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Academic literature on the topic 'Rank of symmetric tensors'

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Published: 14 March 2023

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Journal articles on the topic "Rank of symmetric tensors"

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Ballico, E. "Gaps in the pairs (border rank, symmetric rank) for symmetric tensors." Sarajevo Journal of Mathematics 9, no. 2 (November 2013): 169–81. http://dx.doi.org/10.5644/sjm.09.2.02.

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Comon, Pierre, Gene Golub, Lek-Heng Lim, and Bernard Mourrain. "Symmetric Tensors and Symmetric Tensor Rank." SIAM Journal on Matrix Analysis and Applications 30, no. 3 (January 2008): 1254–79. http://dx.doi.org/10.1137/060661569.

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SEGAL, ARKADY Y. "POINT PARTICLE–SYMMETRIC TENSORS INTERACTION AND GENERALIZED GAUGE PRINCIPLE." International Journal of Modern Physics A 18, no. 27 (October 30, 2003): 5021–38. http://dx.doi.org/10.1142/s0217751x03015842.

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The model of a point particle in the background of external symmetric tensor fields is analyzed from the higher spin theory perspective. It is proposed that the gauge transformations of the infinite collection of symmetric tensor fields may be read off from the covariance properties of the point particle action w.r.t. general canonical transformations. The gauge group turns out to be a semidirect product of all phase space canonical transformations to an Abelian ideal of "hyperWeyl" transformations and includes U(1) and general coordinate symmetries as a subgroup. A general configuration of external fields includes rank-0,1,2 symmetric tensors, so the whole system may be truncated to ordinary particle in Einstein–Maxwell backgrounds by switching off the higher-rank symmetric tensors. When otherwise all the higher rank tensors are switched on, the full gauge group provides a huge gauge symmetry acting on the whole infinite collection of symmetric tensors. We analyze this gauge symmetry and show that the symmetric tensors which couple to the point particle should not be interpreted as Fronsdal gauge fields, but rather as gauge fields of some conformal higher spin theories. It is shown that the Fronsdal fields system possesses twice as many symmetric tensor fields as is contained in the general background of the point particle. Besides, the particle action in general backgrounds is shown to reproduce De Wit–Freedman point particle–symmetric tensors first order interaction suggested many years ago, and extends their result to all orders in interaction, while the generalized equivalence principle completes the first order covariance transformations found in their paper, in all orders.
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Casarotti, Alex, Alex Massarenti, and Massimiliano Mella. "On Comon’s and Strassen’s Conjectures." Mathematics 6, no. 11 (October 25, 2018): 217. http://dx.doi.org/10.3390/math6110217.

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Comon’s conjecture on the equality of the rank and the symmetric rank of a symmetric tensor, and Strassen’s conjecture on the additivity of the rank of tensors are two of the most challenging and guiding problems in the area of tensor decomposition. We survey the main known results on these conjectures, and, under suitable bounds on the rank, we prove them, building on classical techniques used in the case of symmetric tensors, for mixed tensors. Finally, we improve the bound for Comon’s conjecture given by flattenings by producing new equations for secant varieties of Veronese and Segre varieties.
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Bernardi, Alessandra, Alessandro Gimigliano, and Monica Idà. "Computing symmetric rank for symmetric tensors." Journal of Symbolic Computation 46, no. 1 (January 2011): 34–53. http://dx.doi.org/10.1016/j.jsc.2010.08.001.

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De Paris, Alessandro. "Seeking for the Maximum Symmetric Rank." Mathematics 6, no. 11 (November 12, 2018): 247. http://dx.doi.org/10.3390/math6110247.

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We present the state-of-the-art on maximum symmetric tensor rank, for each given dimension and order. After a general discussion on the interplay between symmetric tensors, polynomials and divided powers, we introduce the technical environment and the methods that have been set up in recent times to find new lower and upper bounds.
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Obster, Dennis, and Naoki Sasakura. "Counting Tensor Rank Decompositions." Universe 7, no. 8 (August 15, 2021): 302. http://dx.doi.org/10.3390/universe7080302.

