:
10175:
10172:
10171:
10145:
10141:
10135:
10124:
10119:
10103:
10092:
10087:
10077:
10066:
10061:
10054: and
10052:
10040:
10036:
10030:
10019:
10014:
9998:
9987:
9982:
9972:
9961:
9956:
9950:
9947:
9946:
9909:
9906:
9905:
9883:
9880:
9879:
9862:
9861:
9852:
9846:
9845:
9844:
9842:
9839:
9838:
9818:
9807:
9802:
9786:
9775:
9770:
9760:
9749:
9744:
9737:
9726:
9713:
9707:
9706:
9705:
9703:
9700:
9699:
9680:
9677:
9676:
9654:
9651:
9650:
9631:
9623:
9615:
9601:
9591:
9583:
9569:
9559:
9551:
9537:
9527:
9519:
9504:
9498:
9497:
9496:
9494:
9491:
9490:
9459:
9451:
9445: and
9443:
9429:
9415:
9407:
9398:
9390:
9382:
9374:
9366:
9358:
9350:
9342:
9334:
9325:
9324:
9322:
9319:
9318:
9299:
9296:
9295:
9279:
9276:
9275:
9243:
9239:
9231:
9228:
9227:
9204:
9200:
9194:
9189:
9184:
9168:
9163:
9158:
9148:
9143:
9138:
9131:
9120:
9107:
9106:
9093:
9089:
9088:
9084:
9075:
9070:
9065:
9063:
9057:
9054:
9053:
9028:
9025:
9024:
9007:
9006:
9004:
9001:
9000:
8984:
8981:
8980:
8964:
8961:
8960:
8957:
8928:
8923:
8915:
8912:
8911:
8889:
8886:
8885:
8864:
8860:
8851:
8847:
8838:
8834:
8825:
8821:
8819:
8816:
8815:
8798:
8794:
8785:
8781:
8772:
8768:
8766:
8763:
8762:
8734:
8731:
8730:
8698:
8694:
8679:
8675:
8666:
8662:
8653:
8649:
8641:
8638:
8637:
8613:
8608:
8596:
8592:
8577:
8573:
8564:
8560:
8551:
8547:
8545:
8542:
8541:
8516:
8513:
8512:
8495:
8491:
8482:
8478:
8469:
8465:
8456:
8452:
8443:
8439:
8437:
8434:
8433:
8408:
8405:
8404:
8387:
8383:
8374:
8370:
8361:
8357:
8355:
8352:
8351:
8326:
8323:
8322:
8305:
8301:
8292:
8288:
8279:
8275:
8273:
8270:
8269:
8235:
8232:
8231:
8209:
8206:
8205:
8188:
8184:
8175:
8171:
8162:
8158:
8149:
8145:
8143:
8140:
8139:
8114:
8111:
8110:
8093:
8089:
8080:
8076:
8067:
8063:
8061:
8058:
8057:
8028:
8024:
8015:
8004:
7991:
7987:
7981:
7970:
7958:
7955:
7954:
7928:
7924:
7915:
7904:
7891:
7887:
7881:
7870:
7857:
7853:
7851:
7848:
7847:
7824:
7820:
7805:
7801:
7792:
7788:
7779:
7775:
7767:
7764:
7763:
7738:
7734:
7732:
7729:
7728:
7725:is identifiable
7708:
7704:
7695:
7691:
7682:
7681:
7679:
7676:
7675:
7655:
7651:
7649:
7646:
7645:
7620:
7617:
7616:
7596:
7593:
7592:
7573:
7569:
7568:
7564:
7547:
7543:
7542:
7538:
7524:
7520:
7519:
7515:
7506:
7502:
7500:
7497:
7496:
7473:
7469:
7454:
7450:
7441:
7437:
7435:
7432:
7431:
7408:
7404:
7402:
7399:
7398:
7376:
7373:
7372:
7353:
7349:
7348:
7344:
7327:
7323:
7322:
7318:
7307:
7303:
7302:
7298:
7296:
7293:
7292:
7267:
7264:
7263:
7246:
7243:
7242:
7210:
7202:
7201:
7193:
7190:
7189:
7172:
7161:
7151:
7146:
7145:
7133:
7129:
7124:
7121:
7120:
7103:
7092:
7082:
7077:
7076:
7061:
7057:
7052:
7049:
7048:
7031:
7020:
7010:
7005:
7004:
6993:
6990:
6989:
6972:
6961:
6951:
6946:
6945:
6934:
6931:
6930:
6908:
6905:
6904:
6878:
6870:
6869:
6861:
6858:
6857:
6837:
6832:
6831:
6822:
6817:
6816:
6810:
6799:
6786:
6781:
6776:
6769:
6764:
6763:
6757:
6746:
6733:
6729:
6723:
6719:
6701:
6697:
6682:
6678:
6666:
6661:
6656:
6649:
6644:
6643:
6637:
6626:
6613:
6608:
6607:
6598:
6593:
6592:
6586:
6575:
6562:
6561:
6559:
6556:
6555:
6533:
6530:
6529:
6510:
6506:
6497:
6493:
6492:
6488:
6477:
6473:
6472:
6468:
6457:
6453:
6452:
6448:
6446:
6443:
6442:
6439:
6412:
6409:
6408:
6391:
6390:
6388:
6385:
6384:
6367:
6361:
6360:
6359:
6357:
6354:
6353:
6336:
6335:
6333:
6330:
6329:
6310:
6307:
6306:
6286:
6280:
6279:
6278:
6272:
6261:
6248:
6247:
6245:
6242:
6241:
6225:
6222:
6221:
6204:
6198:
6197:
6196:
6194:
6191:
6190:
6170:
6164:
6163:
6162:
6147:
6141:
6140:
6139:
6130:
6124:
6123:
6122:
6117:
6114:
6113:
6097:
6094:
6093:
6070:
6066:
6065:
6061:
6044:
6040:
6039:
6035:
6024:
6020:
6019:
6015:
6006:
6005:
6003:
6000:
5999:
5983:
5980:
5979:
5962:
5961:
5959:
5956:
5955:
5938:
5927:
5917:
5912:
5911:
5906:
5903:
5902:
5881:
5876:
5875:
5869:
5865:
5856:
5851:
5850:
5848:
5845:
5844:
5821:
5817:
5808:
5804:
5798:
5794:
5792:
5789:
5788:
5771:
5767:
5765:
5762:
5761:
5744:
5739:
5738:
5723:
5718:
5717:
5708:
5703:
5702:
5693:
5688:
5687:
5672:
5667:
5666:
5657:
5652:
5651:
5642:
5641:
5639:
5636:
5635:
5632:
5630:Identifiability
5627:
5596:
5593:
5592:
5575:
5574:
5569:
5561:
5553:
5545:
5541:
5536:
5525:
5517:
5509:
5501:
5493:
5485:
5477:
5469:
5461:
5448:
5440:
5432:
5424:
5416:
5408:
5400:
5392:
5384:
5376:
5367:
5366:
5361:
5353:
5345:
5331:
5321:
5313:
5299:
5289:
5281:
5267:
5257:
5249:
5236:
5230:
5224:
5223:
5222:
5218:
5216:
5213:
5212:
5181:
5173:
5167: and
5165:
5151:
5137:
5129:
5120:
5112:
5104:
5096:
5088:
5080:
5072:
5064:
5056:
5047:
5046:
5044:
5041:
5040:
5008:
5005:
5004:
4983:
4982:
4980:
4977:
4976:
4956:
4953:
4952:
4935:
4934:
4932:
4929:
4928:
4927:whose limit is
4906:
4903:
4902:
4881:
4880:
4878:
4875:
4874:
4858:
4855:
4854:
4851:
4800:
4796:
4794:
4791:
4790:
4761:
4756:
4755:
4753:
4750:
4749:
4730:
4726:
4725:
4721:
4704:
4700:
4699:
4695:
4693:
4690:
4689:
4684:is the (least)
4665:
4661:
4646:
4642:
4634:
4631:
4630:
4599:
4595:
4580:
4576:
4555:
4551:
4545:
4534:
4529:
4525:
4510:
4506:
4491:
4487:
4476:
4472:
4470:
4467:
4466:
4446:
4442:
4427:
4423:
4414:
4410:
4408:
4405:
4404:
4385:
4381:
4380:
4376:
4359:
4355:
4354:
4350:
4348:
4345:
4344:
4324:
4320:
4305:
4301:
4290:
4286:
4284:
4281:
4280:
4272:
4223:
4219:
4184:
4181:
4180:
4146:
4143:
4142:
4126:
4123:
4122:
4100:
4097:
4096:
4058:
4054:
4025:
4022:
4021:
3989:
3985:
3983:
3980:
3979:
3944:
3940:
3938:
3935:
3934:
3908:
3904:
3889:
3885:
3876:
3872:
3860:
3856:
3841:
3837:
3822:
3818:
3810:
3807:
3806:
3766:
3718:
3714:
3712:
3709:
3708:
3670:
3622:
3618:
3616:
3613:
3612:
3589:
3585:
3570:
3566:
3557:
3553:
3541:
3537:
3522:
3518:
3510:
3507:
3506:
3476:
3472:
3457:
3453:
3444:
3440:
3428:
3424:
3409:
3405:
3397:
3394:
3393:
3371:
3367:
3352:
3348:
3347:
3343:
3341:
3338:
3337:
3313:
3309:
3294:
3290:
3281:
3277:
3275:
3272:
3271:
3251:
3247:
3232:
3228:
3219:
3215:
3213:
3210:
3209:
3189:
3186:
3185:
3160:
3156:
3141:
3137:
3136:
3131:
3130:
3128:
3125:
3124:
3092:
3088:
3079:
3068:
3052:
3045:
3041:
3035:
3024:
3012:
3009:
3008:
2971:
2966:
2962:
2950:
2946:
2931:
2927:
2918:
2914:
2912:
2909:
2908:
2898:
2877:
2874:
2873:
2848:
2844:
2829:
2825:
2824:
2820:
2818:
2815:
2814:
2786:
2782:
2776:
2765:
2752:
2748:
2747:
2743:
2728:
2724:
2709:
2705:
2697:
2694:
2693:
2686:
2645:
2641:
2632:
2621:
2608:
2604:
2598:
2587:
2568:
2564:
2562:
2559:
2558:
2534:
2530:
2515:
2511:
2502:
2498:
2496:
2493:
2492:
2473:
2469:
2468:
2464:
2447:
2443:
2442:
2438:
2436:
2433:
2432:
2412:
2407:
2406:
2397:
2392:
2391:
2382:
2377:
2376:
2374:
2371:
2370:
2341:
2338:
2337:
2320:
2315:
2314:
2305:
2300:
2299:
2290:
2285:
2284:
2282:
2279:
2278:
2261:
2256:
2255:
2246:
2241:
2240:
2231:
2226:
2225:
2223:
2220:
2219:
2213:Volker Strassen
2187:
2183:
2168:
2164:
2156:
2153:
2152:
2136:
2133:
2132:
2112:
2109:
2108:
2085:
2081:
2066:
2062:
2054:
2051:
2050:
2031:
2027:
2026:
2022:
2005:
2001:
2000:
1996:
1994:
1991:
1990:
1974:
1971:
1970:
1950:
1947:
1946:
1926:
1922:
1907:
1903:
1895:
1892:
1891:
1885:
1864:
1862:
1859:
1858:
1855:field extension
1834:
1829:
1828:
1816:
1811:
1810:
1801:
1796:
1795:
1793:
1790:
1789:
1763:
1758:
1757:
1748:
1737:
