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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 biologiqueWe 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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Tam, Tin-yau. "A study of induced operators on symmetry classes of tensors /." [Hong Kong] : University of Hong Kong, 1986. http://sunzi.lib.hku.hk/hkuto/record.jsp?B12322593.
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Howarth, Laura. "The existence and structure of constants of geodesic motion admitted by spherically symmetric static space-times." Thesis, University of Hull, 1999. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.310318.
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Cordolino, Sobral Andrews. "Robust low-rank and sparse decomposition for moving object detection : from matrices to tensors." Thesis, La Rochelle, 2017. http://www.theses.fr/2017LAROS007/document.
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Dans ce manuscrit de thèse, nous introduisons les avancées récentes sur la décomposition en matrices (et tenseurs) de rang faible et parcimonieuse ainsi que les contributions pour faire face aux principaux problèmes dans ce domaine. Nous présentons d’abord un aperçu des méthodes matricielles et tensorielles les plus récentes ainsi que ses applications sur la modélisation d’arrière-plan et la segmentation du premier plan. Ensuite, nous abordons le problème de l’initialisation du modèle de fond comme un processus de reconstruction à partir de données manquantes ou corrompues. Une nouvelle méthodologie est présentée montrant un potentiel intéressant pour l’initialisation de la modélisation du fond dans le cadre de VSI. Par la suite, nous proposons une version « double contrainte » de l’ACP robuste pour améliorer la détection de premier plan en milieu marin dans des applications de vidéo-surveillance automatisées. Nous avons aussi développé deux algorithmes incrémentaux basés sur tenseurs afin d’effectuer une séparation entre le fond et le premier plan à partir de données multidimensionnelles. Ces deux travaux abordent le problème de la décomposition de rang faible et parcimonieuse sur des tenseurs. A la fin, nous présentons un travail particulier réalisé en conjonction avec le Centre de Vision Informatique (CVC) de l’Université Autonome de Barcelone (UAB)This thesis introduces the recent advances on decomposition into low-rank plus sparse matrices and tensors, as well as the main contributions to face the principal issues in moving object detection. First, we present an overview of the state-of-the-art methods for low-rank and sparse decomposition, as well as their application to background modeling and foreground segmentation tasks. Next, we address the problem of background model initialization as a reconstruction process from missing/corrupted data. A novel methodology is presented showing an attractive potential for background modeling initialization in video surveillance. Subsequently, we propose a double-constrained version of robust principal component analysis to improve the foreground detection in maritime environments for automated video-surveillance applications. The algorithm makes use of double constraints extracted from spatial saliency maps to enhance object foreground detection in dynamic scenes. We also developed two incremental tensor-based algorithms in order to perform background/foreground separation from multidimensional streaming data. These works address the problem of low-rank and sparse decomposition on tensors. Finally, we present a particular work realized in conjunction with the Computer Vision Center (CVC) at Autonomous University of Barcelona (UAB)
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Eslava, Fernández Laura. "The rank of symmetric random matrices via a graph process." Thesis, McGill University, 2013. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=114602.
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Random matrix theory comprises a broad range of topics and avenues of research, one of them being to understand the probability of singularity for discrete random matrices. This is a fundamental, basic question about discrete matrices. Although is been proven that for random symmetric Bernoulli matrices the probability of singularity decays at least polynomially in the size of the matrix, it is conjectured that the right order of decay is exponential. We are interested in the adjacency matrix Q of the Erdos-Réyni random graph and we study the statistics of the rank of Q as a means of understanding the probability of singularity of Q. We take a stochastic process perspective, looking at the family of matrices Q (parametrized by p) as an increasing family of random matrices. We then investigate the structure of Q at the moment that it becomes non-singular and prove that, similar to some monotone properties of random graphs, the property of being non-singular obeys a so-called 'hitting time theorem'. Broadly speaking, this means that all-zero rows, which are a 'local' property of the matrix, are the only obstruction for non-singularity. This fact, which is the main novel contribution to the thesis, extends previous work by Costello and Vu.La théorie des matrices aléatoires a un large éventail de sujets et de pistes de recherche, l'un d'entre eux étant de comprendre la probabilité de la singularité des matrices aléatoires discrètes. Ca a été prouvé que pour des matrices aléatoires de Bernoulli symétriques la probabilité de singularité a des bornes polynomiales, mais la conjecture est que le bon ordre de décroissance est exponentiel. Nous sommes intéressés par la matrice d'adjacence Q du graphe aléatoire d'Erdos et Réyni et nous étudions les statistiques du rang de Q comme un moyen de comprende la probabilité de singularité de Q. Nous proposons maintenant une perspective de processus stochastique. Dans ce mémoire, nous considérons la famille Q comme une famille croissante de matrices aléatoires et nous étudions la structure de Q au moment oú il devient non singulière et nous prouvons de la même facon pour certaines propriétés monotones des graphes aléatoires, la propriété d'être non singulière obéit à soi-disant 'théorème de temps d'arrêt'. D'une manière globale, cela signifie que les lignes remplies de zéros, qui sont une propriété locale de la matrice, sont la seule obstruction pour la non-singularité. Ce fait, qui est la nouvelle contribution principale de ce mémoire, élargie les résultats antérieurs de Costello et Vu.
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Coloigner, Julie. "Line search and trust region strategies for canonical decomposition of semi-nonnegative semi-symmetric tensors." Rennes 1, 2012. http://www.theses.fr/2012REN1S172.