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Tensor rank decomposition is a useful tool for geometric interpretation of the tensors in the canonical tensor model (CTM) of quantum gravity. In order to understand the stability of this interpretation, it is important to be able to estimate how many tensor rank decompositions can approximate a given tensor. More precisely, finding an approximate symmetric tensor rank decomposition of a symmetric tensor Q with an error allowance Δ is to find vectors ϕi satisfying ∥Q−∑i=1Rϕi⊗ϕi⋯⊗ϕi∥2≤Δ. The volume of all such possible ϕi is an interesting quantity which measures the amount of possible decompositions for a tensor Q within an allowance. While it would be difficult to evaluate this quantity for each Q, we find an explicit formula for a similar quantity by integrating over all Q of unit norm. The expression as a function of Δ is given by the product of a hypergeometric function and a power function. By combining new numerical analysis and previous results, we conjecture a formula for the critical rank, yielding an estimate for the spacetime degrees of freedom of the CTM. We also extend the formula to generic decompositions of non-symmetric tensors in order to make our results more broadly applicable. Interestingly, the derivation depends on the existence (convergence) of the partition function of a matrix model which previously appeared in the context of the CTM.
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Friedland, Shmuel. "Remarks on the Symmetric Rank of Symmetric Tensors." SIAM Journal on Matrix Analysis and Applications 37, no. 1 (January 2016): 320–37. http://dx.doi.org/10.1137/15m1022653.

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Zhang, Xinzhen, Zheng-Hai Huang, and Liqun Qi. "Comon's Conjecture, Rank Decomposition, and Symmetric Rank Decomposition of Symmetric Tensors." SIAM Journal on Matrix Analysis and Applications 37, no. 4 (January 2016): 1719–28. http://dx.doi.org/10.1137/141001470.

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Wen, Jie, Qin Ni, and Wenhuan Zhu. "Rank-r decomposition of symmetric tensors." Frontiers of Mathematics in China 12, no. 6 (May 5, 2017): 1339–55. http://dx.doi.org/10.1007/s11464-017-0632-5.

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Dissertations / Theses on the topic "Rank of symmetric tensors"

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Erdtman, Elias, and Carl Jönsson. "Tensor Rank." Thesis, Linköpings universitet, Matematik och tillämpad matematik, 2012. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-78449.

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This master's thesis addresses numerical methods of computing the typical ranks of tensors over the real numbers and explores some properties of tensors over finite fields. We present three numerical methods to compute typical tensor rank. Two of these have already been published and can be used to calculate the lowest typical ranks of tensors and an approximate percentage of how many tensors have the lowest typical ranks (for some tensor formats), respectively. The third method was developed by the authors with the intent to be able to discern if there is more than one typical rank. Some results from the method are presented but are inconclusive. In the area of tensors over nite filds some new results are shown, namely that there are eight GLq(2) GLq(2) GLq(2)-orbits of 2 2 2 tensors over any finite field and that some tensors over Fq have lower rank when considered as tensors over Fq2 . Furthermore, it is shown that some symmetric tensors over F2 do not have a symmetric rank and that there are tensors over some other finite fields which have a larger symmetric rank than rank.
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mazzon, andrea. "Hilbert functions and symmetric tensors identifiability." Doctoral thesis, Università di Siena, 2021. http://hdl.handle.net/11365/1133145.

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We study the Waring decompositions of a given symmetric tensor using tools of algebraic geometry for the study of finite sets of points. In particular we use the properties of the Hilbert functions and the Cayley-Bacharach property to study the uniqueness of a given decomposition (the identifiability problem), and its minimality, and show how, in some cases, one can effectively determine the uniqueness even in some range in which the Kruskal's criterion does not apply. We give also a more efficient algorithm that, under some hypothesis, certify the identifiability of a given symmetric tensor.
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Wang, Roy Chih Chung. "Adaptive Kernel Functions and Optimization Over a Space of Rank-One Decompositions." Thesis, Université d'Ottawa / University of Ottawa, 2017. http://hdl.handle.net/10393/36975.