1735:
1734:
1733:
1724:
1719:
1718:
1709:
1698:
1696:
1695:
1694:
1685:
1680:
1679:
1670:
1659:
1657:
1656:
1655:
1642:
1633:
1632:
1630:
1627:
1626:
1602:
1597:
1596:
1587:
1582:
1581:
1572:
1567:
1566:
1564:
1561:
1560:
1537:
1532:
1531:
1522:
1517:
1516:
1507:
1502:
1501:
1492:
1487:
1486:
1477:
1472:
1471:
1462:
1457:
1456:
1447:
1442:
1441:
1432:
1427:
1426:
1417:
1412:
1411:
1402:
1397:
1396:
1387:
1382:
1381:
1372:
1367:
1366:
1357:
1356:
1354:
1351:
1350:
1343:
1307:
1303:
1302:
1298:
1289:
1284:
1283:
1281:
1278:
1277:
1260:
1255:
1254:
1239:
1234:
1233:
1231:
1228:
1227:
1191:
1187:
1176:
1172:
1171:
1167:
1156:
1152:
1151:
1147:
1145:
1142:
1141:
1134:
1090:
1087:
1086:
1070:
1067:
1066:
1038:
1035:
1034:
1015:
1011:
1010:
1004:
1003:
1002:
987:
982:
981:
979:
976:
975:
958:
957:
948:
944:
942:
939:
938:
909:
904:
903:
882:
877:
876:
858:
853:
852:
837:
832:
831:
816:
811:
810:
804:
800:
794:
783:
770:
769:
767:
764:
763:
744:
741:
740:
724:
721:
720:
716:
712:
708:
689:
685:
670:
666:
657:
653:
652:
646:
645:
644:
635:
634:
632:
629:
628:
625:
601:
597:
585:
581:
566:
562:
553:
549:
548:
542:
541:
540:
538:
535:
534:
515:
511:
499:
495:
480:
476:
467:
463:
462:
458:
456:
453:
452:
433:
429:
417:
413:
401:
397:
388:
384:
383:
378:
377:
368:
367:
365:
362:
361:
345:
342:
341:
324:
323:
321:
318:
317:
298:
296:
293:
292:
276:
274:
271:
270:
239:
236:
235:
218:
214:
205:
201:
193:
190:
189:
161:
158:
157:
138:
135:
134:
118:
115:
114:
111:
95:computer vision
23:
22:
15:
12:
11:
5:
12978:
12968:
12967:
12962:
12945:
12944:
12942:
12941:
12936:
12934:Woldemar Voigt
12931:
12926:
12921:
12916:
12911:
12906:
12901:
12899:Leonhard Euler
12896:
12891:
12886:
12881:
12875:
12873:
12871:Mathematicians
12867:
12866:
12863:
12862:
12860:
12859:
12854:
12849:
12844:
12839:
12834:
12829:
12824:
12819:
12813:
12811:
12807:
12806:
12804:
12803:
12798:
12796:Torsion tensor
12793:
12788:
12783:
12778:
12773:
12768:
12762:
12760:
12753:
12749:
12748:
12746:
12745:
12740:
12735:
12730:
12725:
12720:
12715:
12710:
12705:
12700:
12695:
12690:
12685:
12680:
12675:
12670:
12665:
12660:
12655:
12649:
12647:
12641:
12640:
12638:
12637:
12631:
12629:Tensor product
12626:
12621:
12619:Symmetrization
12616:
12611:
12609:Lie derivative
12606:
12601:
12596:
12591:
12586:
12580:
12578:
12572:
12571:
12569:
12568:
12563:
12558:
12553:
12548:
12543:
12538:
12533:
12531:Tensor density
12528:
12523:
12517:
12515:
12509:
12508:
12506:
12505:
12503:Voigt notation
12500:
12495:
12490:
12488:Ricci calculus
12485:
12480:
12475:
12473:Index notation
12470:
12465:
12459:
12457:
12453:
12452:
12449:
12448:
12446:
12445:
12440:
12435:
12430:
12425:
12419:
12417:
12415:
12414:
12409:
12403:
12400:
12399:
12397:
12396:
12391:
12389:Tensor algebra
12386:
12381:
12376:
12371:
12369:Dyadic algebra
12366:
12361:
12355:
12353:
12344:
12340:
12339:
12332:
12329:
12328:
12321:
12320:
12313:
12306:
12298:
12292:
12291:
12281:
12276:
12269:
12268:External links
12266:
12265:
12264:
12255:
12219:(3): 455–500.
12203:
12200:
12197:
12196:
12177:
12124:
12071:
12028:
11985:
11950:
11912:
11893:(1–2): 67–90.
11877:
11824:
11787:(2): 636–660.
11771:
11714:
11666:
11613:
11576:(2): 656–681.
11560:
11507:
11451:
11444:
11418:
11407:(4): 692–697.
11378:
11333:Blehkerman, G.
11324:
11311:(2): 295–327.
11281:
11259:
11245:Gimigliano, A.
11228:
11190:
11161:(2): 767–792.
11129:
11100:Gimigliano, A.
11083:
11049:
10990:
10972:
10920:
10917:. ASONAM 2022.
10910:Gujral, Ekta.
10902:
10863:
10844:(3): 283–319.
10824:Carroll, J. D.
10812:
10780:
10761:
10760:
10758:
10755:
10754:
10753:
10748:
10743:
10738:
10733:
10728:
10721:
10718:
10705:
10702:
10699:
10696:
10692:
10688:
10685:
10682:
10671:topic modeling
10655:
10652:
10649:
10646:
10642:
10636:
10632:
10628:
10625:
10622:
10619:
10616:
10613:
10610:
10606:
10600:
10596:
10592:
10589:
10586:
10583:
10580:
10577:
10574:
10570:
10564:
10560:
10556:
10553:
10550:
10547:
10544:
10541:
10538:
10535:
10532:
10527:
10522:
10519:
10516:
10512:
10508:
10505:
10500:
10496:
10492:
10487:
10483:
10479:
10474:
10470:
10466:
10463:
10443:
10438:
10434:
10430:
10425:
10421:
10417:
10412:
10408:
10404:
10401:
10381:
10361:
10339:
10335:
10331:
10326:
10322:
10318:
10313:
10309:
10291:
10288:
10287:
10286:
10277:
10276:
10270:
10264:
10258:
10252:
10242:
10241:
10238:
10231:
10230:
10224:
10213:
10210:
10185:
10182:
10179:
10168:
10167:
10156:
10153:
10148:
10144:
10138:
10133:
10130:
10127:
10122:
10117:
10114:
10111:
10106:
10101:
10098:
10095:
10090:
10085:
10080:
10075:
10072:
10069:
10064:
10059:
10051:
10048:
10043:
10039:
10033:
10028:
10025:
10022:
10017:
10012:
10009:
10006:
10001:
9996:
9993:
9990:
9985:
9980:
9975:
9970:
9967:
9964:
9959:
9954:
9931:
9928:
9925:
9922:
9919:
9916:
9913:
9893:
9890:
9887:
9865:
9860:
9855:
9849:
9835:
9834:
9821:
9816:
9813:
9810:
9805:
9800:
9797:
9794:
9789:
9784:
9781:
9778:
9773:
9768:
9763:
9758:
9755:
9752:
9747:
9740:
9735:
9732:
9729:
9725:
9721:
9716:
9710:
9684:
9664:
9661:
9658:
9647:
9646:
9634:
9630:
9626:
9622:
9618:
9614:
9611:
9608:
9604:
9598:
9595:
9590:
9586:
9582:
9579:
9576:
9572:
9566:
9563:
9558:
9554:
9550:
9547:
9544:
9540:
9534:
9531:
9526:
9522:
9518:
9515:
9512:
9507:
9501:
9484:
9483:
9472:
9469:
9466:
9462:
9458:
9454:
9450:
9442:
9439:
9436:
9432:
9428:
9425:
9422:
9418:
9414:
9405:
9401:
9397:
9393:
9389:
9385:
9381:
9377:
9373:
9369:
9365:
9361:
9357:
9353:
9349:
9345:
9341:
9337:
9333:
9328:
9303:
9283:
9272:border tensors
9261:Frobenius norm
9246:
9242:
9238:
9235:
9224:
9223:
9212:
9207:
9203:
9197:
9192:
9187:
9182:
9179:
9176:
9171:
9166:
9161:
9156:
9151:
9146:
9141:
9134:
9129:
9126:
9123:
9119:
9115:
9110:
9105:
9096:
9092:
9087:
9083:
9078:
9073:
9068:
9062:
9038:
9035:
9032:
9010:
8988:
8968:
8956:
8953:
8937:
8934:
8931:
8927:
8922:
8919:
8899:
8896:
8893:
8882:
8881:
8867:
8863:
8859:
8854:
8850:
8846:
8841:
8837:
8833:
8828:
8824:
8801:
8797:
8793:
8788:
8784:
8780:
8775:
8771:
8759:
8756:
8753:
8750:
8738:
8719:
8718:
8706:
8701:
8697:
8693:
8690:
8687:
8682:
8678:
8674:
8669:
8665:
8661:
8656:
8652:
8648:
8645:
8622:
8619:
8616:
8612:
8607:
8604:
8599:
8595:
8591:
8588:
8585:
8580:
8576:
8572:
8567:
8563:
8559:
8554:
8550:
8538:
8526:
8523:
8520:
8498:
8494:
8490:
8485:
8481:
8477:
8472:
8468:
8464:
8459:
8455:
8451:
8446:
8442:
8430:
8418:
8415:
8412:
8390:
8386:
8382:
8377:
8373:
8369:
8364:
8360:
8348:
8336:
8333:
8330:
8308:
8304:
8300:
8295:
8291:
8287:
8282:
8278:
8266:
8254:
8251:
8248:
8245:
8242:
8239:
8219:
8216:
8213:
8191:
8187:
8183:
8178:
8174:
8170:
8165:
8161:
8157:
8152:
8148:
8136:
8124:
8121:
8118:
8096:
8092:
8088:
8083:
8079:
8075:
8070:
8066:
8054:
8042:
8039:
8036:
8031:
8027:
8023:
8018:
8013:
8010:
8007:
8003:
7999:
7994:
7990:
7984:
7979:
7976:
7973:
7969:
7965:
7962:
7942:
7939:
7936:
7931:
7927:
7923:
7918:
7913:
7910:
7907:
7903:
7899:
7894:
7890:
7884:
7879:
7876:
7873:
7869:
7865:
7860:
7856:
7844:
7832:
7827:
7823:
7819:
7816:
7813:
7808:
7804:
7800:
7795:
7791:
7787:
7782:
7778:
7774:
7771:
7741:
7737:
7711:
7707:
7703:
7698:
7694:
7690:
7685:
7658:
