## Symmetric tensors and symmetric tensor rank (2006)

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Venue: | Scientific Computing and Computational Mathematics (SCCM |

Citations: | 41 - 18 self |

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@TECHREPORT{Comon06symmetrictensors,

author = {Pierre Comon and Gene Golub and Lek-heng Lim and Bernard Mourrain},

title = {Symmetric tensors and symmetric tensor rank},

institution = {Scientific Computing and Computational Mathematics (SCCM},

year = {2006}

}

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### Abstract

Abstract. A symmetric tensor is a higher order generalization of a symmetric matrix. In this paper, we study various properties of symmetric tensors in relation to a decomposition into a symmetric sum of outer product of vectors. A rank-1 order-k tensor is the outer product of k non-zero vectors. Any symmetric tensor can be decomposed into a linear combination of rank-1 tensors, each of them being symmetric or not. The rank of a symmetric tensor is the minimal number of rank-1 tensors that is necessary to reconstruct it. The symmetric rank is obtained when the constituting rank-1 tensors are imposed to be themselves symmetric. It is shown that rank and symmetric rank are equal in a number of cases, and that they always exist in an algebraically closed field. We will discuss the notion of the generic symmetric rank, which, due to the work of Alexander and Hirschowitz, is now known for any values of dimension and order. We will also show that the set of symmetric tensors of symmetric rank at most r is not closed, unless r = 1. Key words. Tensors, multiway arrays, outer product decomposition, symmetric outer product decomposition, candecomp, parafac, tensor rank, symmetric rank, symmetric tensor rank, generic symmetric rank, maximal symmetric rank, quantics AMS subject classifications. 15A03, 15A21, 15A72, 15A69, 15A18 1. Introduction. We

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68 |
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Citation Context ... the coordinate array representing the tensor be invariant under all permutations of indices, as in Definition 3.1. Many authors have persistently mislabeled the latter a ‘supersymmetric tensor’ (cf. =-=[10, 34, 45]-=-). In fact, we have found that even the classical definition of a symmetric tensor is not as well-known as it should be. We see this as an indication of the need to inform our target readership. It is... |

56 |
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Citation Context ...ructive algorithm to compute the symmetric outer product decomposition has only been proposed recently [35]. The simplest case of binary quantics (n = 2) has also been known for more than two decades =-=[61, 16, 38]-=- — a result that is used in real world engineering problems [15]. 5. Rank and symmetric rank. Let RS(k, n) be the generic symmetric rank and RS(k, n) be the maximally attainable symmetric rank in the ... |

56 |
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Citation Context ...osition of degree-k quantics into a sum of linear forms raised to the kth power. For a long time, it was believed that there was no explicit expression for the generic rank. As Reznick pointed out in =-=[47]-=-, Clebsh proved that even when the numbers of free parameters are the same on both sides of the symmetric outer product decomposition, the generic rank may not be equal to 1 � � n+k−1 n k . For exampl... |

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Using algebraic geometry. Graduate Texts
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Citation Context ...asic result in algebraic geometry, stating that the linear space generated by points of an algebraic variety that is not included in a hyperplane, i.e. a subspace of codimension 1, is the whole space =-=[27, 18, 48]-=-. For completeness, a proof of our special case is given above. Note that it follows from the proof that � � n + k − 1 rankS(A) ≤ k for all A ∈ S k (C n ). On the other hand, given a symmetric tensor ... |

50 |
Supersymmetry for Mathematicians an Introduction
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Citation Context ...34, 45] among others); this terminology should be avoided since it refers to an entirely different class of tensors [7]. The word ‘supersymmetric’ has always been used in both mathematics and physics =-=[25, 60, 62]-=- to describe objects with a Z2-grading and so using it in the sense of [10, 34, 45] is both inconsistent and confusing (the correct usage will be one in the sense of [7]). In fact, we will show below ... |

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42 |
and optimal computation of generic tensors
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Citation Context ...≡ 0. Thus either f ≡ 0 or g ≡ 0 on Yr or equivalently on Yr, which proves that Yr is an irreducible variety. For more details on properties of parameterized varieties, see [18]. See also the proof of =-=[51, 9]-=- for third order tensors. Lemma 6.4. We have RS = min{r | Yr = Yr+1}. Proof. Suppose that there exists r < RS such that Yr = Yr+1. Then since Yr ⊆ Yr + Y1 ⊆ Yr+1 = Yr, we have Yr = Yr + Y1 = Yr + Y1 +... |

37 | Elements of Mathematics - Bourbaki - 1966 |

35 | Rank, decomposition, and uniqueness for 3-way and n-way arrays - Kruskal - 1989 |

34 |
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Citation Context ... . For example, in the case (k, n) = (4, 3), there are � � 6 4 = 15 degrees of freedom but the generic symmetric rank RS(4, 3) = 6 �= 5 = 1 � � 6 3 4 . In fact, this holds true over both R [47] and C =-=[22]-=-. In Section 7, we will see that the generic rank in Sk (Cn ) is now known for any order and dimension due to the ground breaking work of Alexander and Hirschowitz. The special case of cubics (k = 3) ... |

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Citation Context ...34, 45] among others); this terminology should be avoided since it refers to an entirely different class of tensors [7]. The word ‘supersymmetric’ has always been used in both mathematics and physics =-=[25, 60, 62]-=- to describe objects with a Z2-grading and so using it in the sense of [10, 34, 45] is both inconsistent and confusing (the correct usage will be one in the sense of [7]). In fact, we will show below ... |

29 |
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Citation Context ...gnal processing. The decomposition of a tensor into an (asymmetric) outer product of vectors and the corresponding notion of tensor rank was first introduced and studied by Frank L. Hitchcock in 1927 =-=[29, 30]-=-. This same decomposition was rediscovered in the 1970s by psychometricians in their attempts to define data analytic models that generalize factor analysis to multiway data [59]. The name candecomp, ... |