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The representer theorem from the reproducing kernel Hilbert space theory is the origin of many kernel-based machine learning and signal modelling techniques that are popular today. Most kernel functions used in practical applications behave in a homogeneous manner across the domain of the signal of interest, and they are called stationary kernels. One open problem in the literature is the specification of a non-stationary kernel that is computationally tractable. Some recent works solve large-scale optimization problems to obtain such kernels, and they often suffer from non-identifiability issues in their optimization problem formulation. Many practical problems can benefit from using application-specific prior knowledge on the signal of interest. For example, if one can adequately encode the prior assumption that edge contours are smooth, one does not need to learn a finite-dimensional dictionary from a database of sampled image patches that each contains a circular object in order to up-convert images that contain circular edges. In the first portion of this thesis, we present a novel method for constructing non-stationary kernels that incorporates prior knowledge. A theorem is presented that ensures the result of this construction yields a symmetric and positive-definite kernel function. This construction does not require one to solve any non-identifiable optimization problems. It does require one to manually design some portions of the kernel while deferring the specification of the remaining portions to when an observation of the signal is available. In this sense, the resultant kernel is adaptive to the data observed. We give two examples of this construction technique via the grayscale image up-conversion task where we chose to incorporate the prior assumption that edge contours are smooth. Both examples use a novel local analysis algorithm that summarizes the p-most dominant directions for a given grayscale image patch. The non-stationary properties of these two types of kernels are empirically demonstrated on the Kodak image database that is popular within the image processing research community. Tensors and tensor decomposition methods are gaining popularity in the signal processing and machine learning literature, and most of the recently proposed tensor decomposition methods are based on the tensor power and alternating least-squares algorithms, which were both originally devised over a decade ago. The algebraic approach for the canonical polyadic (CP) symmetric tensor decomposition problem is an exception. This approach exploits the bijective relationship between symmetric tensors and homogeneous polynomials. The solution of a CP symmetric tensor decomposition problem is a set of p rank-one tensors, where p is fixed. In this thesis, we refer to such a set of tensors as a rank-one decomposition with cardinality p. Existing works show that the CP symmetric tensor decomposition problem is non-unique in the general case, so there is no bijective mapping between a rank-one decomposition and a symmetric tensor. However, a proposition in this thesis shows that a particular space of rank-one decompositions, SE, is isomorphic to a space of moment matrices that are called quasi-Hankel matrices in the literature. Optimization over Riemannian manifolds is an area of optimization literature that is also gaining popularity within the signal processing and machine learning community. Under some settings, one can formulate optimization problems over differentiable manifolds where each point is an equivalence class. Such manifolds are called quotient manifolds. This type of formulation can reduce or eliminate some of the sources of non-identifiability issues for certain optimization problems. An example is the learning of a basis for a subspace by formulating the solution space as a type of quotient manifold called the Grassmann manifold, while the conventional formulation is to optimize over a space of full column rank matrices. The second portion of this thesis is about the development of a general-purpose numerical optimization framework over SE. A general-purpose numerical optimizer can solve different approximations or regularized versions of the CP decomposition problem, and they can be applied to tensor-related applications that do not use a tensor decomposition formulation. The proposed optimizer uses many concepts from the Riemannian optimization literature. We present a novel formulation of SE as an embedded differentiable submanifold of the space of real-valued matrices with full column rank, and as a quotient manifold. Riemannian manifold structures and tangent space projectors are derived as well. The CP symmetric tensor decomposition problem is used to empirically demonstrate that the proposed scheme is indeed a numerical optimization framework over SE. Future investigations will concentrate on extending the proposed optimization framework to handle decompositions that correspond to non-symmetric tensors.
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Harmouch, Jouhayna. "Décomposition de petit rang, problèmes de complétion et applications : décomposition de matrices de Hankel et des tenseurs de rang faible." Thesis, Université Côte d'Azur (ComUE), 2018. http://www.theses.fr/2018AZUR4236/document.