7654:
7630:
7627:
7624:
7600:
7576:
7572:
7567:
7563:
7560:
7557:
7550:
7546:
7541:
7537:
7534:
7527:
7523:
7518:
7514:
7509:
7505:
7484:
7481:
7476:
7472:
7468:
7465:
7462:
7457:
7453:
7449:
7444:
7440:
7419:
7416:
7411:
7407:
7386:
7383:
7380:
7356:
7352:
7347:
7343:
7340:
7337:
7330:
7326:
7321:
7317:
7310:
7306:
7301:
7277:
7274:
7271:
7250:
7230:
7227:
7224:
7221:
7218:
7213:
7208:
7205:
7200:
7197:
7175:
7170:
7167:
7164:
7160:
7154:
7149:
7144:
7141:
7136:
7132:
7128:
7106:
7101:
7098:
7095:
7091:
7085:
7080:
7075:
7072:
7067:
7064:
7060:
7056:
7034:
7029:
7026:
7023:
7019:
7013:
7008:
7003:
7000:
6997:
6975:
6970:
6967:
6964:
6960:
6954:
6949:
6944:
6941:
6938:
6918:
6915:
6912:
6892:
6889:
6886:
6881:
6876:
6873:
6868:
6865:
6845:
6840:
6835:
6830:
6825:
6820:
6813:
6808:
6805:
6802:
6798:
6794:
6789:
6784:
6779:
6772:
6767:
6760:
6755:
6752:
6749:
6745:
6741:
6736:
6732:
6726:
6722:
6718:
6715:
6712:
6707:
6704:
6700:
6696:
6693:
6690:
6685:
6681:
6677:
6674:
6669:
6664:
6659:
6652:
6647:
6640:
6635:
6632:
6629:
6625:
6621:
6616:
6611:
6606:
6601:
6596:
6589:
6584:
6581:
6578:
6574:
6570:
6565:
6543:
6540:
6537:
6513:
6509:
6505:
6500:
6496:
6491:
6487:
6480:
6476:
6471:
6467:
6460:
6456:
6451:
6438:
6435:
6422:
6419:
6416:
6394:
6370:
6364:
6339:
6317:
6314:
6294:
6289:
6283:
6275:
6270:
6267:
6264:
6260:
6256:
6251:
6229:
6207:
6201:
6178:
6173:
6167:
6161:
6158:
6155:
6150:
6144:
6138:
6133:
6127:
6121:
6101:
6073:
6069:
6064:
6060:
6057:
6054:
6047:
6043:
6038:
6034:
6027:
6023:
6018:
6014:
6009:
5987:
5965:
5941:
5936:
5933:
5930:
5926:
5920:
5915:
5910:
5884:
5879:
5872:
5868:
5864:
5859:
5854:
5832:
5829:
5824:
5820:
5816:
5811:
5807:
5801:
5797:
5774:
5770:
5747:
5742:
5737:
5734:
5731:
5726:
5721:
5716:
5711:
5706:
5701:
5696:
5691:
5686:
5683:
5680:
5675:
5670:
5665:
5660:
5655:
5650:
5645:
5631:
5628:
5626:
5623:
5606:
5603:
5600:
5589:
5588:
5572:
5568:
5564:
5560:
5556:
5548:
5544:
5540:
5535:
5532:
5528:
5524:
5520:
5516:
5512:
5508:
5504:
5500:
5496:
5492:
5488:
5484:
5480:
5476:
5472:
5468:
5464:
5460:
5455:
5452:
5447:
5443:
5439:
5435:
5431:
5427:
5423:
5419:
5415:
5411:
5407:
5403:
5399:
5395:
5391:
5387:
5383:
5379:
5375:
5372:
5370:
5368:
5364:
5360:
5356:
5352:
5348:
5344:
5341:
5338:
5334:
5328:
5325:
5320:
5316:
5312:
5309:
5306:
5302:
5296:
5293:
5288:
5284:
5280:
5277:
5274:
5270:
5264:
5261:
5256:
5252:
5248:
5245:
5242:
5239:
5237:
5233:
5227:
5221:
5220:
5206:
5205:
5194:
5191:
5188:
5184:
5180:
5176:
5172:
5164:
5161:
5158:
5154:
5150:
5147:
5144:
5140:
5136:
5127:
5123:
5119:
5115:
5111:
5107:
5103:
5099:
5095:
5091:
5087:
5083:
5079:
5075:
5071:
5067:
5063:
5059:
5055:
5050:
5015:
5012:
4986:
4960:
4938:
4916:
4913:
4910:
4884:
4862:
4850:
4847:
4834:
4831:
4828:
4825:
4822:
4819:
4816:
4813:
4810:
4799:
4776:
4773:
4770:
4767:
4764:
4759:
4733:
4729:
4724:
4720:
4717:
4714:
4707:
4703:
4698:
4673:
4668:
4664:
4660:
4657:
4654:
4649:
4645:
4641:
4638:
4627:
4626:
4615:
4611:
4607:
4602:
4598:
4594:
4591:
4588:
4583:
4579:
4575:
4572:
4569:
4566:
4563:
4558:
4554:
4548:
4543:
4540:
4537:
4533:
4528:
4524:
4521:
4518:
4513:
4509:
4505:
4502:
4499:
4494:
4490:
4486:
4475:
4449:
4445:
4441:
4438:
4435:
4430:
4426:
4422:
4417:
4413:
4388:
4384:
4379:
4375:
4372:
4369:
4362:
4358:
4353:
4332:
4327:
4323:
4319:
4316:
4313:
4308:
4304:
4300:
4289:
4271:
4268:
4255:
4252:
4249:
4246:
4243:
4240:
4237:
4234:
4231:
4226:
4222:
4218:
4215:
4212:
4209:
4206:
4203:
4200:
4197:
4194:
4191:
4188:
4168:
4165:
4162:
4159:
4156:
4153:
4150:
4130:
4110:
4107:
4104:
4084:
4081:
4078:
4075:
4072:
4069:
4066:
4061:
4057:
4053:
4050:
4047:
4044:
4041:
4038:
4035:
4032:
4029:
4004:
4001:
3998:
3995:
3992:
3988:
3965:
3962:
3959:
3956:
3953:
3950:
3947:
3943:
3922:
3919:
3916:
3911:
3907:
3903:
3900:
3897:
3892:
3888:
3884:
3879:
3875:
3871:
3868:
3863:
3859:
3855:
3852:
3849:
3844:
3840:
3836:
3833:
3830:
3825:
3821:
3817:
3814:
3803:
3802:
3791:
3788:
3785:
3782:
3779:
3776:
3773:
3763:
3760:
3757:
3754:
3751:
3748:
3745:
3742:
3739:
3736:
3733:
3730:
3727:
3724:
3721:
3717:
3706:
3695:
3692:
3689:
3686:
3683:
3680:
3677:
3667:
3664:
3661:
3658:
3655:
3652:
3649:
3646:
3643:
3640:
3637:
3634:
3631:
3628:
3625:
3621:
3597:
3592:
3588:
3584:
3581:
3578:
3573:
3569:
3565:
3560:
3556:
3552:
3549:
3544:
3540:
3536:
3533:
3530:
3525:
3521:
3517:
3514:
3499:
3498:
3487:
3484:
3479:
3475:
3471:
3468:
3465:
3460:
3456:
3452:
3447:
3443:
3439:
3436:
3431:
3427:
3423:
3420:
3417:
3412:
3408:
3404:
3401:
3374:
3370:
3366:
3363:
3360:
3355:
3351:
3346:
3321:
3316:
3312:
3308:
3305:
3302:
3297:
3293:
3289:
3284:
3280:
3259:
3254:
3250:
3246:
3243:
3240:
3235:
3231:
3227:
3222:
3218:
3193:
3173:
3170:
3163:
3159:
3155:
3152:
3149:
3144:
3140:
3134:
3121:
3120:
3109:
3106:
3103:
3100:
3095:
3091:
3087:
3082:
3077:
3074:
3071:
3067:
3063:
3060:
3048:
3044:
3038:
3033:
3030:
3027:
3023:
3019:
3016:
2999:
2998:
2986:
2980:
2977:
2974:
2970:
2965:
2961:
2958:
2953:
2949:
2945:
2942:
2939:
2934:
2930:
2926:
2921:
2917:
2897:
2894:
2881:
2861:
2858:
2851:
2847:
2843:
2840:
2837:
2832:
2828:
2823:
2808:
2807:
2795:
2789:
2785:
2779:
2774:
2771:
2768:
2764:
2760:
2755:
2751:
2746:
2742:
2739:
2736:
2731:
2727:
2723:
2720:
2717:
2712:
2708:
2704:
2701:
2685:
2682:
2674:
2673:
2662:
2659:
2656:
2653:
2648:
2644:
2640:
2635:
2630:
2627:
2624:
2620:
2616:
2611:
2607:
2601:
2596:
2593:
2590:
2586:
2582:
2579:
2576:
2571:
2567:
2537:
2533:
2529:
2526:
2523:
2518:
2514:
2510:
2505:
2501:
2476:
2472:
2467:
2463:
2460:
2457:
2450:
2446:
2441:
2415:
2410:
2405:
2400:
2395:
2390:
2385:
2380:
2357:
2354:
2351:
2348:
2345:
2323:
2318:
2313:
2308:
2303:
2298:
2293:
2288:
2264:
2259:
2254:
2249:
2244:
2239:
2234:
2229:
2195:
2190:
2186:
2182:
2179:
2176:
2171:
2167:
2163:
2160:
2140:
2116:
2093:
2088:
2084:
2080:
2077:
2074:
2069:
2065:
2061:
2058:
2034:
2030:
2025:
2021:
2018:
2015:
2008:
2004:
1999:
1978:
1954:
1934:
1929:
1925:
1921:
1918:
1915:
1910:
1906:
1902:
1899:
1884:
1881:
1867:
1837:
1832:
1827:
1824:
1819:
1814:
1809:
1804:
1799:
1786:
1785:
1774:
1771:
1766:
1761:
1756:
1751:
1744:
1740:
1732:
1727:
1722:
1717:
1712:
1705:
1701:
1693:
1688:
1683:
1678:
1673:
1666:
1662:
1654:
1649:
1646:
1641:
1636:
1605:
1600:
1595:
1590:
1585:
1580:
1575:
1570:
1557:
1556:
1545:
1540:
1535:
1530:
1525:
1520:
1515:
1510:
1505:
1500:
1495:
1490:
1485:
1480:
1475:
1470:
1465:
1460:
1455:
1450:
1445:
1440:
1435:
1430:
1425:
1420:
1415:
1410:
1405:
1400:
1395:
1390:
1385:
1380:
1375:
1370:
1365:
1360:
1342:
1339:
1326:
1323:
1320:
1317:
1310:
1306:
1301:
1297:
1292:
1287:
1263:
1258:
1253:
1250:
1247:
1242:
1237:
1194:
1190:
1186:
1179:
1175:
1170:
1166:
1159:
1155:
1150:
1133:
1130:
1105:is called the
1094:
1074:
1054:
1051:
1048:
1045:
1042:
1018:
1014:
1007:
1001:
996:
993:
990:
985:
961:
956:
951:
947:
935:
934:
923:
918:
915:
912:
907:
902:
899:
896:
891:
888:
885:
880:
875:
872:
867:
864:
861:
856:
851:
846:
843:
840:
835:
830:
825:
822:
819:
814:
807:
803:
797:
792:
789:
786:
782:
778:
773:
748:
728:
692:
688:
684:
681:
678:
673:
669:
665:
660:
656:
649:
643:
638:
