## SYMMETRIC TENSOR DECOMPOSITION (2010)

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Venue: | LINEAR ALGEBRA AND APPLICATIONS 433, 11-12 (2010) 1851-1872 |

Citations: | 5 - 0 self |

### BibTeX

@MISC{Brachat10symmetrictensor,

author = {Jerome Brachat and Pierre Comon and Bernard Mourrain and Elias P. Tsigaridas},

title = { SYMMETRIC TENSOR DECOMPOSITION},

year = {2010}

}

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### Abstract

We present an algorithm for decomposing a symmetric tensor, of dimension n and order d as a sum of rank-1 symmetric tensors, extending the algorithm of Sylvester devised in 1886 for binary forms. We recall the correspondence between the decomposition of a homogeneous polynomial in n variables of total degree d as a sum of powers of linear forms (Waring’s problem), incidence properties on secant varieties of the Veronese Variety and the representation of linear forms as a linear combination of evaluations at distinct points. Then we reformulate Sylvester’s approach from the dual point of view. Exploiting this duality, we propose necessary and sufficient conditions for the existence of such a decomposition of a given rank, using the properties of Hankel (and quasi-Hankel) matrices, derived from multivariate polynomials and normal form computations. This leads to the resolution of polynomial equations of small degree in non-generic cases. We propose a new algorithm for symmetric tensor decomposition, based on this characterization and on linear algebra computations with these Hankel matrices. The impact of this contribution is two-fold. First it permits an efficient computation of the decomposition of any tensor of sub-generic rank, as opposed to widely used iterative algorithms with unproved global convergence (e.g. Alternate Least Squares or gradient descents). Second, it gives tools for understanding

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Citation Context ...sition (SVD) problem for symmetric matrices. This former method is an important tool in (numerical) linear algebra, which received at lot of attention and which is routinely used in many applications =-=[27]-=-. As exhibited above, the extension to general symmetric tensors also appears in many application domains. However, many issues either theoretical or algorithmic remains unsolved. Among solved problem... |

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Citation Context ...etrics [9]. Another important application field is Data Analysis. For instance, Independent Component Analysis was originally introduced for symmetric tensors whose rank did not exceed dimension [10] =-=[11]-=-. Now, it has become possible to estimate more factors than the dimension. Further references may be found in [12] [13], and numerous applications of tensor decompositions may be found in [14] [15]. T... |

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Citation Context ...ance, Independent Component Analysis, originally introduced for symmetric tensors whose rank did not exceed dimension [12] [6]. Now, it has become possible to estimate more factors than the dimension =-=[23]-=- [32]. In some applications, tensors may be symmetric only in some modes [14], or may not be symmetric nor have equal dimensions [10] [49]. Further numerous applications of tensor decompositions may b... |

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Citation Context ...[35] [3] [50] [37]. Another important application field is Data Analysis. For instance, Independent Component Analysis, originally introduced for symmetric tensors whose rank did not exceed dimension =-=[12]-=- [6]. Now, it has become possible to estimate more factors than the dimension [23] [32]. In some applications, tensors may be symmetric only in some modes [14], or may not be symmetric nor have equal ... |

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Citation Context ...trics [33]. Arithmetic complexity is also an important field where the understanding of tensor decompositions has made a lot of progress, especially third order tensors, which represent bilinear maps =-=[35]-=- [3] [50] [37]. Another important application field is Data Analysis. For instance, Independent Component Analysis, originally introduced for symmetric tensors whose rank did not exceed dimension [12]... |

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The Algebraic Theory of Modular Systems
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Citation Context ...Equivalently, A is Gorenstein iff A ∗ is a free A-module generated by one element Λ ∈ A ∗ : A ∗ = A ⋆ Λ. See e.g. [24] for more details. The set A ⋆ Λ is also called the inverse system generated by Λ =-=[41]-=-. Proposition 3.3. The dual space A ∗ Λ of AΛ, can be identified with the set D = {q ⋆ Λ | q ∈ R} and AΛ is a Gorenstein algebra. Proof. Let D = {q ⋆ Λ; q ∈ R} be the inverse system generated by Λ. By... |

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Citation Context ... tensors are derivatives of the second characteristic function [1]. Tensors have been widely utilized in Electrical Engineering since the nineties, because of the use of High-Order Statistics [2] [3] =-=[4]-=- [5] [6] [7]. Even earlier in the seventies, tensors have been used in Chemometrics [8] or psychometrics [9]. Another important application field is Data Analysis. For instance, Independent Component ... |