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On étudie la décomposition de matrice de Hankel comme une somme des matrices de Hankel de rang faible en corrélation avec la décomposition de son symbole σ comme une somme des séries exponentielles polynomiales. On présente un nouvel algorithme qui calcule la décomposition d’un opérateur de Hankel de petit rang et sa décomposition de son symbole en exploitant les propriétés de l’algèbre quotient de Gorenstein . La base de est calculée à partir la décomposition en valeurs singuliers d’une sous-matrice de matrice de Hankel . Les fréquences et les poids se déduisent des vecteurs propres généralisés des sous matrices de Hankel déplacés de . On présente une formule pour calculer les poids en fonction des vecteurs propres généralisés au lieu de résoudre un système de Vandermonde. Cette nouvelle méthode est une généralisation de Pencil méthode déjà utilisée pour résoudre un problème de décomposition de type de Prony. On analyse son comportement numérique en présence des moments contaminés et on décrit une technique de redimensionnement qui améliore la qualité numérique des fréquences d’une grande amplitude. On présente une nouvelle technique de Newton qui converge localement vers la matrice de Hankel de rang faible la plus proche au matrice initiale et on montre son effet à corriger les erreurs sur les moments. On étudie la décomposition d’un tenseur multi-symétrique T comme une somme des puissances de produit des formes linéaires en corrélation avec la décomposition de son dual comme une somme pondérée des évaluations. On utilise les propriétés de l’algèbre de Gorenstein associée pour calculer la décomposition de son dual qui est définie à partir d’une série formelle τ. On utilise la décomposition d’un opérateur de Hankel de rang faible associé au symbole τ comme une somme des opérateurs indécomposables de rang faible. La base d’ est choisie de façon que la multiplication par certains variables soit possible. On calcule les coordonnées des points et leurs poids correspondants à partir la structure propre des matrices de multiplication. Ce nouvel algorithme qu’on propose marche bien pour les matrices de Hankel de rang faible. On propose une approche théorique de la méthode dans un espace de dimension n. On donne un exemple numérique de la décomposition d’un tenseur multilinéaire de rang 3 en dimension 3 et un autre exemple de la décomposition d’un tenseur multi-symétrique de rang 3 en dimension 3. On étudie le problème de complétion de matrice de Hankel comme un problème de minimisation. On utilise la relaxation du problème basé sur la minimisation de la norme nucléaire de la matrice de Hankel. On adapte le SVT algorithme pour le cas d’une matrice de Hankel et on calcule l’opérateur linéaire qui décrit les contraintes du problème de minimisation de norme nucléaire. On montre l’utilité du problème de décomposition à dissocier un modèle statistique ou biologique
We study the decomposition of a multivariate Hankel matrix as a sum of Hankel matrices of small rank in correlation with the decomposition of its symbol σ as a sum of polynomialexponential series. We present a new algorithm to compute the low rank decomposition of the Hankel operator and the decomposition of its symbol exploiting the properties of the associated Artinian Gorenstein quotient algebra . A basis of is computed from the Singular Value Decomposition of a sub-matrix of the Hankel matrix . The frequencies and the weights are deduced from the generalized eigenvectors of pencils of shifted sub-matrices of Explicit formula for the weights in terms of the eigenvectors avoid us to solve a Vandermonde system. This new method is a multivariate generalization of the so-called Pencil method for solving Pronytype decomposition problems. We analyse its numerical behaviour in the presence of noisy input moments, and describe a rescaling technique which improves the numerical quality of the reconstruction for frequencies of high amplitudes. We also present a new Newton iteration, which converges locally to the closest multivariate Hankel matrix of low rank and show its impact for correcting errors on input moments. We study the decomposition of a multi-symmetric tensor T as a sum of powers of product of linear forms in correlation with the decomposition of its dual as a weighted sum of evaluations. We use the properties of the associated Artinian Gorenstein Algebra to compute the decomposition of its dual which is defined via a formal power series τ. We use the low rank decomposition of the Hankel operator associated to the symbol τ into a sum of indecomposable operators of low rank. A basis of is chosen such that the multiplication by some variables is possible. We compute the sub-coordinates of the evaluation points and their weights using the eigen-structure of multiplication matrices. The new algorithm that we propose works for small rank. We give a theoretical generalized approach of the method in n dimensional space. We show a numerical example of the decomposition of a multi-linear tensor of rank 3 in 3 dimensional space. We show a numerical example of the decomposition of a multi-symmetric tensor of rank 3 in 3 dimensional space. We study the completion problem of the low rank Hankel matrix as a minimization problem. We use the relaxation of it as a minimization problem of the nuclear norm of Hankel matrix. We adapt the SVT algorithm to the case of Hankel matrix and we compute the linear operator which describes the constraints of the problem and its adjoint. We try to show the utility of the decomposition algorithm in some applications such that the LDA model and the ODF model
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Savas, Berkant. "Algorithms in data mining using matrix and tensor methods." Doctoral thesis, Linköpings universitet, Beräkningsvetenskap, 2008. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-11597.