627:A data tensor
624:
621:
604:
600:
596:
593:
588:
584:
580:
577:
574:
569:
565:
561:
556:
552:
545:
518:
514:
510:
507:
502:
498:
494:
491:
488:
483:
479:
475:
470:
466:
461:
451:is denoted by
436:
432:
428:
425:
420:
416:
412:
409:
404:
400:
396:
391:
387:
381:
376:
371:
360:-order tensor
349:
327:
301:
279:
255:
252:
249:
246:
243:
221:
217:
213:
208:
204:
200:
197:
177:
174:
171:
168:
165:
142:
122:
110:
107:
9:
6:
4:
3:
2:
12977:
12966:
12963:
12961:
12958:
12957:
12955:
12940:
12937:
12935:
12932:
12930:
12927:
12925:
12922:
12920:
12917:
12915:
12912:
12910:
12907:
12905:
12902:
12900:
12897:
12895:
12892:
12890:
12887:
12885:
12882:
12880:
12877:
12876:
12874:
12872:
12868:
12858:
12855:
12853:
12850:
12848:
12845:
12843:
12840:
12838:
12835:
12833:
12830:
12828:
12825:
12823:
12820:
12818:
12815:
12814:
12812:
12808:
12802:
12799:
12797:
12794:
12792:
12789:
12787:
12784:
12782:
12779:
12777:
12776:Metric tensor
12774:
12772:
12769:
12767:
12764:
12763:
12761:
12757:
12754:
12750:
12744:
12741:
12739:
12736:
12734:
12731:
12729:
12726:
12724:
12721:
12719:
12716:
12714:
12711:
12709:
12706:
12704:
12701:
12699:
12696:
12694:
12691:
12689:
12688:Exterior form
12686:
12684:
12681:
12679:
12676:
12674:
12671:
12669:
12666:
12664:
12661:
12659:
12656:
12654:
12651:
12650:
12648:
12642:
12635:
12632:
12630:
12627:
12625:
12622:
12620:
12617:
12615:
12612:
12610:
12607:
12605:
12602:
12600:
12597:
12595:
12592:
12590:
12587:
12585:
12582:
12581:
12579:
12577:
12573:
12567:
12564:
12562:
12561:Tensor bundle
12559:
12557:
12554:
12552:
12549:
12547:
12544:
12542:
12539:
12537:
12534:
12532:
12529:
12527:
12524:
12522:
12519:
12518:
12516:
12510:
12504:
12501:
12499:
12496:
12494:
12491:
12489:
12486:
12484:
12481:
12479:
12476:
12474:
12471:
12469:
12466:
12464:
12461:
12460:
12458:
12454:
12444:
12441:
12439:
12436:
12434:
12431:
12429:
12426:
12424:
12421:
12420:
12418:
12413:
12410:
12408:
12405:
12404:
12401:
12395:
12392:
12390:
12387:
12385:
12382:
12380:
12377:
12375:
12372:
12370:
12367:
12365:
12362:
12360:
12357:
12356:
12354:
12352:
12348:
12345:
12341:
12337:
12336:
12330:
12326:
12319:
12314:
12312:
12307:
12305:
12300:
12299:
12296:
12289:
12285:
12282:
12280:
12277:
12275:
12272:
12271:
12261:
12256:
12252:
12248:
12244:
12240:
12235:
12230:
12226:
12222:
12218:
12214:
12210:
12206:
12205:
12192:
12188:
12181:
12173:
12169:
12165:
12161:
12157:
12153:
12148:
12143:
12139:
12135:
12128:
12120:
12116:
12112:
12108:
12104:
12100:
12095:
12090:
12086:
12082:
12075:
12067:
12063:
12059:
12055:
12051:
12047:
12043:
12039:
12038:Psychometrika
12032:
12024:
12020:
12016:
12012:
12008:
12004:
12000:
11996:
11989:
11981:
11977:
11973:
11969:
11965:
11961:
11954:
11946:
11942:
11938:
11934:
11930:
11926:
11922:
11916:
11908:
11904:
11900:
11896:
11892:
11888:
11881:
11873:
11869:
11865:
11861:
11857:
11853:
11848:
11843:
11839:
11835:
11828:
11820:
11816:
11812:
11808:
11804:
11800:
11795:
11790:
11786:
11782:
11775:
11767:
11763:
11759:
11755:
11751:
11747:
11742:
11737:
11733:
11729:
11725:
11718:
11710:
11706:
11702:
11698:
11693:
11688:
11685:(753): 1–22.
11684:
11680:
11673:
11671:
11662:
11658:
11654:
11650:
11646:
11642:
11637:
11632:
11628:
11624:
11617:
11609:
11605:
11601:
11597:
11593:
11589:
11584:
11579:
11575:
11571:
11564:
11556:
11552:
11548:
11544:
11540:
11536:
11531:
11526:
11522:
11518:
11511:
11503:
11499:
11495:
11491:
11487:
11483:
11478:
11473:
11469:
11465:
11458:
11456:
11447:
11441:
11437:
11433:
11429:
11422:
11414:
11410:
11406:
11402:
11401:
11396:
11392:
11388:
11382:
11374:
11370:
11366:
11362:
11357:
11352:
11349:(3–4): 1–11.
11348:
11344:
11343:
11338:
11334:
11328:
11319:
11314:
11310:
11306:
11302:
11300:
11296:
11246:
11242:
11238:
11232:
11223:
11218:
11214:
11210:
11209:
11204:
11200:
11194:
11186:
11182:
11178:
11174:
11169:
11164:
11160:
11156:
11155:
11150:
11146:
11145:Ottaviani, G.
11142:
11136:
11134:
11124:
11119:
11115:
11111:
11110:
11105:
11101:
11097:
11093:
11087:
11078:
11073:
11069:
11068:
11063:
11059:
11053:
11045:
11041:
11037:
11033:
11028:
11023:
11019:
11015:
11014:
11009:
11005:
10999:
10997:
10995:
10986:
10982:
10976:
10968:
10964:
10960:
10956:
10951:
10946:
10942:
10938:
10934:
10930:
10929:Hillar, C. J.
10924:
10913:
10906:
10895:
10891:
10887:
10880:
10876:
10870:
10868:
10859:
10855:
10851:
10847:
10843:
10839:
10838:
10837:Psychometrika
10833:
10829:
10825:
10819:
10817:
10808:
10804:
10800:
10796:
10795:
10790:
10784:
10773:
10766:
10762:
10752:
10749:
10747:
10744:
10742:
10739:
10737:
10734:
10732:
10729:
10727:
10724:
10723:
10717:
10700:
10697:
10694:
10686:
10680:
10672:
10667:
10650:
10647:
10644:
10634:
10630:
10623:
10620:
10614:
10611:
10608:
10598:
10594:
10587:
10584:
10578:
10575:
10572:
10562:
10558:
10551:
10545:
10542:
10539:
10533:
10530:
10525:
10520:
10517:
10514:
10510:
10506:
10498:
10494:
10490:
10485:
10481:
10477:
10472:
10468:
10461:
10436:
10432:
10428:
10423:
10419:
10415:
10410:
10406:
10399:
10379:
10359:
10337:
10333:
10329:
10324:
10320:
10316:
10311:
10307:
10298:
10285:
10282:
10281:
10280:
10274:
10271:
10268:
10265:
10262:
10259:
10256:
10253:
10250:
10247:
10246:
10245:
10239:
10236:
10235:
10234:
10228:
10225:
10222:
10219:
10218:
10217:
10209:
10207:
10206:inner product
10204:or a bounded
10201:
10199:
10177:
10146:
10136:
10131:
10128:
10125:
10115:
10112:
10109:
10104:
10099:
10096:
10093:
10083:
10078:
10073:
10070:
10067:
10041:
10031:
10026:
10023:
10020:
10010:
10007:
10004:
9999:
9994:
9991:
9988:
9978:
9973:
9968:
9965:
9962:
9945:
9944:
9943:
9929:
9926:
9923:
9920:
9917:
9914:
9911:
9885:
9853:
9819:
9814:
9811:
9808:
9798:
9795:
9792:
9787:
9782:
9779:
9776:
9766:
9761:
9756:
9753:
9750:
9738:
9733:
9730:
9727:
9723:
9719:
9714:
9698:
9697:
9696:
9682:
9656:
9628:
9620:
9612:
9609:
9596:
9593:
9588:
9577:
9564:
9561:
9556:
9545:
9532:
9529:
9524:
9513:
9510:
9505:
9489:
9488:
9487:
9470:
9467:
9456:
9440:
9437:
9423:
9403:
9395:
9387:
9379:
9371:
9363:
9355:
9347:
9339:
9331:
9317:
9316:
9315:
9301:
9281:
9273:
9269:
9264:
9262:
9244:
9236:
9210:
9205:
9195:
9190:
9180:
9177:
9174:
9169:
9164:
9154:
9149:
9144:
9132:
9127:
9124:
9121:
9117:
9113:
9094:
9090:
9085:
9081:
9076:
9071:
9052:
9051:
9050:
9036:
9033:
9030:
8986:
8966:
8952:
8935:
8932:
8920:
8917:
8897:
8894:
8891:
8865:
8861:
8857:
8852:
8848:
8844:
8839:
8835:
8831:
8826:
8822:
8799:
8795:
8791:
8786:
8782:
8778:
8773:
8769:
8760:
8757:
8754:
8751:
8729:proved to be
8728:
8727:
8726:
8724:
8699:
8695:
8691:
8688:
8685:
8680:
8676:
8672:
8667:
8663:
8654:
8650:
8646:
8643:
8620:
8617:
8605:
8597:
8593:
8589:
8586:
8583:
8578:
8574:
8570:
8565:
8561:
8552:
8548:
8539:
8524:
8521:
8518:
8496:
8492:
8488:
8483:
8479:
8475:
8470:
8466:
8462:
8457:
8453:
8449:
8444:
8440:
8432:The space is
8431:
8416:
8413:
8410:
8388:
8384:
8380:
8375:
8371:
8367:
8362:
8358:
8350:The space is
8349:
8334:
8331:
8328:
8306:
8302:
8298:
8293:
8289:
8285:
8280:
8276:
8268:The space is
8267:
8252:
8249:
8246:
8243:
8240:
8237:
8217:
8214:
8211:
8189:
8185:
8181:
8176:
8172:
8168:
8163:
8159:
8155:
8150:
