## 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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Citation Context ...posed problem in general, since the set of symmetric tensors of symmetric rank not more than r is not closed, unless r = 1 (see Sections 6 and 8) — a fact that parallels the illposedness discussed in =-=[21]-=-. The focus of this paper is mainly on symmetric tensors. The asymmetric case will be addressed in a companion paper, and will use similar tools borrowed from algebraic geometry. Symmetric tensors for... |

73 |
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Citation Context ...ecomposition is particularly important in the process of blind identification of under-determined mixtures (UDM), i.e. linear mixtures with more inputs than observable outputs. We refer the reader to =-=[14, 17, 20, 49, 50]-=- and references therein for a list of other application areas, including speech, mobile communications, machine learning, factor analysis of k-way arrays, biomedical engineering, psychometrics, and ch... |

69 |
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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... |

69 | Giannakis, “Parallel factor analysis in sensor array processing
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Citation Context ...ecomposition is particularly important in the process of blind identification of under-determined mixtures (UDM), i.e. linear mixtures with more inputs than observable outputs. We refer the reader to =-=[14, 17, 20, 49, 50]-=- and references therein for a list of other application areas, including speech, mobile communications, machine learning, factor analysis of k-way arrays, biomedical engineering, psychometrics, and ch... |

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Citation Context ...p(j) , (3.4) where for every j = (j1, . . . , jk), one associates bijectively the nonnegative integer vector p(j) = (p1(j), . . . , pn(j)) with pj(j) counting the number of times index j appears in j =-=[16, 14]-=-. We have in particular |p(j)| = k. The converse is true as well, and the correspondence between symmetric tensors and homogeneous polynomials is obviously bijective. Thus S k (C n ) ∼ = C[x1, . . . ,... |

58 | Super-symmetric decomposition of the fourthorder cumulant tensor. Blind identification of more sources than sensors
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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... |

57 |
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(Show Context)
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 | On the best rank-1 approximation of higher-order supersymmetric tensors
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(Show Context)
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 |
sums of even powers of real linear forms
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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 ... |

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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 +... |

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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- 2001
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Citation Context |

30 | Blind identification of underdetermined mixtures based on the characteristic function - COMON, RAJIH - 2006 |

30 |
Five lectures on supersymmetry
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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 ... |

30 | Eigenvalues of a real supersymmetric tensor
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- 2005
(Show Context)
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... |

29 |
The Expression of a Tensor or a Polyadic as a Sum of Products
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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
- Hitchcock
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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, ... |

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

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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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- 1992
(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 ... |

22 |
An Introduction to Homological Algebra
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(Show Context)
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
- Eldén, Savas
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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
- Kogan, Maza
- 2002
(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
- COMAS, SEIGUER
- 2001
(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... |

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

9 |
Bounds on the ranks of some 3-tensors
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- 1980
(Show Context)
Citation Context ... This is useful in signal processing for instance, where cumulant tensors are estimated from actual data, and are asymptotically Gaussian distributed [6, 43]. These statements extend previous results =-=[3]-=-, and prove that there can be only one subset Zr of non-empty interior, and that the latter is dense in S k (C n ); this result, however, requires that we work over an algebraically closed field such ... |

9 |
Multilinear algebra, 2nd Ed
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(Show Context)
Citation Context |

7 |
An Addendum to Kronecker’s Theory of Pencils
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(Show Context)
Citation Context ...essed as A ′ = A•1L•2M•3N. An alternative notation for (2.1) from the theory of group actions is A ′ = (L, M, N) · A, which may be viewed as multiplying A on ‘three sides’ by the matrices L, M, and N =-=[21, 32]-=-. 3. Symmetric arrays and symmetric tensors. We shall say that a k-way array is cubical if all its k dimensions are identical, i.e. n1 = · · · = nk = n. A cubical array will be called symmetric if its... |

7 |
Optimal solutions to non-negative parafac/multilinear nmf always exist
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(Show Context)
Citation Context ...aintss10 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 that A = �s i . i=1 y⊗k Proof. What we actu... |

7 |
Sums of powers of complex linear forms,” preprint, private correspondence
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(Show Context)
Citation Context ...ank k by Proposition 5.6. Example 6.7. Let n = 3 and k = 3. Then Y5 ⊂ Y3, whereas 3 < RS. In fact, take the symmetric tensor associated with the ternary cubic p(x, y, z) = x 2 y − xz 2 . According to =-=[16, 46]-=-, this tensor has rank 5. On the other hand, it is the limit of the sequence pε(x, y, z) = x 2 y − xz 2 + εz 3 as ε tends to zero. According to a result in [16], the latter polynomial is associated wi... |

6 |
The umbral symbolic method for supersymmetric tensors
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(Show Context)
Citation Context .... , n}. Such arrays have been improperly labeled ‘supersymmetric’ tensors (cf. [10, 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 an... |

6 |
Linear systems of cubics singular at general points of projective space
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(Show Context)
Citation Context ...orem 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 in terms of symmetric tensors. The exception (k, n) = (4, 3) has been kn... |

4 |
uniqueness for 3-way and N-way arrays,” pp
- Kruskal, “Rank
- 1989
(Show Context)
Citation Context ...ading to the same qualitative conclusions. This procedure is not new [53, pp. 13] and has already been proposed in the past to illustrate the existence of several typical ranks for asymmetric tensors =-=[37, 55]-=-. An interesting result obtained by ten Berge [56] is that p×p×2 real asymmetric tensors have typical ranks {p, p + 1}. The problems pertaining to rank and decompositions of real symmetric tensors hav... |

3 |
Algebraic Complexity Theory, 315
- Burgisser, Clausen, et al.
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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, ... |

3 |
and Functional Analysis, 3rd ed., Graduate Texts inMathematics
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- 1993
(Show Context)
Citation Context ...ill use similar tools borrowed from algebraic geometry. Symmetric tensors form a singularly important class of tensors. Examples where these arise include higher order derivatives of smooth functions =-=[40]-=-, and moments and cumulants of random vectors [43]. The decomposition of such symmetric tensors into simpler ones, as in the symmetric outer product decomposition, plays an important role in independe... |

2 | Lectures on the geometry of syzygies,” with a chapter by - Eisenbud - 2004 |

2 |
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Citation Context ...e. Thus S k (C n ) ∼ = C[x1, . . . , xn]k. (3.5) This justifies the use of the Zariski topology, where the elementary closed subsets are the common zeros of a finite number of homogeneous polynomials =-=[48]-=-. Note that for asymmetric tensors, the same association is not possible (although they can still be associated with polynomials via another bijection). As will be subsequently seen,sSYMMETRIC TENSORS... |