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In many fields of science, engineering, and economics large amounts of data are stored and there is a need to analyze these data in order to extract information for various purposes. Data mining is a general concept involving different tools for performing this kind of analysis. The development of mathematical models and efficient algorithms is of key importance. In this thesis we discuss algorithms for the reduced rank regression problem and algorithms for the computation of the best multilinear rank approximation of tensors. The first two papers deal with the reduced rank regression problem, which is encountered in the field of state-space subspace system identification. More specifically the problem is \[ \min_{\rank(X) = k} \det (B - X A)(B - X A)\tp, \] where $A$ and $B$ are given matrices and we want to find $X$ under a certain rank condition that minimizes the determinant. This problem is not properly stated since it involves implicit assumptions on $A$ and $B$ so that $(B - X A)(B - X A)\tp$ is never singular. This deficiency of the determinant criterion is fixed by generalizing the minimization criterion to rank reduction and volume minimization of the objective matrix. The volume of a matrix is defined as the product of its nonzero singular values. We give an algorithm that solves the generalized problem and identify properties of the input and output signals causing a singular objective matrix. Classification problems occur in many applications. The task is to determine the label or class of an unknown object. The third paper concerns with classification of handwritten digits in the context of tensors or multidimensional data arrays. Tensor and multilinear algebra is an area that attracts more and more attention because of the multidimensional structure of the collected data in various applications. Two classification algorithms are given based on the higher order singular value decomposition (HOSVD). The main algorithm makes a data reduction using HOSVD of 98--99 \% prior the construction of the class models. The models are computed as a set of orthonormal bases spanning the dominant subspaces for the different classes. An unknown digit is expressed as a linear combination of the basis vectors. The resulting algorithm achieves 5\% in classification error with fairly low amount of computations. The remaining two papers discuss computational methods for the best multilinear rank approximation problem \[ \min_{\cB} \| \cA - \cB\| \] where $\cA$ is a given tensor and we seek the best low multilinear rank approximation tensor $\cB$. This is a generalization of the best low rank matrix approximation problem. It is well known that for matrices the solution is given by truncating the singular values in the singular value decomposition (SVD) of the matrix. But for tensors in general the truncated HOSVD does not give an optimal approximation. For example, a third order tensor $\cB \in \RR^{I \x J \x K}$ with rank$(\cB) = (r_1,r_2,r_3)$ can be written as the product \[ \cB = \tml{X,Y,Z}{\cC}, \qquad b_{ijk}=\sum_{\lambda,\mu,\nu} x_{i\lambda} y_{j\mu} z_{k\nu} c_{\lambda\mu\nu}, \] where $\cC \in \RR^{r_1 \x r_2 \x r_3}$ and $X \in \RR^{I \times r_1}$, $Y \in \RR^{J \times r_2}$, and $Z \in \RR^{K \times r_3}$ are matrices of full column rank. Since it is no restriction to assume that $X$, $Y$, and $Z$ have orthonormal columns and due to these constraints, the approximation problem can be considered as a nonlinear optimization problem defined on a product of Grassmann manifolds. We introduce novel techniques for multilinear algebraic manipulations enabling means for theoretical analysis and algorithmic implementation. These techniques are used to solve the approximation problem using Newton and Quasi-Newton methods specifically adapted to operate on products of Grassmann manifolds. The presented algorithms are suited for small, large and sparse problems and, when applied on difficult problems, they clearly outperform alternating least squares methods, which are standard in the field.
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Santarsiero, Pierpaola. "Identifiability of small rank tensors and related problems." Doctoral thesis, Università degli studi di Trento, 2022. https://hdl.handle.net/11572/335243.

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In this thesis we work on problems related to tensor decomposition from a geometrical perspective. In the first part of the thesis we focus on the identifiability problem, which amounts to understand in how many ways a tensor can be decomposed as a minimal sum of elementary tensors. In particular we completely classify the identifiability of any tensor up to rank 3. In the second part of the thesis we continue to work with specific elementsand we introduce the notion of r-thTerracini locus of a Segre variety. This is the locus containing all points for which the differential of the map between the r-th abstarct secant variety and the r-th secant variety of a Segre variety drops rank. We completely determine the r-th Terracini locus of any Segre variety in the case of r = 2, 3.
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Turner, Kenneth James. "Higher-order filtering for nonlinear systems using symmetric tensors." Thesis, Queensland University of Technology, 1999.

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Hjelm, Andersson Hampus. "Classification of second order symmetric tensors in the Lorentz metric." Thesis, Linköpings universitet, Matematiska institutionen, 2010. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-57197.

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This bachelor thesis shows a way to classify second order symmetric tensors in the Lorentz metric. Some basic prerequisite about indefinite and definite algebra is introduced, such as the Jordan form, indefinite inner products, the Segre type, and the Minkowski space. There are also some results concerning the invariant 2-spaces of a symmetric tensor and a different approach on how to classify second order symmetric tensor.
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Rovi, Ana. "Analysis of 2 x 2 x 2 Tensors." Thesis, Linköping University, Department of Mathematics, 2010. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-56762.