8146:
8137:
8122:
8119:
8116:
8094:
8090:
8086:
8081:
8077:
8073:
8068:
8064:
8055:
8037:
8034:
8029:
8025:
8016:
8011:
8008:
8005:
8001:
7997:
7992:
7988:
7982:
7977:
7974:
7971:
7967:
7963:
7960:
7937:
7934:
7929:
7925:
7916:
7911:
7908:
7905:
7901:
7897:
7892:
7888:
7882:
7877:
7874:
7871:
7867:
7863:
7858:
7854:
7845:
7825:
7821:
7817:
7814:
7811:
7806:
7802:
7798:
7793:
7789:
7780:
7776:
7772:
7769:
7761:
7760:
7759:
7757:
7739:
7735:
7726:
7709:
7705:
7696:
7692:
7688:
7674:
7656:
7652:
7642:
7628:
7625:
7614:
7598:
7574:
7570:
7565:
7561:
7558:
7555:
7548:
7544:
7539:
7535:
7532:
7525:
7521:
7516:
7512:
7507:
7503:
7482:
7479:
7474:
7470:
7466:
7463:
7460:
7455:
7451:
7447:
7442:
7438:
7417:
7414:
7409:
7405:
7384:
7381:
7378:
7354:
7350:
7345:
7341:
7338:
7335:
7328:
7324:
7319:
7315:
7308:
7304:
7299:
7289:
7275:
7272:
7269:
7248:
7228:
7219:
7211:
7198:
7195:
7173:
7168:
7165:
7162:
7152:
7139:
7134:
7130:
7126:
7104:
7099:
7096:
7093:
7083:
7070:
7065:
7062:
7058:
7054:
7032:
7027:
7024:
7021:
7011:
6998:
6995:
6973:
6968:
6965:
6962:
6952:
6939:
6936:
6916:
6913:
6910:
6887:
6879:
6866:
6863:
6843:
6838:
6828:
6823:
6811:
6806:
6803:
6800:
6796:
6792:
6787:
6782:
6770:
6758:
6753:
6750:
6747:
6743:
6739:
6734:
6724:
6720:
6716:
6705:
6702:
6698:
6694:
6688:
6683:
6679:
6675:
6672:
6667:
6662:
6650:
6638:
6633:
6630:
6627:
6623:
6619:
6614:
6604:
6599:
6587:
6582:
6579:
6576:
6572:
6568:
6541:
6538:
6535:
6511:
6507:
6503:
6498:
6494:
6489:
6485:
6478:
6474:
6469:
6465:
6458:
6454:
6449:
6434:
6420:
6417:
6414:
6368:
6315:
6312:
6292:
6287:
6273:
6268:
6265:
6262:
6258:
6254:
6227:
6205:
6171:
6159:
6156:
6153:
6148:
6136:
6131:
6099:
6091:
6071:
6067:
6062:
6058:
6055:
6052:
6045:
6041:
6036:
6032:
6025:
6021:
6016:
6012:
5985:
5939:
5934:
5931:
5928:
5918:
5900:
5882:
5870:
5866:
5862:
5857:
5830:
5827:
5822:
5818:
5814:
5809:
5805:
5799:
5795:
5772:
5768:
5745:
5735:
5732:
5729:
5724:
5714:
5709:
5699:
5694:
5684:
5681:
5678:
5673:
5663:
5658:
5648:
5622:
5620:
5598:
5566:
5558:
5546:
5542:
5538:
5533:
5522:
5514:
5506:
5498:
5490:
5482:
5474:
5466:
5453:
5450:
5445:
5437:
5429:
5421:
5413:
5405:
5397:
5389:
5381:
5373:
5371:
5358:
5350:
5342:
5339:
5326:
5323:
5318:
5307:
5294:
5291:
5286:
5275:
5262:
5259:
5254:
5243:
5240:
5238:
5231:
5211:
5210:
5209:
5192:
5189:
5178:
5162:
5159:
5145:
5125:
5117:
5109:
5101:
5093:
5085:
5077:
5069:
5061:
5053:
5039:
5038:
5037:
5034:
5032:
5029:
5013:
5010:
5002:
4974:
4958:
4914:
4911:
4908:
4900:
4899:border tensor
4860:
4846:
4832:
4829:
4823:
4820:
4817:
4814:
4811:
4797:
4774:
4771:
4768:
4765:
4762:
4731:
4727:
4722:
4718:
4715:
4712:
4705:
4701:
4696:
4687:
4666:
4662:
4658:
4655:
4652:
4647:
4643:
4636:
4613:
4609:
4600:
4596:
4592:
4589:
4586:
4581:
4577:
4570:
4567:
4564:
4561:
4556:
4552:
4546:
4541:
4538:
4535:
4531:
4526:
4519:
4511:
4507:
4503:
4500:
4497:
4492:
4488:
4473:
4465:
4464:
4463:
4447:
4443:
4439:
4436:
4433:
4428:
4424:
4420:
4415:
4411:
4386:
4382:
4377:
4373:
4370:
4367:
4360:
4356:
4351:
4325:
4321:
4317:
4314:
4311:
4306:
4302:
4287:
4277:
4267:
4250:
4247:
4244:
4241:
4238:
4235:
4232:
4224:
4220:
4216:
4210:
4207:
4204:
4201:
4198:
4195:
4192:
4186:
4166:
4163:
4160:
4157:
4154:
4151:
4148:
4128:
4108:
4105:
4102:
4079:
4076:
4073:
4070:
4067:
4059:
4055:
4051:
4045:
4042:
4039:
4036:
4033:
4027:
4018:
4002:
3999:
3996:
3993:
3990:
3986:
3963:
3960:
3957:
3954:
3951:
3948:
3945:
3941:
3920:
3917:
3909:
3905:
3901:
3898:
3895:
3890:
3886:
3877:
3873:
3869:
3861:
3857:
3853:
3850:
3847:
3842:
3838:
3834:
3831:
3828:
3823:
3819:
3812:
3789:
3786:
3783:
3780:
3777:
3774:
3771:
3761:
3758:
3755:
3752:
3746:
3743:
3740:
3734:
3728:
3725:
3722:
3715:
3707:
3693:
3690:
3687:
3684:
3681:
3678:
3675:
3665:
3662:
3656:
3653:
3650:
3647:
3641:
3635:
3632:
3629:
3626:
3619:
3611:
3610:
3609:
3590:
3586:
3582:
3579:
3576:
3571:
3567:
3558:
3554:
3550:
3542:
3538:
3534:
3531:
3528:
3523:
3519:
3512:
3504:
3485:
3477:
3473:
3469:
3466:
3463:
3458:
3454:
3445:
3441:
3437:
3429:
3425:
3421:
3418:
3415:
3410:
3406:
3399:
3392:
3391:
3390:
3372:
3368:
3364:
3361:
3358:
3353:
3349:
3344:
3335:
3314:
3310:
3306:
3303:
3300:
3295:
3291:
3282:
3278:
3252:
3248:
3244:
3241:
3238:
3233:
3229:
3220:
3216:
3207:
3191:
3171:
3161:
3157:
3153:
3150:
3147:
3142:
3138:
3107:
3101:
3098:
3093:
3089:
3080:
3075:
3072:
3069:
3065:
3061:
3046:
3042:
3036:
3031:
3028:
3025:
3021:
3017:
3007:
3006:
3005:
3003:
2984:
2978:
2975:
2963:
2959:
2951:
2947:
2943:
2940:
2937:
2932:
2928:
2919:
2915:
2907:
2906:
2905:
2903:
2893:
2879:
2859:
2849:
2845:
2841:
2838:
2835:
2830:
2826:
2821:
2812:
2793:
2787:
2783:
2777:
2772:
2769:
2766:
2762:
2758:
2753:
2749:
2744:
2737:
2729:
2725:
2721:
2718:
2715:
2710:
2706:
2699:
2692:
2691:
2690:
2681:
2679:
2660:
2654:
2651:
2646:
2642:
2633:
2628:
2625:
2622:
2618:
2614:
2609:
2605:
2599:
2594:
2591:
2588:
2584:
2580:
2577:
2574:
2569:
2565:
2557:
2556:
2555:
2553:
2535:
2531:
2527:
2524:
2521:
2516:
2512:
2508:
2503:
2499:
2474:
2470:
2465:
2461:
2458:
2455:
2448:
2444:
2439:
2429:
2413:
2403:
2398:
2388:
2383:
2355:
2352:
2349:
2346:
2343:
2321:
2311:
2306:
2296:
2291:
2262:
2252:
2247:
2237:
2232:
2216:
2214:
2210:
2209:typical ranks
2188:
2184:
2180:
2177:
2174:
2169:
2165:
2158:
2138:
2130:
2114:
2107:
2086:
2082:
2078:
2075:
2072:
2067:
2063:
2056:
2032:
2028:
2023:
2019:
2016:
2013:
2006:
2002:
1997:
1976:
1968:
1952:
1927:
1923:
1919:
1916:
1913:
1908:
1904:
1897:
1890:
1880:
1856:
1851:
1835:
1825:
1822:
1817:
1807:
1802:
1772:
1764:
1754:
1749:
1730:
1725:
1715:
1710:
1691:
1686:
1676:
1671:
1647:
1644:
1639:
1625:
1624:
1623:
1621:
1603:
1593:
1588:
1578:
1573:
1543:
1538:
1528:
1523:
1513:
1508:
1498:
1493:
1483:
1478:
1468:
1463:
1453:
1448:
1438:
1433:
1423:
1418:
1408:
1403:
1393:
1388:
1378:
1373:
1363:
1349:
1348:
1347:
1338:
1321:
1308:
1304:
1299:
1295:
1290:
1261:
1251:
1248:
1245:
1240:
1224:
1222:
1218:
1217:matrix pencil
1214:
1210:
1192:
1188:
1184:
1177:
1173:
1168:
1164:
1157:
1153:
1148:
1139:
1129:
1128:
1124:
1120:
1116:
1112:
1108:
1092:
1072:
1052:
1049:
1046:
1043:
1040:
1016:
1012:
999:
994:
991:
988:
954:
949:
945:
921:
916:
913:
910:
900:
897:
894:
889:
886:
883:
873:
870:
865:
862:
859:
849:
844:
841:
838:
828:
823:
820:
817:
805:
801:
795:
790:
787:
784:
780:
776:
762:
761:
760:
746:
726:
690:
686:
682:
679:
676:
671:
667:
663:
658:
654:
641:
620:
602:
598:
594:
591:
586:
582:
578:
575:
572:
567:
563:
559:
554:
550:
516:
512:
508:
505:
500:
496:
492:
489:
486:
481:
477:
473:
468:
464:
459:
434:
430:
426:
423:
418:
414:
410:
407:
402:
398:
394:
389:
385:
374:
347:
314:
267:
253:
250:
247:
244:
241:
219:
215:
211:
206:
202:
198:
195:
175:
172:
169:
166:
163:
154:
140:
120:
106:
104:
103:psychometrics
100:
96:
92:
88:
84:
79:
76:
72:
68:
64:
60:
56:
52:
50:
46:
42:
41:decomposition
40:
34:
30:
19:
12939:Hermann Weyl
12743:Vector space
12728:Pseudotensor
12693:Fiber bundle
12646:abstractions
12541:Mixed tensor
12526:Tensor field
12333:
12259:
12216:
12212:
12190:
12186:
12180:
12137:
12133:
12127:
12084:
12080:
12074:
12044:(1): 39–67.