29 |
Multiple invariants and generalized rank of a p-way matrix or tensor
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Citation Context ...gnal processing. The decomposition of a tensor into an (asymmetric) outer product of vectors and the corresponding notion of tensor rank was first introduced and studied by Frank L. Hitchcock in 1927 =-=[29, 30]-=-. This same decomposition was rediscovered in the 1970s by psychometricians in their attempts to define data analytic models that generalize factor analysis to multiway data [59]. The name candecomp, ... |

29 | C.R.: Characterization problems in Mathematical Statistics - Kagan, Linmik, et al. - 1973 |

29 | Eigenvalues of a real supersymmetric tensor
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Citation Context ... the coordinate array representing the tensor be invariant under all permutations of indices, as in Definition 3.1. Many authors have persistently mislabeled the latter a ‘supersymmetric tensor’ (cf. =-=[10, 34, 45]-=-). In fact, we have found that even the classical definition of a symmetric tensor is not as well-known as it should be. We see this as an indication of the need to inform our target readership. It is... |

26 | O(n2.7799) complexity for n × n approximate matrix multiplication - Bini, Capovani, et al. - 1979 |

26 | Basic Algebraic Geometry - Shafarevitch - 1974 |

23 |
Algebra (Chapters 1-3), Elements of Mathematics
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Citation Context ...d the asymmetric version of the outer product decomposition defined in (4.1) is central to multiway factor analysis [50]. In Sections 2 and 3, we discuss some classical results in multilinear algebra =-=[5, 26, 39, 42, 44, 63]-=- and algebraic geometry [27, 64]. While these background materials are well-known to many pure mathematicians, we found that practitioners and applied mathematicians (in signal processing, neuroimagin... |

22 |
La méthode d’Horace éclatée: Application à l’interpolation en degré quatre
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(Show Context)
Citation Context ..., 5), (4, 3), (4, 4), (4, 5)}, where it should be increased by 1. This theorem is extremely complicated to prove, and the interested reader should refer to the two papers of Alexander and Hirschowitz =-=[1, 2]-=-. Simplifications to this proof have also been recently proposed in [12]. It is worth noting that these results have been proved in terms of multivariate polynomials and interpolation theory, and not ... |

21 |
An Introduction to Homological Algebra
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Citation Context ...d the asymmetric version of the outer product decomposition defined in (4.1) is central to multiway factor analysis [50]. In Sections 2 and 3, we discuss some classical results in multilinear algebra =-=[5, 26, 39, 42, 44, 63]-=- and algebraic geometry [27, 64]. While these background materials are well-known to many pure mathematicians, we found that practitioners and applied mathematicians (in signal processing, neuroimagin... |

18 | A Newton-Grassmann Method for Computing the Best Multilinear Rank-(r1, r2, r3) Approximation of a Tensor
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(Show Context)
Citation Context ... More precisely, if A and B are of orders k and ℓ respectively, this yields for p = 1 the array C = A•1B of order k + ℓ − 2: � = ci 2 ···ikj 2 ···jℓ α aαi 2 ···ik bαj 2 ···jℓ . Note that some authors =-=[20, 24, 59]-=- denoted this contraction product as A ×p B or 〈A, B〉p. By convention, when the contraction is between a tensor and a matrix, it is convenient to assume that the summation is always done on the second... |

18 | Global properties of tensor rank - Howell - 1978 |

18 | Computation of canonical forms for ternary cubics
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(Show Context)
Citation Context ...ubics (k = 3) is much better known — a complete classification is known since 1964 though a constructive algorithm to compute the symmetric outer product decomposition has only been proposed recently =-=[35]-=-. The simplest case of binary quantics (n = 2) has also been known for more than two decades [61, 16, 38] — a result that is used in real world engineering problems [15]. 5. Rank and symmetric rank. L... |

17 | On the rank of a binary form
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(Show Context)
Citation Context ...able by symmetric tensors of order k and dimension n = 2 is k, i.e. RS(k, 2) = k. One can say that such symmetric tensors lie on a tangent line to the Veronese variety of symmetric rank-1 tensors. In =-=[13]-=-, an algorithm has been proposed to decompose binary forms when their rank is not larger than k/2; however, this algorithm would not have found the decompositions above since the symmetric ranks of A3... |

17 |
Simplicity of core arrays in three-way principal component analysis and the typical rank of p× q × 2 arrays
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(Show Context)
Citation Context ...rank if Zr is dense with the Zariski topology, i.e. if Zr = S k (C n ). When a typical rank is unique, it may be called generic. We used the wording “typical” in agreement with previous terminologies =-=[9, 55, 57]-=-. Since two dense algebraic sets always intersect over C, there can only be one typical rank over C, and hence is generic. In the remainder of this section, we will write RS = RS(k, n) and RS = RS(k, ... |

17 |
Tensor Space and Exterior Algebra
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(Show Context)
Citation Context |

12 |
Typical rank and Indscal dimensionality for symmetric three-way arrays of order
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- 2004
(Show Context)
Citation Context ...ding a symmetric rank, rankS(A). Constraintss10 P. COMON, G.H. GOLUB, L.-H. LIM, B. MOURRAIN other than full symmetry may be relevant in some application areas, such as partial symmetry as in indscal =-=[11, 57]-=-, or positivity/non-negativity [41, 50, 54]. The definition of symmetric rank is not vacuous because of the following result. Lemma 4.2. Let A ∈ S k (C n ). Then there exist y1, . . . , ys ∈ C n such ... |

11 | identification and source separation in 2 × 3 under-determined mixtures - Comon, Blind - 2004 |

11 |
The typical rank of tall three-way arrays
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(Show Context)
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