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The question about how to determine the rank of a tensor has been widely studied in the literature. However the analytical methods to compute the decomposition of tensors have not been so much developed even for low-rank tensors.

In this report we present analytical methods for finding real and complex PARAFAC decompositions of 2 x 2 x 2 tensors before computing the actual rank of the tensor.

These methods are also implemented in MATLAB.

We also consider the question of how best lower-rank approximation gives rise to problems of degeneracy, and give some analytical explanations for these issues.

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譚天佑 and Tin-yau Tam. "A study of induced operators on symmetry classes of tensors." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 1986. http://hub.hku.hk/bib/B31230738.

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Books on the topic "Rank of symmetric tensors"

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Baerheim, Reidar. Coordinate free representation of the hierarchically symmetric tensor of rank 4 in determination of symmetry. [Utrecht: Faculteit Aardwetenschappen, Universiteit Utrecht], 1998.

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Garcia, Miguel Angel Garrido. Characterization of the Fluctuations in a Symmetric Ensemble of Rank-Based Interacting Particles. [New York, N.Y.?]: [publisher not identified], 2021.

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Werner, Müller. L²-index of elliptic operators on manifolds with cusps of rank one. Berlin: Akademie der Wissenschaften der DDR, Karl-Weierstrass-Institut für Mathematik, 1985.

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Terras, Audrey. Harmonic Analysis on Symmetric Spaces—Higher Rank Spaces, Positive Definite Matrix Space and Generalizations. New York, NY: Springer New York, 2016. http://dx.doi.org/10.1007/978-1-4939-3408-9.

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Cai, Jianqing. Statistical inference of the eigenspace components of a symmetric random deformation tensor. Munchen: Verlag der Bayerischen Akademie der Wissenschaften in Kommission beim Verlags C.H. Beck, 2004.

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Terras, Audrey. Harmonic Analysis on Symmetric Spaces--Higher Rank Spaces, Positive Definite Matrix Space and Generalizations. Springer London, Limited, 2016.

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Harmonic Analysis on Symmetric Spaces--Higher Rank Spaces, Positive Definite Matrix Space and Generalizations. Springer New York, 2016.

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Terras, Audrey. Harmonic Analysis on Symmetric Spaces―Higher Rank Spaces, Positive Definite Matrix Space and Generalizations. Springer, 2018.

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Buchler, Justin. Voter Preferences over Bundles of Roll Call Votes. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780190865580.003.0002.

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Legislators do not adopt locations in the policy space with a single action. Instead, they cast roll call votes. Thus, rational voters should evaluate legislative candidates, not based on their locations in the policy space, but based on the bundles of roll call votes implied by those locations. Voters with single-peaked, symmetric preferences over policy can prefer a distant candidate to a more proximate candidate when they rank legislative candidates based on the bundles of roll call votes implied by their locations. When the most substantively important votes on the legislative agenda are the votes that divide the party factions cleanly, extreme incumbents from both parties can defeat moderate challengers from the opposing party given the same legislative agenda.
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Lukas, Andre. The Oxford Linear Algebra for Scientists. Oxford University PressOxford, 2022. http://dx.doi.org/10.1093/oso/9780198844914.001.0001.

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Abstract This book provides a introduction into linear algebra which covers the mathematical set-up as well as applications to science. After the introductory material on sets, functions, groups and fields, the basic features of vector spaces are developed, including linear independence, bases, dimension, vector subspaces and linear maps. Practical methods for calculating with dot, cross and triple products are introduced early on. The theory of linear maps and their relation to matrices is developed in detail, culminating in the rank theorem. Algorithmic methods bases on row reduction and determinants are discussed an applied to computing the rank and the inverse of matrices and to solve systems of linear equations. Eigenvalues and eigenvectors and the application to diagonalising linear maps, as well as scalar products and unitary linear maps are covered in detail. Advanced topics included are the Jordon normal form, normal linear maps, the singular value decomposition, bi-linear and sesqui-linear forms, duality and tensors. The book also included short expositions of diverse scientific applications of linear algebra, including to internet search, classical mechanics, graph theory, cryptography, coding theory, data compression, special relativity, quantum mechanics and quantum computing.
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Book chapters on the topic "Rank of symmetric tensors"

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Hess, Siegfried. "Symmetric Second Rank Tensors." In Tensors for Physics, 55–74. Cham: Springer International Publishing, 2015. http://dx.doi.org/10.1007/978-3-319-12787-3_5.

