## Symmetric tensors and symmetric tensor rank (2006)

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Venue: | Scientific Computing and Computational Mathematics (SCCM |

Citations: | 40 - 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 ...ribution (e.g. Gaussian) is RS with probability 1. 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... |

346 | Using Algebraic Geometry - Cox, Little, et al. - 2005 |

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Citation Context ...sychometricians in their attempts to define data analytic models that generalize factor analysis to multiway data [59]. The name candecomp, for ‘canonical decomposition’, was used by Carrol and Chang =-=[11]-=- while the name parafac, for ‘parallel factor analysis’, was used by Harshman [28] for their respective models. The symmetric outer product decomposition is particularly important in the process of bl... |

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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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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 ...addressed in the general literature, and even less so in the engineering literature. For several years, the alternating least squares algorithm has been used to fit data arrays to a multilinear model =-=[36, 50]-=-. Yet, the minimization of this matching error is an ill-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 ... |

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Citation Context ...etry. 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 independent component analysis [14] and constitutes a probl... |

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Citation Context ...ecomposition 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, neuroimaging, numerical analysis, optimizat... |

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Citation Context ...known in the case of binary quantics (n = 2) and ternary cubics (k = 3) [22, 16, 47, 35].s20 P. COMON, G.H. GOLUB, L.-H. LIM, B. MOURRAIN 7.1. Alexander-Hirschowitz Theorem. It was not until the work =-=[1]-=- of Alexander and Hirschowitz in 1995 that the generic symmetric rank problem was completely settled. Nevertheless, the relevance of their result has remained largely unknown in the applied and comput... |

69 | Tensor rank and the ill-posedness of the best low-rank approximation problem
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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... |

67 |
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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... |

65 | 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 |
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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... |

57 |
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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 ... |

57 |
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Citation Context ...ecomposition 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, neuroimaging, numerical analysis, optimizat... |

55 |
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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... |

54 | Super-symmetric decomposition of the fourth-order 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... |

50 |
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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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46 |
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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 ... |

46 | On the best rank-1 approximation of higher-order supersymmetric tensors
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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... |

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 +... |

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) ... |

32 | Rank, decomposition, and uniqueness for 3-way and N-way arrays - Kruskal - 1989 |

31 | Commutative Algebra (Elements of Mathematics - Bourbaki - 1989 |

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Citation Context |

28 | Blind identification of under-determined mixtures based on the characteristic function. Signal Process - Comon, Rajih - 2006 |

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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 |
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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 ... |

25 |
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, ... |

25 | Basic algebraic geometry - Shafarevitch - 1974 |

24 | O(n 2.77 ) Complexity for n × n approximate matrix multiplication - Bini, Capovani, et al. - 1979 |

24 | Characterization problems in Mathematical Statistics - Kagan, Linnik, et al. - 1973 |

24 | 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... |

21 |
Algebra I, 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... |

20 |
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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 ... |

19 | Computation of canonical forms for ternary cubics
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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... |

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

18 |
Multilinear 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... |

17 | On the rank of a binary form
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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 | A Newton–Grassmann Method for Computing the Best Multi-Linear Rank-(r1,r2,r3) Approximation of a Tensor
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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... |

17 |
Simplicity of core arrays in three-way principal component analysis and the typical rank of p×q×2 arrays. Linear Algebra and its Applications
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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, ... |

16 |
Tensor Spaces and Exterior Algebra
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Citation Context |

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)
Citation Context ...ure 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 have not received as much attention as their complex ... |

11 |
2004) Typical rank and indscal dimensionality for symmetric three-way arrays of order
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(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 ... |