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Citation Context ...ronese variety. It corresponds to the set of (possibly non-symmetric) tensors of rank 1. For any f ∈ Sd − {0}, the smallest r such that f ∈ S r−1 (Vn,d) is called the typical rank or border rank of f =-=[5, 15, 53]-=-. 2.3. Decomposition using duality. Let f, g ∈ Sd, where f = ∑ |α|=d We define the apolar inner product on Sd as 〈f, g〉 = ∑ |α|=d fα gα ( d α0, . . .,αn ) −1 . fαx α0 0 · · ·xαn n and g = ∑ α0 |α|=d g... |

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Citation Context ... are derivatives of the second characteristic function [1]. Tensors have been widely utilized in Electrical Engineering since the nineties, because of the use of High-Order Statistics [2] [3] [4] [5] =-=[6]-=- [7]. Even earlier in the seventies, tensors have been used in Chemometrics [8] or psychometrics [9]. Another important application field is Data Analysis. For instance, Independent Component Analysis... |

67 |
Multi-way Analysis
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Citation Context ...0] [11]. Now, it has become possible to estimate more factors than the dimension. Further references may be found in [12] [13], and numerous applications of tensor decompositions may be found in [14] =-=[15]-=-. The goal of this paper is to devise an algebraic technique able to decompose a symmetric tensor of arbitrary order and dimension in an essentially unique manner (i.e. up to scale and permutation) in... |

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Citation Context ... than generic [21]. On the other hand, the algorithm based on Sylvester’s theorem [22], recalled in section 2, provides a complete answer to the questions of uniqueness and computation, for any order =-=[23]-=-. However, the latter is devoted to 2-dimensional symmetric tensors, and techniques based on pairwise processing have a very limited range of use when the rank exceeds the dimension. For the decomposi... |

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Citation Context ...ng either necessary and sufficient conditions of the quotient algebra, e.g. it holds that xi xj − xj xi = [43] for any i, j ∈ {1, . . ., n}. There are other algorithms to extend a moment matrix, e.g. =-=[18, 38, 39]-=-. (6) We solve the equation (∆1 − λ∆0)X = 0. We solve the generalized eigenvalue/eigenvector problem using one of the well-known techniques [28]. We normalize the elements of the eigenvectors so that ... |

56 |
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Citation Context ...idely utilized in Electrical Engineering since the nineties, because of the use of High-Order Statistics [2] [3] [4] [5] [6] [7]. Even earlier in the seventies, tensors have been used in Chemometrics =-=[8]-=- or psychometrics [9]. Another important application field is Data Analysis. For instance, Independent Component Analysis was originally introduced for symmetric tensors whose rank did not exceed dime... |

56 |
The invariant theory of binary forms
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Citation Context ...tive and lack a guarantee of global convergence [19] [20]. In addition, they often request the rank to be much smaller than generic [21]. On the other hand, the algorithm based on Sylvester’s theorem =-=[22]-=-, recalled in section 2, provides a complete answer to the questions of uniqueness and computation, for any order [23]. However, the latter is devoted to 2-dimensional symmetric tensors, and technique... |

49 |
sums, Gorenstein algebras, and determinantal loci Appendix C by Iarrobino and
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Citation Context ...he rank. See however [11] for an answer in the binary case. For a detailed presentation of the symmetric tensor decomposition problem, from a projective algebraic geometric point of view, we refer to =-=[31]-=-. The properties of so-called catalecticant matrices, related to the apolar duality induced by the symmetric tensor associated with homogeneous polynomials of a given degree, are extensively studied. ... |

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Citation Context ...the variable x0 is denoted fa := f(1, x1, . . . , xn). Duality is an important ingredient of our approach. For a comprehensive treatment of duality on multivariate polynomials, we refer the reader to =-=[44]-=-. Hereafter, for a K-vector space E its dual E∗ = HomK(E, K) is the set of K-linear forms form E to K. A basis of the dual space R∗ d , is the set of linear forms that compute the coefficients of a po... |

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Citation Context ... Λ(h), with H˜ Λ of rank r then the tables of multiplications by the variables xi are Mi = (HB Λ )−1HB xi⋆Λ (proposition 3.9) and they commute. Conversely suppose that these matrices commute. Then by =-=[43]-=-, we have K[x] = 〈B〉 ⊕ (K), where K is the vector space generated by the border relations xim − Mi(m) for m ∈ B and i = 1, . . .,n. Let πB be the projection of R on 〈B〉 along (K). We define ˜ Λ ∈ R∗ a... |