12041:
12037:
12031:
12001:(1): 29–45.
11998:
11994:
11988:
11963:
11959:
11953:
11928:
11924:
11915:
11890:
11886:
11880:
11837:
11833:
11827:
11784:
11780:
11774:
11731:
11727:
11717:
11682:
11678:
11626:
11622:
11616:
11573:
11569:
11563:
11520:
11516:
11510:
11467:
11463:
11427:
11421:
11404:
11398:
11381:
11346:
11340:
11327:
11308:
11304:
11298:
11294:
11231:
11212:
11206:
11193:
11168:math/0607191
11158:
11152:
11149:Peterson, C.
11113:
11107:
11086:
11065:
11058:Strassen, V.
11052:
11027:math/0607647
11017:
11011:
11004:de Silva, V.
10984:
10975:
10940:
10936:
10923:
10905:
10894:the original
10889:
10885:
10841:
10835:
10831:
10798:
10792:
10783:
10765:
10668:
10293:
10290:Applications
10278:
10243:
10232:
10215:
10202:
10197:
10169:
9836:
9648:
9485:
9271:
9267:
9265:
9225:
8958:
8883:
8722:
8720:
7755:
7724:
7673:every tensor
7672:
7643:
7290:
6440:
6090:identifiable
6089:
5898:
5633:
5590:
5207:
5035:
5027:
5000:
4972:
4898:
4897:is called a
4852:
4686:generic rank
4685:
4628:
4462:, satisfies
4276:maximum rank
4275:
4273:
4270:Maximum rank
4019:
3804:
3502:
3500:
3333:
3122:
3000:
2901:
2899:
2809:
2687:
2677:
2675:
2551:
2550:, is called
2430:
2217:
2208:
1889:generic rank
1888:
1886:
1883:Generic rank
1852:
1787:
1558:
1344:
1225:
1135:
1126:
1122:
1118:
1114:
1110:
1106:
936:
626:
315:
268:
155:
112:
87:econometrics
80:
71:chemometrics
62:
58:
54:
53:
48:
44:
38:
36:
32:
26:
12879:Élie Cartan
12827:Spin tensor
12801:Weyl tensor
12759:Mathematics
12723:Multivector
12514:definitions
12412:Engineering
12351:Mathematics
11840:: 342–375.
11734:(1): 1–11.
11337:Teitler, Z.
10943:(6): 1–39.
5028:approximate
4973:border rank
4849:Border rank
2680:otherwise.
1213:Weierstrass
1132:Tensor rank
67:linguistics
12954:Categories
12708:Linear map
12576:Operations
12284:FactoMineR
12147:1512.04312
12140:: 78–105.
11847:1501.07251
11692:1501.00090
11583:1609.00123
11395:Romani, F.
11215:: 95–120.
10757:References
9942:such that
9409:with
7754:is called
6189:where the
6088:is called
5787:such that
5625:Properties
5131:with
2552:unbalanced
623:Definition
12847:EM tensor
12683:Dimension
12634:Transpose
12229:CiteSeerX
12172:119147635
12164:0926-2245
12111:0747-7171
12094:1105.1229
12087:: 51–71.
12066:121003817
12058:0033-3123
12023:120459386
12015:0886-9383
11980:0003-2700
11945:0895-4798
11907:0169-7439
11872:119729978
11864:0024-3795
11811:0895-4798
11794:1312.2848
11766:119671913
11758:1056-3911
11741:1105.3643
11653:0895-4798
11636:1103.2696
11600:0895-4798
11555:119721371
11547:0373-3114
11530:1303.6915
11494:0895-4798
11477:1403.4157
11391:Lotti, G.
11356:1402.2371
10950:0911.1393
10828:Chang, J.
10621:⊗
10585:⊗
10511:∑
10491:⊗
10478:⊗
10429:⊗
10416:⊗
10184:∞
10181:→
10155:∞
10152:→
10143:‖
10116:⊗
10113:⋯
10110:⊗
10084:⊗
10058:‖
10050:∞
10047:→
10038:‖
10011:⊗
10008:⋯
10005:⊗
9979:⊗
9953:‖
9927:≤
9921:≠
9915:≤
9892:∞
9889:→
9859:→
9799:⊗
9796:⋯
9793:⊗
9767:⊗
9724:∑
9663:∞
9660:→
9629:⊗
9621:⊗
9610:−
9578:⊗
9546:⊗
9468:≠
9465:⟩
9449:⟨
9435:‖
9427:‖
9421:‖
9413:‖
9396:⊗
9388:⊗
9372:⊗
9364:⊗
9348:⊗
9340:⊗
9268:ill-posed
9241:‖
9237:⋅
9234:‖
9202:‖
9181:⊗
9178:⋯
9175:⊗
9155:⊗
9118:∑
9114:−
9104:‖
9082:∈
8930:Σ
8926:Π
8910:and rank
8858:⊗
8845:⊗
8832:⊗
8792:⊗
8779:⊗
8737:∞
8689:…
8615:Σ
8611:Π
8587:…
8489:⊗
8476:⊗
8463:⊗
8450:⊗
8381:⊗
8368:⊗
8299:⊗
8286:⊗
8250:−
8215:≥
8182:⊗
8169:⊗
8156:⊗
8087:⊗
8074:⊗
8035:−
8002:∑
7998:−
7968:∏
7964:≥
7935:−
7902:∑
7898:−
7868:∏
7815:…
7702:∖
7689:∈
7623:Π
7562:⊗
7559:⋯
7556:⊗
7536:⋯
7533:⊗
7513:⊂
7480:≥
7467:≥
7464:⋯
7461:≥
7448:≥
7415:≥
7342:⊗
7339:⋯
7336:⊗
7316:⊗
7226:∖
7199:∈
7063:−
6914:×
6867:∈
6829:⊗
6797:∑
6744:∑
6703:−
6624:∑
6605:⊗
6573:∑
6504:×
6486:≃
6466:⊗
6418:−
6259:∑
6157:…
6059:⊗
6056:⋯
6053:⊗
6033:⊗
6013:∈
5867:λ
5819:λ
5815:⋯
5806:λ
5796:λ
5769:λ
5736:⊗
5733:⋯
5730:⊗
5715:⊗
5685:⊗
5682:⋯
5679:⊗
5664:⊗
5605:∞
5602:→
5567:⊗
5559:⊗
5523:⊗
5515:⊗
5499:⊗
5491:⊗
5475:⊗
5467:⊗
5438:⊗
5430:⊗
5414:⊗
5406:⊗
5390:⊗
5382:⊗
5359:⊗
5351:⊗
5340:−
5308:⊗
5276:⊗
5190:≠
5187:⟩
5171:⟨
5157:‖
5149:‖
5143:‖
5135:‖
5118:⊗
5110:⊗
5094:⊗
5086:⊗
5070:⊗
5062:⊗
5011:≥
4830:≤
4772:×
4766:×
4719:⊗
4716:⋯
4713:⊗
4656:…
4590:…
4568:⋅
4532:∏
4520:≤
4501:…
4440:≥
4437:⋯
4434:≥
4421:≥
4374:⊗
4371:⋯
4368:⊗
4315:…
4245:…
4205:…
4164:×
4161:⋯
4158:×
4152:×
4106:≠
4000:×
3994:×
3961:×
3955:×
3949:×
3899:…
3851:…
3832:…
3790:…
3759:×
3753:×
3735:×
3694:…
3663:×
3642:×
3580:…
3532:…
3467:…
3438:≥
3419:…
3365:×
3362:⋯
3359:×
3304:…
3242:…
3169:∖
3154:×
3151:⋯
3148:×
3099:−
3066:∑
3059:Σ
3022:∏
3015:Π
2973:Σ
2969:Π
2941:…
2857:∖
2842:×
2839:⋯
2836:×
2763:∏
2719:…
2652:−
2619:∑
2615:−
2585:∏
2554:whenever
2528:≥
2525:⋯
2522:≥
2509:≥
2462:⊗
2459:⋯
2456:⊗
2404:⊗
2389:⊗
2353:×
2347:×
2312:⊗
2297:⊗
2253:⊗
2238:⊗
2178:…
2106:dense set
2076:…
2020:⊗
2017:⋯
2014:⊗
1917:…
1755:⊗
1743:¯
1731:⊗
1704:¯
1692:⊗
1677:⊗
1665:¯
1622:, namely
1594:∈
1529:⊗
1514:⊗
1484:⊗
1469:⊗
1454:−
1439:⊗
1424:⊗
1394:⊗
1379:⊗
1316:∖
1296:∈
1252:⊗
1249:⋯
1246:⊗
1209:Kronecker
1185:⊗
1165:⊗
1050:≤
1044:≤
1000:∈
955:∈
946:λ
901:⊗
898:⋯
895:⊗
874:⊗
871:⋯
850:⊗
829:⊗
802:λ
781:∑
683:×
680:…
677:×
664:×
642:∈
595:…
576:…
509:…
490:…
427:…
424:×
411:…
408:×
395:×
375:∈
251:≤
245:≤
212:≤
199:≤
173:≤
167:≤
12713:Manifold
12698:Geodesic
12456:Notation
12251:16074195
12213:SIAM Rev
12119:14181289
11819:14851072
11709:16324593
11661:43781880
11608:23983015
11502:28478606
11387:Bini, D.
11373:14309435
11271:× ··· ×
11247:(2011).
11201:(1985).
11185:59069541
11102:(2002).
11060:(1983).
10983:(2012).
10877:(1970).
10858:50364581
10720:See also
10275:(L-BFGS)
9023:, where
8204:, where
7397:and all
7241:, where
6929:matrix,
6305:and all
5897:for all
4403:, where
3184:, where
2985:⌉
2964:⌈
2902:expected
2872:, where
2678:balanced
2491:, where
109:Notation
12965:Tensors
12810:Physics
12644:Related
12407:Physics
12325:Tensors
12221:Bibcode
11141:Abo, H.
11044:7159193
11008:Lim, L.
10967:1460452
10933:Lim, L.
9259:is the
8999:tensor
8723:complex
5998:tensor
5619:W state
4873:tensor
4853:A rank-
2104:form a
1138:NP-hard
234:where
12738:Vector
12733:Spinor
12718:Matrix
12512:Tensor
12262:. AMS.