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Malgrange, Cécile, Christian Ricolleau, and Michel Schlenker. "Second-rank tensors." In Symmetry and Physical Properties of Crystals, 205–23. Dordrecht: Springer Netherlands, 2014. http://dx.doi.org/10.1007/978-94-017-8993-6_10.

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Hess, Siegfried. "Symmetry of Second Rank Tensors, Cross Product." In Tensors for Physics, 33–46. Cham: Springer International Publishing, 2015. http://dx.doi.org/10.1007/978-3-319-12787-3_3.

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Harmouch, Jouhayna, Bernard Mourrain, and Houssam Khalil. "Decomposition of Low Rank Multi-symmetric Tensor." In Mathematical Aspects of Computer and Information Sciences, 51–66. Cham: Springer International Publishing, 2017. http://dx.doi.org/10.1007/978-3-319-72453-9_4.

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Liu, Haixia, Lizhang Miao, and Yang Wang. "Synchronized Recovery Method for Multi-Rank Symmetric Tensor Decomposition." In Mathematics and Visualization, 241–51. Cham: Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-319-91274-5_11.

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Kaimakamis, George, and Konstantina Panagiotidou. "The *-Ricci Tensor of Real Hypersurfaces in Symmetric Spaces of Rank One or Two." In Springer Proceedings in Mathematics & Statistics, 199–210. Tokyo: Springer Japan, 2014. http://dx.doi.org/10.1007/978-4-431-55215-4_18.

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Ballet, Stéphane, Jean Chaumine, and Julia Pieltant. "Shimura Modular Curves and Asymptotic Symmetric Tensor Rank of Multiplication in any Finite Field." In Algebraic Informatics, 160–72. Berlin, Heidelberg: Springer Berlin Heidelberg, 2013. http://dx.doi.org/10.1007/978-3-642-40663-8_16.

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Bocci, Cristiano, and Luca Chiantini. "Symmetric Tensors." In UNITEXT, 105–16. Cham: Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-24624-2_7.

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Tinder, Richard F. "Third- and Fourth-Rank Tensor Properties—Symmetry Considerations." In Tensor Properties of Solids, 95–122. Cham: Springer International Publishing, 2007. http://dx.doi.org/10.1007/978-3-031-79306-6_6.

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Hess, Siegfried. "Summary: Decomposition of Second Rank Tensors." In Tensors for Physics, 75. Cham: Springer International Publishing, 2015. http://dx.doi.org/10.1007/978-3-319-12787-3_6.

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Conference papers on the topic "Rank of symmetric tensors"

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Merino-Caviedes, Susana, and Marcos Martin-Fernandez. "A general interpolation method for symmetric second-rank tensors in two dimensions." In 2008 5th IEEE International Symposium on Biomedical Imaging (ISBI 2008). IEEE, 2008. http://dx.doi.org/10.1109/isbi.2008.4541150.

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Gaith, Mohamed, and Cevdet Akgoz. "On the Properties of Anisotropic Piezoelectric and Fiber Reinforced Composite Materials." In ASME 2006 International Mechanical Engineering Congress and Exposition. ASMEDC, 2006. http://dx.doi.org/10.1115/imece2006-14075.

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A new procedure based on constructing orthonormal tensor basis using the form-invariant expressions which can easily be extended to any tensor of rank n. A new decomposition, which is not in literature, of the stress tensor is presented. An innovational general form and more explicit physical property of the symmetric fourth rank elastic tensors is presented. The new method allows to measure the stiffness and piezoelectricity in the elastic fiber reinforced composite and piezoelectric ceramic materials, respecively, using a proposed norm concept on the crystal scale. This method will allow to investigate the effects of fiber orientaion, number of plies, material properties of matrix and fibers, and degree of anisotropy on the stiffness of the structure. The results are compared with those available in the literature for semiconductor compounds, piezoelectric ceramics and fiber reinforced composite materials.
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Marmin, Arthur, Marc Castella, and Jean-Christophe Pesquet. "Detecting the Rank of a Symmetric Tensor." In 2019 27th European Signal Processing Conference (EUSIPCO). IEEE, 2019. http://dx.doi.org/10.23919/eusipco.2019.8902781.

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Kyrgyzov, Olexiy, and Deniz Erdogmus. "Geometric structure of sum-of-rank-1 decompositions for n-dimensional order-p symmetric tensors." In 2008 IEEE International Symposium on Circuits and Systems - ISCAS 2008. IEEE, 2008. http://dx.doi.org/10.1109/iscas.2008.4541674.