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Citation Context ...]. Arithmetic complexity is also an important field where the understanding of tensor decompositions has made a lot of progress, especially third order tensors, which represent bilinear maps [35] [3] =-=[50]-=- [37]. Another important application field is Data Analysis. For instance, Independent Component Analysis, originally introduced for symmetric tensors whose rank did not exceed dimension [12] [6]. Now... |

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Citation Context ...mial q(x1,x2) = ∑r ℓ=0 qℓ xℓ 1 x r−ℓ 2 admits ∏ r distinct roots, i.e. it can be written as q(x1,x2) = r j=1 (β∗ j x1 − α∗ j x2). It turns out that the proof of this theorem is constructive [24] [23] =-=[25]-=- and yields the algorithm below. 1. Initialize r = 0 2. Increment r ← r + 1 3. If the row rank of H[r] is full, then go to step 2 4. Else compute a basis {k1,...,kl} of the right kernel of H[r]. 5. Sp... |

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The multilinear engine: A table-driven, least squares program for solving multilinear problems, including the n-way parallel factor analysis model
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Citation Context ...al, in the sense that they either do not fully exploit symmetries [16], minimize different successive criteria sequentially [17] [18], or are iterative and lack a guarantee of global convergence [19] =-=[20]-=-. In addition, they often request the rank to be much smaller than generic [21]. On the other hand, the algorithm based on Sylvester’s theorem [22], recalled in section 2, provides a complete answer t... |

33 |
Recursiveness, positivity, and truncated moment problems
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Citation Context ...ng either necessary and sufficient conditions of the quotient algebra, e.g. it holds that xi xj − xj xi = [43] for any i, j ∈ {1, . . ., n}. There are other algorithms to extend a moment matrix, e.g. =-=[18, 38, 39]-=-. (6) We solve the equation (∆1 − λ∆0)X = 0. We solve the generalized eigenvalue/eigenvector problem using one of the well-known techniques [28]. We normalize the elements of the eigenvectors so that ... |

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Citation Context ...sors whose rank did not exceed dimension [12] [6]. Now, it has become possible to estimate more factors than the dimension [23] [32]. In some applications, tensors may be symmetric only in some modes =-=[14]-=-, or may not be symmetric nor have equal dimensions [10] [49]. Further numerous applications of tensor decompositions may be found in [10] [49]. Note that in some cases, tensors are encountered in the... |

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Citation Context ...e different successive criteria sequentially [17] [18], or are iterative and lack a guarantee of global convergence [19] [20]. In addition, they often request the rank to be much smaller than generic =-=[21]-=-. On the other hand, the algorithm based on Sylvester’s theorem [22], recalled in section 2, provides a complete answer to the questions of uniqueness and computation, for any order [23]. However, the... |

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Citation Context ...cumulant tensors are derivatives of the second characteristic function [1]. Tensors have been widely utilized in Electrical Engineering since the nineties, because of the use of High-Order Statistics =-=[2]-=- [3] [4] [5] [6] [7]. Even earlier in the seventies, tensors have been used in Chemometrics [8] or psychometrics [9]. Another important application field is Data Analysis. For instance, Independent Co... |

24 |
O(n 2.7799 ) complexity for n × n approximate matrix multiplication
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Citation Context ... [33]. Arithmetic complexity is also an important field where the understanding of tensor decompositions has made a lot of progress, especially third order tensors, which represent bilinear maps [35] =-=[3]-=- [50] [37]. Another important application field is Data Analysis. For instance, Independent Component Analysis, originally introduced for symmetric tensors whose rank did not exceed dimension [12] [6]... |

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Citation Context ... [21] [26] [54] [47] [53], in which case they may enjoy symmetries in some modes but not in others. Conversely, some algorithms treat symmetric tensors as a collection of symmetric matrix slices [55] =-=[57]-=- [20]. The problem of decomposition of a symmetric tensor, that we consider in this paper, is a rank determinant problem which extends the Singular Value Decomposition (SVD) problem for symmetric matr... |

21 |
Algorithms. Undergraduate Texts in Mathematics
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Citation Context ...) ∈ K[∂1, . . . , ∂n] and • the eigenvalues of the operators Ma and M t a, are given by {a(ζ1), . . . , a(ζr)}. • the common eigenvectors of the operators (M t xi )1≤i≤n are (up to scalar) ζi. Proof. =-=[16, 17, 24]-=- Using the previous proposition, one can recover the points ζi ∈ K n by eigenvector computation as follows. Assume that B ⊂ R with |B| = rank(HΛ), then equation (6) and its transposition yield B a⋆Λ =... |