12249:
12231:
12170:
12162:
12117:
12109:
12064:
12056:
12021:
12013:
11978:
11943:
11905:
11870:
11862:
11817:
11809:
11764:
11756:
11707:
11659:
11651:
11606:
11598:
11553:
11545:
11500:
11492:
11442:
11371:
11183:
11042:
10987:. AMS.
10965:
10856:
10257:(SGSD)
9226:where
7495:. Let
6856:where
5001:always
4629:where
1788:where
1559:where
1033:where
937:where
101:, and
31:, the
12658:Basis
12343:Scope
12247:S2CID
12168:S2CID
12142:arXiv
12115:S2CID
12089:arXiv
12062:S2CID
12019:S2CID
11868:S2CID
11842:arXiv
11815:S2CID
11789:arXiv
11762:S2CID
11736:arXiv
11705:S2CID
11687:arXiv
11657:S2CID
11631:arXiv
11604:S2CID
11578:arXiv
11551:S2CID
11525:arXiv
11498:S2CID
11472:arXiv
11369:S2CID
11351:arXiv
11181:S2CID
11163:arXiv
11040:S2CID
11022:arXiv
10963:S2CID
10945:arXiv
10915:(PDF)
10897:(PDF)
10882:(PDF)
10854:S2CID
10775:(PDF)
10352:of a
10269:(NCG)
10229:(ASD)
10223:(ALS)
7629:15000
7371:with
4951:. If
1117:, or
37:rank-
12160:ISSN
12107:ISSN
12054:ISSN
12011:ISSN
11976:ISSN
11941:ISSN
11903:ISSN
11860:ISSN
11807:ISSN
11754:ISSN
11683:2019
11649:ISSN
11596:ISSN
11543:ISSN
11490:ISSN
11440:ISBN
11301:≥ 5"
10263:(LM)
10251:(SD)
9034:<
8921:<
8895:>
8814:and
8429:; or
7864:>
7773:>
7626:<
7382:>
7273:>
7119:and
6539:>
5843:and
4912:<
4274:The
3501:The
2900:The
2575:>
1887:The
1107:rank
974:and
69:and
12239:doi
12152:doi
12099:doi
12046:doi
12003:doi
11968:doi
11933:doi
11895:doi
11852:doi
11838:513
11799:doi
11746:doi
11697:doi
11641:doi
11588:doi
11535:doi
11521:193
11482:doi
11432:doi
11409:doi
11361:doi
11347:362
11313:doi
11217:doi
11173:doi
11159:361
11118:doi
11114:355
11072:doi
11032:doi
10955:doi
10846:doi
10803:doi
10170:as
9649:as
9061:min
5591:as
4975:of
4803:max
4688:of
4523:min
4479:max
4343:of
4293:max
3054:and
2741:min
1857:to
533:or
105:.
35:or
27:In
12956::
12245:.
12237:.
12227:.
12217:51
12215:.
12191:15
12189:.
12166:.
12158:.
12150:.
12138:55
12136:.
12113:.
12105:.
12097:.
12085:52
12083:.
12060:.
12052:.
12042:45
12040:.
12017:.
12009:.
11997:.
11974:.
11964:57
11962:.
11939:.
11929:14
11927:.
11901:.
11891:55
11889:.
11866:.
11858:.
11850:.
11836:.
11813:.
11805:.
11797:.
11785:35
11783:.
11760:.
11752:.
11744:.
11732:22
11730:.
11726:.
11703:.
11695:.
11681:.
11669:^
11655:.
11647:.
11639:.
11627:33
11625:.
11602:.
11594:.
11586:.
11574:38
11572:.
11549:.
11541:.
11533:.
11519:.
11496:.
11488:.
11480:.
11468:35
11466:.
11454:^
11438:.
11403:.
11393:;
11389:;
11367:.
11359:.
11345:.
11335:;
11309:20
11307:.
11303:.
11243:;
11239:;
11213:69
11211:.
11205:.
11179:.
11171:.
11157:.
11147:;
11143:;
11132:^
11112:.
11106:.
11098:;
11094:;
11064:.
11038:.
11030:.
11018:30
11016:.
11006:;
10993:^
10961:.
10953:.
10941:60
10939:.
10931:;
10890:16
10888:.
10884:.
10866:^
10852:.
10842:35
10840:.
10826:;
10815:^
10797:.
10666:.
9263:.
7047:,
6988:,
6433:.
5621:.
5193:1.
2215:.
1850:.
1337:.
1223:.
1125:,
1113:,
619:.
313:.
266:.
153:.
97:,
93:,
89:,
12317:e
12310:t
12303:v
12290:)
12288:R
12253:.
12241::
12223::
12174:.
12154::
12144::
12121:.
12101::
12091::
12068:.
12048::
12025:.
12005::
11999:4
11982:.
11970::
11947:.
11935::
11909:.
11897::
11874:.
11854::
11844::
11821:.
11801::
11791::
11768:.
11748::
11738::
11711:.
11699::
11689::
11663:.
11643::
11633::
11610:.
11590::
11580::
11557:.
11537::
11527::
11504:.
11484::
11474::
11448:.
11434::
11415:.
11411::
11405:9
11375:.
11363::
11353::
11321:.
11315::
11299:n
11295:n
11293:(
11280:P
11258:P
11225:.
11219::
11187:.
11175::
11165::
11126:.
11120::
11080:.
11074::
11046:.
11034::
11024::
10969:.
10957::
10947::
10860:.
10848::
10832:n
10809:.
10805::
10799:6
10777:.
10704:]
10701:i
10698:=
10695:h
10691:|
10687:x
10684:[
10681:E
10654:]
10651:i
10648:=
10645:h
10641:|
10635:3
10631:x
10627:[
10624:E
10618:]
10615:i
10612:=
10609:h
10605:|
10599:2
10595:x
10591:[
10588:E
10582:]
10579:i
10576:=
10573:h
10569:|
10563:1
10559:x
10555:[
10552:E
10549:)
10546:i
10543:=
10540:h
10537:(
10534:r
10531:P
10526:k
10521:1
10518:=
10515:i
10507:=
10504:]
10499:3
10495:x
10486:2
10482:x
10473:1
10469:x
10465:[
10462:E
10442:]
10437:3
10433:x
10424:2
10420:x
10411:1
10407:x
10403:[
10400:E
10380:h
10360:k
10338:3
10334:x
10330:,
10325:2
10321:x
10317:,
10312:1
10308:x
10178:n
10147:F
10137:M
10132:n
10129:,
10126:j
10121:a
10105:2
10100:n
10097:,
10094:j
10089:a
10079:1
10074:n
10071:,
10068:j
10063:a
10042:F
10032:M
10027:n
10024:,
10021:i
10016:a
10000:2
9995:n
9992:,
9989:i
9984:a
9974:1
9969:n
9966:,
9963:i
9958:a
9930:r
9924:j
9918:i
9912:1
9886:n
9864:A
9854:n
9848:A
9820:M
9815:n
9812:,
9809:i
9804:a
9788:2
9783:n
9780:,
9777:i
9772:a
9762:1
9757:n
9754:,
9751:i
9746:a
9739:r
9734:1
9731:=
9728:i
9720:=
9715:n
9709:A
9683:r
9657:n
9633:u
9625:u
9617:u
9613:n
9607:)
9603:v
9597:n
9594:1
9589:+
9585:u
9581:(
9575:)
9571:v
9565:n
9562:1
9557:+
9553:u
9549:(
9543:)
9539:v
9533:n
9530:1
9525:+
9521:u
9517:(
9514:n
9511:=
9506:n
9500:A
9471:1
9461:v
9457:,
9453:u
9441:1
9438:=
9431:v
9424:=
9417:u
9404:,
9400:u
9392:u
9384:v
9380:+
9376:u
9368:v
9360:u
9356:+
9352:v
9344:u
9336:u
9332:=
9327:A
9302:r
9282:r
9245:F
9211:,
9206:F
9196:M
9191:i
9186:a
9170:2
9165:i
9160:a
9150:1
9145:i
9140:a
9133:r
9128:1
9125:=
9122:i
9109:A
9095:m
9091:I
9086:F
9077:m
9072:i
9067:a
9037:s
9031:r
9009:A
8987:s
8967:r
8936:1
8933:+
8918:r
8898:2
8892:M
8880:.
8866:2
8862:F
8853:2
8849:F
8840:2
8836:F
8827:3
8823:F
8800:3
8796:F
8787:4
8783:F
8774:5
8770:F
8717:.
8705:)
8700:M
8696:I
8692:,
8686:,
8681:2
8677:I
8673:,
8668:1
8664:I
8660:(
8655:E
8651:r
8647:=
8644:r
8621:1
8618:+
8606:=
8603:)
8598:M
8594:I
8590:,
8584:,
8579:2
8575:I
8571:,
8566:1
8562:I
8558:(
8553:E
8549:r
8537:.
8525:5
8522:=
8519:r
8497:2
8493:F
8484:2
8480:F
8471:2
8467:F
8458:2
8454:F
8445:2
8441:F
8417:8
8414:=
8411:r
8389:3
8385:F
8376:6
8372:F
8363:6
8359:F
8347:;
8335:6
8332:=
8329:r
8307:4
8303:F
8294:4
8290:F
8281:4
8277:F
8265:;
8253:1
8247:n
8244:2
8241:=
8238:r
8218:2
8212:n
8190:2
8186:F
8177:2
8173:F
8164:n
8160:F
8151:n
8147:F
8135:;
8123:5
8120:=
8117:r
8095:3
8091:F
8082:4
8078:F
8069:4
8065:F
8053:;
8041:)
8038:1
8030:m
8026:I
8022:(
8017:M
8012:2
8009:=
8006:m
7993:m
7989:I
7983:M
7978:2
7975:=
7972:m
7961:r
7941:)
7938:1
7930:m
7926:I
7922:(
7917:M
7912:2
7909:=
7906:m
7893:m
7889:i
7883:M
7878:2
7875:=
7872:m
7859:1
7855:I
7843:;
7831:)
7826:M
7822:I
7818:,
7812:,
7807:2
7803:I
7799:,
7794:1
7790:I
7786:(
7781:E
7777:r
7770:r
7740:r
7736:S
7727:(
7710:r
7706:Z
7697:r
7693:S
7684:A
7657:r
7653:Z
7599:r
7575:M
7571:I
7566:F
7549:m
7545:I
7540:F
7526:1
7522:I
7517:F
7508:r
7504:S
7483:2
7475:M
7471:I
7456:2
7452:I
7443:1
7439:I
7418:2
7410:m
7406:I
7385:2
7379:M
7355:M
7351:I
7346:F
7329:2
7325:I
7320:F
7309:1
7305:I
7300:F
7276:1
7270:r
7249:Z
7229:Z
7223:)
7220:F
7217:(
7212:n
7207:L
7204:G
7196:X
7174:r
7169:1
7166:=
7163:i
7159:]
7153:i
7148:d
7143:[
7140:=
7135:T
7131:X
7127:B
7105:r
7100:1
7097:=
7094:i
7090:]
7084:i
7079:c
7074:[
7071:=
7066:1
7059:X
7055:A
7033:r
7028:1
7025:=
7022:i
7018:]
7012:i
7007:b
7002:[
6999:=
6996:B
6974:r
6969:1
6966:=
6963:i
6959:]
6953:i
6948:a
6943:[
6940:=
6937:A
6917:r
6911:r
6891:)
6888:F
6885:(
6880:r
6875:L
6872:G
6864:X
6844:,
6839:i
6834:d
6824:i
6819:c
6812:r
6807:1
6804:=
6801:i
6793:=
6788:T
6783:i
6778:d
6771:i
6766:c
6759:r
6754:1
6751:=
6748:i
6740:=
6735:T
6731:)
6725:T
6721:X
6717:B
6714:(
6711:)
6706:1
6699:X
6695:A
6692:(
6689:=
6684:T
6680:B
6676:A
6673:=
6668:T
6663:i
6658:b
6651:i
6646:a
6639:r
6634:1
6631:=
6628:i
6620:=
6615:i
6610:b
6600:i
6595:a
6588:r
6583:1
6580:=
6577:i
6569:=
6564:A
6542:1
6536:r
6512:2
6508:I
6499:1
6495:I
6490:F
6479:2
6475:I
6470:F
6459:1
6455:I
6450:F
6421:1
6415:r
6393:A
6369:i
6363:A
6338:A
6316:!