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Barbier, Jean, Clement Luneau, and Nicolas Macris. "Mutual Information for Low-Rank Even-Order Symmetric Tensor Factorization." In 2019 IEEE Information Theory Workshop (ITW). IEEE, 2019. http://dx.doi.org/10.1109/itw44776.2019.8989408.

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Chen, Bin, and John Moreland. "Human Brain Diffusion Tensor Imaging Visualization With Virtual Reality." In ASME 2010 World Conference on Innovative Virtual Reality. ASMEDC, 2010. http://dx.doi.org/10.1115/winvr2010-3761.

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Magnetic resonance diffusion tensor imaging (DTI) is sensitive to the anisotropic diffusion of water exerted by its macromolecular environment and has been shown useful in characterizing structures of ordered tissues such as the brain white matter, myocardium, and cartilage. The water diffusivity inside of biological tissues is characterized by the diffusion tensor, a rank-2 symmetrical 3×3 matrix, which consists of six independent variables. The diffusion tensor contains much information of diffusion anisotropy. However, it is difficult to perceive the characteristics of diffusion tensors by looking at the tensor elements even with the aid of traditional three dimensional visualization techniques. There is a need to fully explore the important characteristics of diffusion tensors in a straightforward and quantitative way. In this study, a virtual reality (VR) based MR DTI visualization with high resolution anatomical image segmentation and registration, ROI definition and neuronal white matter fiber tractography visualization and fMRI activation map integration is proposed. The VR application will utilize brain image visualization techniques including surface, volume, streamline and streamtube rendering, and use head tracking and wand for navigation and interaction, the application will allow the user to switch between different modalities and visualization techniques, as well making point and choose queries. The main purpose of the application is for basic research and clinical applications with quantitative and accurate measurements to depict the diffusivity or the degree of anisotropy derived from the diffusion tensor.
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Wilson, Daniel W., Elias N. Glytsis, Nile F. Hartman, and Thomas K. Gaylord. "Bulk photovoltaic tensor and polarization conversion in LiNbO3." In OSA Annual Meeting. Washington, D.C.: Optica Publishing Group, 1990. http://dx.doi.org/10.1364/oam.1990.tud8.

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Using methods that are applicable to any crystal class, the form of the third-rank bulk photovoltaic tensor for LiNbO3 is found to be composed of a symmetric part (similar to the piezoelectric tensor) and an antisymmetric part. In LiNbCO3 the antisymmetric part contains two positive elements and two negative elements, all of the same magnitude, and these are shown to lead to polarization conversion in bulk crystals and in waveguides. The polarization conversion1 is examined in terms of the propagation direction, the initial polarization direction, the electro-optic effect, and the nonlinear interaction. For propagation perpendicular to the optic axis, a space-oscillating photocurrent is produced, forming an index grating in the direction of propagation. The nonlinear coupled-wave equations that result from this grating show a unidirectional conversion to an extraordinarily polarizated output wave.2,3 For propagation parallel to the optic axis, a uniform photocurrent is produced, which can couple orthogonal polarizations by a rotation of the index ellipsoid. Experimental results for bulk LiNbO3 and Ti:LiNbO3 waveguides are presented.
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Kiraly, Franz J., and Andreas Ziehe. "Approximate rank-detecting factorization of low-rank tensors." In ICASSP 2013 - 2013 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). IEEE, 2013. http://dx.doi.org/10.1109/icassp.2013.6638397.

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Wang, Xiaofei, and Carmeliza Navasca. "Adaptive Low Rank Approximation for Tensors." In 2015 IEEE International Conference on Computer Vision Workshop (ICCVW). IEEE, 2015. http://dx.doi.org/10.1109/iccvw.2015.124.

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Rajbhandari, Samyam, Akshay Nikam, Pai-Wei Lai, Kevin Stock, Sriram Krishnamoorthy, and P. Sadayappan. "CAST: Contraction Algorithm for Symmetric Tensors." In 2014 43nd International Conference on Parallel Processing (ICPP). IEEE, 2014. http://dx.doi.org/10.1109/icpp.2014.35.

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Reports on the topic "Rank of symmetric tensors"

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Khalfan, H., R. H. Byrd, and R. B. Schnabel. A Theoretical and Experimental Study of the Symmetric Rank One Update. Fort Belvoir, VA: Defense Technical Information Center, December 1990. http://dx.doi.org/10.21236/ada233965.

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