20 |
La méthode d’Horace éclatée : application à l’interpolation en degré quatre
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Citation Context ...owever, many issues either theoretical or algorithmic remains unsolved. Among solved problems, let us mention the determination of the minimal number of terms in the decomposition of a generic tensor =-=[2]-=-, which is stated there in terms of a dual interpolation problem. See [31, chap. 2] and section 2 for the link between these two points of view. Among open problems are the determination of the maxima... |

19 |
Tensor Methods
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Citation Context ...ymmetric tensors show up in applications mainly as high-order derivatives of multivariate functions. For instance in Statistics, cumulant tensors are derivatives of the second characteristic function =-=[1]-=-. Tensors have been widely utilized in Electrical Engineering since the nineties, because of the use of High-Order Statistics [2] [3] [4] [5] [6] [7]. Even earlier in the seventies, tensors have been ... |

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Citation Context ...teristic function [42]. Tensors have been widely utilized in Electrical Engineering since the nineties [51], and in particular in Antenna Array Processing [22] [9] or Telecommunications [54] [8] [48] =-=[25]-=- [19]. Even earlier in the seventies, tensors have been used in Chemometrics [4] or Psychometrics [33]. Arithmetic complexity is also an important field where the understanding of tensor decomposition... |

18 |
Common principal components in k groups
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Citation Context ...ns [10] [49]. Further numerous applications of tensor decompositions may be found in [10] [49]. Note that in some cases, tensors are encountered in the form of a collection of symmetric matrices [21] =-=[26]-=- [54] [47] [53], in which case they may enjoy symmetries in some modes but not in others. Conversely, some algorithms treat symmetric tensors as a collection of symmetric matrix slices [55] [57] [20].... |

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Citation Context ...lant tensors are derivatives of the second characteristic function [1]. Tensors have been widely utilized in Electrical Engineering since the nineties, because of the use of High-Order Statistics [2] =-=[3]-=- [4] [5] [6] [7]. Even earlier in the seventies, tensors have been used in Chemometrics [8] or psychometrics [9]. Another important application field is Data Analysis. For instance, Independent Compon... |

17 | On the rank of a binary form
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Citation Context ...GORITHM The algorithm that we will present for decomposing a symmetric tensor as sum of rank 1 symmetric tensors generalizes the algorithm of Sylvester [24], devised for dimension 2 tensors, see also =-=[29]-=-. 4.1 Overview Algorithm 1: Symmetric tensor decomposition Input: A homogeneous polynomial f(x0,x1,...,xn) of degree d. Output: A decomposition of f as f = ∑r i=1 λi ki(x) d with r minimal. ( d α – Co... |

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Citation Context ...ng either necessary and sufficient conditions of the quotient algebra, e.g. it holds that xi xj − xj xi = [43] for any i, j ∈ {1, . . ., n}. There are other algorithms to extend a moment matrix, e.g. =-=[18, 38, 39]-=-. (6) We solve the equation (∆1 − λ∆0)X = 0. We solve the generalized eigenvalue/eigenvector problem using one of the well-known techniques [28]. We normalize the elements of the eigenvectors so that ... |

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Citation Context ...optimal, in the sense that they either do not fully exploit symmetries [16], minimize different successive criteria sequentially [17] [18], or are iterative and lack a guarantee of global convergence =-=[19]-=- [20]. In addition, they often request the rank to be much smaller than generic [21]. On the other hand, the algorithm based on Sylvester’s theorem [22], recalled in section 2, provides a complete ans... |

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Citation Context ...practical interest as they can deal with inexact data, numerical algorithms presently used in most scientific communities are suboptimal, in the sense that they either do not fully exploit symmetries =-=[16]-=-, minimize different successive criteria sequentially [17] [18], or are iterative and lack a guarantee of global convergence [19] [20]. In addition, they often request the rank to be much smaller than... |

14 |
B.: Introduction la résolution des systèmes polynomiaux. Volume 57 of Mathmatiques et Applications
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Citation Context ... In other words, the common roots of all the polynomials in IΦ define the linear terms in the tensor decomposition of F. In order to compute the zeros of IΦ, we may use a well-known theorem (see e.g. =-=[26, 27, 28]-=-), which we apply to the zero-dimensional ideal IΦ: Theorem 3.3 The eigenvalues of the matrices a and ⊤ a , of the linear operators that correspond to the multiplication by a in R modulo IΦ, and its t... |

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Citation Context ...ithmetic complexity is also an important field where the understanding of tensor decompositions has made a lot of progress, especially third order tensors, which represent bilinear maps [35] [3] [50] =-=[37]-=-. Another important application field is Data Analysis. For instance, Independent Component Analysis, originally introduced for symmetric tensors whose rank did not exceed dimension [12] [6]. Now, it ... |