6313:r
6293:,
6288:i
6282:A
6274:r
6269:1
6266:=
6263:i
6255:=
6250:A
6228:r
6206:i
6200:A
6177:}
6172:r
6166:A
6160:,
6154:,
6149:2
6143:A
6137:,
6132:1
6126:A
6120:{
6100:r
6072:M
6068:I
6063:F
6046:2
6042:I
6037:F
6026:1
6022:I
6017:F
6008:A
5986:r
5964:A
5940:M
5935:1
5932:=
5929:m
5925:}
5919:m
5914:a
5909:{
5899:m
5883:m
5878:b
5871:m
5863:=
5858:m
5853:a
5831:1
5828:=
5823:M
5810:2
5800:1
5773:k
5746:M
5741:b
5725:2
5720:b
5710:1
5705:b
5700:=
5695:M
5690:a
5674:2
5669:a
5659:1
5654:a
5649:=
5644:A
5599:m
5571:v
5563:v
5555:v
5547:2
5543:m
5539:1
5534:+
5531:)
5527:u
5519:v
5511:v
5507:+
5503:v
5495:u
5487:v
5483:+
5479:v
5471:v
5463:u
5459:(
5454:m
5451:1
5446:+
5442:u
5434:u
5426:v
5422:+
5418:u
5410:v
5402:u
5398:+
5394:v
5386:u
5378:u
5374:=
5363:u
5355:u
5347:u
5343:m
5337:)
5333:v
5327:m
5324:1
5319:+
5315:u
5311:(
5305:)
5301:v
5295:m
5292:1
5287:+
5283:u
5279:(
5273:)
5269:v
5263:m
5260:1
5255:+
5251:u
5247:(
5244:m
5241:=
5232:m
5226:A
5183:v
5179:,
5175:u
5163:1
5160:=
5153:v
5146:=
5139:u
5126:,
5122:u
5114:u
5106:v
5102:+
5098:u
5090:v
5082:u
5078:+
5074:v
5066:u
5058:u
5054:=
5049:A
5014:3
4985:A
4959:r
4937:A
4915:s
4909:r
4883:A
4861:s
4833:4
4827:)
4824:2
4821:,
4818:2
4815:,
4812:2
4809:(
4798:r
4775:2
4769:2
4763:2
4758:R
4732:M
4728:I
4723:F
4706:1
4702:I
4697:F
4672:)
4667:M
4663:I
4659:,
4653:,
4648:1
4644:I
4640:(
4637:r
4614:,
4610:}
4606:)
4601:M
4597:I
4593:,
4587:,
4582:1
4578:I
4574:(
4571:r
4565:2
4562:,
4557:m
4553:I
4547:M
4542:2
4539:=
4536:m
4527:{
4517:)
4512:M
4508:I
4504:,
4498:,
4493:1
4489:I
4485:(
4474:r
4448:M
4444:I
4429:2
4425:I
4416:1
4412:I
4387:M
4383:I
4378:F
4361:1
4357:I
4352:F
4331:)
4326:M
4322:I
4318:,
4312:,
4307:1
4303:I
4299:(
4288:r
4254:)
4251:2
4248:,
4242:,
4239:2
4236:,
4233:2
4230:(
4225:E
4221:r
4217:=
4214:)
4211:2
4208:,
4202:,
4199:2
4196:,
4193:2
4190:(
4187:r
4167:2
4155:2
4149:2
4129:s
4109:3
4103:n
4083:)
4080:n
4077:,
4074:n
4071:,
4068:n
4065:(
4060:E
4056:r
4052:=
4049:)
4046:n
4043:,
4040:n
4037:,
4034:n
4031:(
4028:r
4003:3
3997:4
3991:4
3987:F
3964:2
3958:2
3952:2
3946:2
3942:F
3921:1
3918:+
3915:)
3910:M
3906:I
3902:,
3896:,
3891:1
3887:I
3883:(
3878:E
3874:r
3870:=
3867:)
3862:M
3858:I
3854:,
3848:,
3843:m
3839:I
3835:,
3829:,
3824:1
3820:I
3816:(
3813:r
3787:,
3784:3
3781:,
3778:2
3775:=
3772:m
3762:2
3756:2
3750:)
3747:1
3744:+
3741:m
3738:(
3732:)
3729:1
3726:+
3723:m
3720:(
3716:F
3691:,
3688:2
3685:,
3682:1
3679:=
3676:m
3666:3
3660:)
3657:1
3654:+
3651:m
3648:2
3645:(
3639:)
3636:1
3633:+
3630:m
3627:2
3624:(
3620:F
3596:)
3591:M
3587:I
3583:,
3577:,
3572:1
3568:I
3564:(
3559:E
3555:r
3551:=
3548:)
3543:M
3539:I
3535:,
3529:,
3524:1
3520:I
3516:(
3513:r
3486:.
3483:)
3478:M
3474:I
3470:,
3464:,
3459:1
3455:I
3451:(
3446:E
3442:r
3435:)
3430:M
3426:I
3422:,
3416:,
3411:1
3407:I
3403:(
3400:r
3373:M
3369:I
3354:1
3350:I
3345:F
3320:)
3315:M
3311:I
3307:,
3301:,
3296:1
3292:I
3288:(
3283:E
3279:r
3258:)
3253:M
3249:I
3245:,
3239:,
3234:1
3230:I
3226:(
3221:E
3217:r
3192:Z
3172:Z
3162:M
3158:I
3143:1
3139:I
3133:C
3108:.
3105:)
3102:1
3094:m
3090:I
3086:(
3081:M
3076:1
3073:=
3070:m
3062:=
3047:m
3043:I
3037:M
3032:1
3029:=
3026:m
3018:=
2979:1
2976:+
2960:=
2957:)
2952:M
2948:I
2944:,
2938:,
2933:1
2929:I
2925:(
2920:E
2916:r
2880:Z
2860:Z
2850:M
2846:I
2831:1
2827:I
2822:F
2794:}
2788:m
2784:I
2778:M
2773:2
2770:=
2767:m
2759:,
2754:1
2750:I
2745:{
2738:=
2735:)
2730:M
2726:I
2722:,
2716:,
2711:1
2707:I
2703:(
2700:r
2661:,
2658:)
2655:1
2647:m
2643:I
2639:(
2634:M
2629:2
2626:=
2623:m
2610:m
2606:I
2600:M
2595:2
2592:=
2589:m
2581:+
2578:1
2570:1
2566:I
2536:M
2532:I
2517:2
2513:I
2504:1
2500:I
2475:M
2471:I
2466:F
2449:1
2445:I
2440:F
2414:2
2409:R
2399:2
2394:R
2384:2
2379:R
2356:2
2350:2
2344:2
2322:2
2317:C
2307:2
2302:C
2292:2
2287:C
2263:2
2258:R
2248:2
2243:R
2233:2
2228:R
2194:)
2189:M
2185:I
2181:,
2175:,
2170:1
2166:I
2162:(
2159:r
2139:S
2115:S
2092:)
2087:M
2083:I
2079:,
2073:,
2068:1
2064:I
2060:(
2057:r
2033:M
2029:I
2024:F
2007:1
2003:I
1998:F
1977:r
1953:r
1933:)
1928:M
1924:I
1920:,
1914:,
1909:1
1905:I
1901:(
1898:r
1866:C
1836:k
1831:y
1826:i
1823:+
1818:k
1813:x
1808:=
1803:k
1798:z
1773:,
1770:)
1765:3
1760:z
1750:2
1739:z
1726:1
1721:z
1716:+
1711:3
1700:z
1687:2
1682:z
1672:1
1661:z
1653:(
1648:2
1645:1
1640:=
1635:A
1604:2
1599:R
1589:j
1584:y
1579:,
1574:i
1569:x
1544:,
1539:3
1534:x
1524:2
1519:y
1509:1
1504:y
1499:+
1494:3
1489:y
1479:2
1474:x
1464:1
1459:y
1449:3
1444:y
1434:2
1429:y
1419:1
1414:x
1409:+
1404:3
1399:x
1389:2
1384:x
1374:1
1369:x
1364:=
1359:A
1325:}
1322:0
1319:{
1309:m
1305:I
1300:F
1291:m
1286:a
1262:M
1257:a
1241:1
1236:a
1211:–
1193:2
1189:F
1178:n
1174:I
1169:F
1158:m
1154:I
1149:F
1093:R
1073:R
1053:M
1047:m
1041:1
1017:m
1013:I
1006:F
995:r
992:,
989:m
984:a
960:R
950:r
922:,
917:r
914:,
911:C
906:a
890:r
887:,
884:c
879:a
866:r
863:,
860:2
855:a
845:r
842:,
839:1
834:a
824:r
821:,
818:0
813:a
806:r
796:R
791:1
788:=
785:r
777:=
772:A
747:r
727:R
717:C
715:=
713:M
709:M
691:C
687:I
672:1
668:I
659:0
655:I
648:F
637:A
603:M
599:i
592:,
587:m
583:i
579:,
573:,
568:2
564:i
560:,
555:1
551:i
544:A
517:M
513:i
506:,
501:m
497:i
493:,
487:,
482:2
478:i
474:,
469:1
465:i
460:a
435:M
431:I
419:m
415:I
403:2
399:I
390:1
386:I
380:C
370:A
348:M
326:A
300:A
278:a
254:M
248:m
242:1
220:m
216:I
207:m
203:i
196:1
176:I
170:i
164:1
141:A
121:a
63:K
59:K
49:R
45:R
39:R
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.