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Symmetric tensors and symmetric tensor rank
 Scientific Computing and Computational Mathematics (SCCM
, 2006
"... 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 rank1 orderk tensor is the outer product of k nonzero vectors. An ..."
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Cited by 101 (22 self)
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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 rank1 orderk tensor is the outer product of k nonzero vectors. Any symmetric tensor can be decomposed into a linear combination of rank1 tensors, each of them being symmetric or not. The rank of a symmetric tensor is the minimal number of rank1 tensors that is necessary to reconstruct it. The symmetric rank is obtained when the constituting rank1 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
Canonical Tensor Decompositions
 ARCC WORKSHOP ON TENSOR DECOMPOSITION
, 2004
"... The Singular Value Decomposition (SVD) may be extended to tensors at least in two very different ways. One is the HighOrder SVD (HOSVD), and the other is the Canonical Decomposition (CanD). Only the latter is closely related to the tensor rank. Important basic questions are raised in this short pap ..."
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Cited by 42 (16 self)
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The Singular Value Decomposition (SVD) may be extended to tensors at least in two very different ways. One is the HighOrder SVD (HOSVD), and the other is the Canonical Decomposition (CanD). Only the latter is closely related to the tensor rank. Important basic questions are raised in this short paper, such as the maximal achievable rank of a tensor of given dimensions, or the computation of a CanD. Some questions are answered, and it turns out that the answers depend on the choice of the underlying field, and on tensor symmetry structure, which outlines a major difference compared to matrices.
Blind channel identification and extraction of more sources than sensors
, 1998
"... It is often admitted that a static system with more inputs (sources) than outputs (sensors, or channels) cannot be blindly identified, that is, identified only from the observation of its outputs, and without any a priori knowledge on the source statistics but their independence. By resorting to Hig ..."
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Cited by 21 (7 self)
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It is often admitted that a static system with more inputs (sources) than outputs (sensors, or channels) cannot be blindly identified, that is, identified only from the observation of its outputs, and without any a priori knowledge on the source statistics but their independence. By resorting to HighOrder Statistics, it turns out that static MIMO systems with fewer outputs than inputs can be identified, as demonstrated in the present paper. The principle, already described in a recent rather theoretical paper, had not yet been applied to a concrete blind identification problem. Here, in order to demonstrate its feasibility, the procedure is detailed in the case of a 2sensor 3source mixture; a numerical algorithm is devised, that blindly identifies a 3input 2output mixture. Computer results show its behavior as a function of the data length when sources are QPSKmodulated signals, widely used in digital communications. Then another algorithm is proposed to extract the 3 sources from the 2 observations, once the mixture has been identified. Contrary to the first algorithm, this one assumes that the sources have a known discrete distribution. Computer experiments are run in the case of three BPSK sources in presence of Gaussian noise.
Blind Identification and Source Separation In 2 x 3 Underdetermined Mixtures
, 2004
"... Underdetermined mixtures are characterized by the fact that they have more inputs than outputs, or, with the antenna array processing terminology, more sources than sensors. The problem addressed is that of identifying and inverting the mixture, which obviously does not admit a linear inverse. Id ..."
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Cited by 16 (4 self)
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Underdetermined mixtures are characterized by the fact that they have more inputs than outputs, or, with the antenna array processing terminology, more sources than sensors. The problem addressed is that of identifying and inverting the mixture, which obviously does not admit a linear inverse. Identification is carried out with the help of tensor canonical decompositions. On the other hand, the discrete distribution of the sources is utilized for performing the source extraction, the underdetermined mixture being either known or unknown. The results presented in this paper are limited to 2dimensional mixtures of 3 sources.
Genericity and Rank Deficiency of High Order Symmetric Tensors
 Proc. IEEE Int. Conference on Acoustics, Speech, and Signal Processing (ICASSP
, 2006
"... Blind Identification of UnderDetermined Mixtures (UDM) is involved in numerous applications, including MultiWay factor Analysis (MWA) and Signal Processing. In the latter case, the use of HighOrder Statistics (HOS) like Cumulants leads to the decomposition of symmetric tensors. Yet, little has be ..."
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Cited by 9 (6 self)
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Blind Identification of UnderDetermined Mixtures (UDM) is involved in numerous applications, including MultiWay factor Analysis (MWA) and Signal Processing. In the latter case, the use of HighOrder Statistics (HOS) like Cumulants leads to the decomposition of symmetric tensors. Yet, little has been published about rankrevealing decompositions of symmetric tensors. Definitions of rank are discussed, and useful results on Generic Rank are proved, with the help of tools borrowed from Algebraic Geometry. 1.
SCIENTIFIQUE Canonical Tensor Decompositions
, 2004
"... The Singular Value Decomposition (SVD) may be extended to tensors at least in two very different ways. One is the HighOrder SVD (HOSVD), and the other is the Canonical Decomposition (CanD). Only the latter is closely related to the tensor rank. Important basic questions are raised in this short pap ..."
Abstract
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The Singular Value Decomposition (SVD) may be extended to tensors at least in two very different ways. One is the HighOrder SVD (HOSVD), and the other is the Canonical Decomposition (CanD). Only the latter is closely related to the tensor rank. Important basic questions are raised in this short paper, such as the maximal achievable rank of a tensor of given dimensions, or the computation of a CanD. Some questions are answered, and it turns out that the answers depend on the choice of the underlying field, and on tensor symmetry structure, which outlines a major difference compared to matrices.