12 |
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Citation Context ...s construction as a map k ↦→ k(x) d from the projective space P n−1 to the projective space of symmetric tensors: ν : P(S1) → P(Sd) k(x) ↦→ k(x) d . The image of ν is called the Veronese variety Vn,d =-=[29, 56]-=-. Following this point of view, a tensor is of rank 1 if it corresponds to a point on the Veronese variety. A tensor is of rank ≤ r if it is a linear combination of r tensors of rank 1. In other words... |

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Citation Context ... derivatives of the second characteristic function [1]. Tensors have been widely utilized in Electrical Engineering since the nineties, because of the use of High-Order Statistics [2] [3] [4] [5] [6] =-=[7]-=-. Even earlier in the seventies, tensors have been used in Chemometrics [8] or psychometrics [9]. Another important application field is Data Analysis. For instance, Independent Component Analysis was... |

11 |
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Citation Context ...urther numerous applications of tensor decompositions may be found in [10] [49]. Note that in some cases, tensors are encountered in the form of a collection of symmetric matrices [21] [26] [54] [47] =-=[53]-=-, in which case they may enjoy symmetries in some modes but not in others. Conversely, some algorithms treat symmetric tensors as a collection of symmetric matrix slices [55] [57] [20]. The problem of... |

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Citation Context ...sors are derivatives of the second characteristic function [1]. Tensors have been widely utilized in Electrical Engineering since the nineties, because of the use of High-Order Statistics [2] [3] [4] =-=[5]-=- [6] [7]. Even earlier in the seventies, tensors have been used in Chemometrics [8] or psychometrics [9]. Another important application field is Data Analysis. For instance, Independent Component Anal... |

9 |
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Citation Context ... tensors are derivatives of the second characteristic function [42]. Tensors have been widely utilized in Electrical Engineering since the nineties [51], and in particular in Antenna Array Processing =-=[22]-=- [9] or Telecommunications [54] [8] [48] [25] [19]. Even earlier in the seventies, tensors have been used in Chemometrics [4] or Psychometrics [33]. Arithmetic complexity is also an important field wh... |

9 |
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Citation Context ...ymmetric tensors show up in applications mainly as high-order derivatives of multivariate functions. For instance in Statistics, cumulant tensors are derivatives of the second characteristic function =-=[42]-=-. Tensors have been widely utilized in Electrical Engineering since the nineties [51], and in particular in Antenna Array Processing [22] [9] or Telecommunications [54] [8] [48] [25] [19]. Even earlie... |

8 |
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Citation Context ...s construction as a map k ↦→ k(x) d from the projective space P n−1 to the projective space of symmetric tensors: ν : P(S1) → P(Sd) k(x) ↦→ k(x) d . The image of ν is called the Veronese variety Vn,d =-=[29, 56]-=-. Following this point of view, a tensor is of rank 1 if it corresponds to a point on the Veronese variety. A tensor is of rank ≤ r if it is a linear combination of r tensors of rank 1. In other words... |

7 |
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Citation Context ... In other words, the common roots of all the polynomials in IΦ define the linear terms in the tensor decomposition of F. In order to compute the zeros of IΦ, we may use a well-known theorem (see e.g. =-=[26, 27, 28]-=-), which we apply to the zero-dimensional ideal IΦ: Theorem 3.3 The eigenvalues of the matrices a and ⊤ a , of the linear operators that correspond to the multiplication by a in R modulo IΦ, and its t... |

6 |
Non-orthogonal joint diagoinalization in the LS sense with application in blind source separation
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Citation Context ...erical algorithms presently used in most scientific communities are suboptimal, in the sense that they either do not fully exploit symmetries [16], minimize different successive criteria sequentially =-=[17]-=- [18], or are iterative and lack a guarantee of global convergence [19] [20]. In addition, they often request the rank to be much smaller than generic [21]. On the other hand, the algorithm based on S... |

6 | Reducing the number of variables of a polynomial
- Carlini
- 2005
(Show Context)
Citation Context ... is of rank 4. (2) Compute the actual number of variables needed. For algorithms computing the so-called number of essential variables, the reader may refer to the work of Oldenburger [45] or Carlini =-=[7]-=-. In our example the number of essential variable is 3, so we have nothing to do. (3) Compute the matrix of the quotient algebra. We form a ( ) ( ) n+d−1 n+d−1 d × d matrix, the rows and the columns o... |