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Several numerical methods are proposed in this thesis in order to compute the canonical polyadic decomposition of semi-nonnegative semi-symmetric three-way arrays, say with two identical nonnegative loading matrices. Such multi-way arrays are encountered in blind source separation when a set of data cumulant matrices have to be jointly diagonalized in order to identify a nonnegative mixture of independent sources. The proposed solutions belong to two fundamental strategies of optimization: line search and trust region strategies. Each optimization takes into account the semi-symmetry but also the semi-nonnegativity of the processed tensor. The latter constraint is imposed by means of square or exponentional changes of variable, leading to an unconstrained problem. A matrix computation of derivatives is performed for most of the proposed methods, allowing for a straightforward implementation in matrix programming environments. Computer results show a better behaviour of the proposed methods in comparison with the classical Levenberg-Marquardt technique, which uses no a priori information about the considered array. It appears that a joint use of semi-symmetry and semi-nonnegativity improves the performance for low signal to noise ratios but also for rank values greater than dimensions. Our algorithms are also tested, through the semi-nonnegative ICA, on simulated magnetic resonance spectroscopy data and compared to classical independent component analysis and nonnegative matrix factorization methodsPendant cette thèse, des méthodes numériques pour décomposer canoniquement des tableaux d'ordre 3 semi-nonnégatifs et semi-symétriques ont été proposées. Ces tableaux possèdent deux matrices de facteurs identiques à composantes positives. Ils apparaissent en séparation aveugle de sources lorsque l'on souhaite diagonaliser conjointement par congruence un ensemble de tranches matricielles de tableaux d'un mélange nonnégatif de sources independantes. Nous nous sommes intéressés à deux familles d'optimisation : la première est celle de la recherche linéaire, combinant le calcul d'une direction de descente basée sur des dérivées de premier et deuxième ordre à la recherche d'un pas optimal ; la seconde est celle de la région de confiance. Ces familles prennent en compte non seulement l'égalité mais aussi la nonnégativité de deux des trois matrices de facteurs par un changement de variable, carré ou exponentiel, permettant ainsi de se ramener à un problème d'optimisation sans contrainte. Le calcul des dérivées est effectué matriciellement pour la plupart des methodes proposées, ce qui permet une implémentation efficace de ces dernières dans un langage de programmation matricielle. Nos simulations sur des données aléatoires montrent un gain en performance par comparaison avec des méthodes n'exploitant aucun a priori notamment dans des contextes difficiles : faibles valeurs de rapport signal à bruit, collinearité des facteurs, et valeurs de rang excédant la plus grande des dimensions. Nos algorithmes sont aussi testés sur données simulées et semi-simulées de spectroscopie à résonance magnétique dans le cadre de l'analyse en composantes indépendantes (ICA) et comparés à des méthodes classiques d'ICA et de factorisation matricielle nonnégative
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Malloy, Nicole Andrea. "Minimum Rank Problems for Cographs." BYU ScholarsArchive, 2013. https://scholarsarchive.byu.edu/etd/3873.
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Let G be a simple graph on n vertices, and let S(G) be the class of all real-valued symmetric nxn matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. The smallest rank achieved by a matrix in S(G) is called the minimum rank of G, denoted mr(G). The maximum nullity achieved by a matrix in S(G) is denoted M(G). For each graph G, there is an associated minimum rank class, MR(G) consisting of all matrices A in S(G) with rank A = mr(G). Although no restrictions are applied to the diagonal entries of matrices in S(G), sometimes diagonal entries corresponding to specific vertices of G must be zero for all matrices in MR(G). These vertices are known as nil vertices (see [6]). In this paper I discuss some basic results about nil vertices in general and nil vertices in cographs and prove that cographs with a nil vertex of a particular form contain two other nil vertices symmetric to the first. I discuss several open questions relating to these results and a counterexample. I prove that for all cographs G without an induced complete tripartite graph with independent sets all of size 3, the zero-forcing number Z(G), a graph theoretic parameter, is equal to M(G). In fact this result holds for a slightly larger class of cographs and in particular holds for all threshold graphs. Lastly, I prove that the maximum of the minimum ranks of all cographs on n vertices is the floor of 2n/3.
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Walach, Hanna Maria [Verfasser], and Christian [Akademischer Betreuer] Lubich. "Time integration for the dynamical low-rank approximation of matrices and tensors / Hanna Maria Walach ; Betreuer: Christian Lubich." Tübingen : Universitätsbibliothek Tübingen, 2019. http://d-nb.info/1190639831/34.
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Sexton, William Nelson. "The Minimum Rank of Schemes on Graphs." BYU ScholarsArchive, 2014. https://scholarsarchive.byu.edu/etd/4402.
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Let G be an undirected graph on n vertices and let S(G) be the class of all real-valued symmetric n × n matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. Let V = {1, 2, . . . , n} be the vertex set of G. A scheme on G is a function f : V → {0, 1}. Given a scheme f on G, there is an associated class of matrices Sf (G) = {A ∈ S(G)|aii = 0 if and only if f(i) = 0}. A scheme f is said to be constructible if there exists a matrix A ∈ Sf (G) with rank A = min{rank M|M ∈ S(G)}. We explore properties of constructible schemes and give a complete classification of which schemes are constructible for paths and cycles. We also consider schemes on complete graphs and show the existence of a graph for which every possible scheme is constructible.
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Majidzadeh, Garjani Babak. "On the Rank of the Reduced Density Operator for the Laughlin State and Symmetric Polynomials." Licentiate thesis, Stockholms universitet, Fysikum, 2015. http://urn.kb.se/resolve?urn=urn:nbn:se:su:diva-118807.
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One effective tool to probe a system revealing topological order is to biparti- tion the system in some way and look at the properties of the reduced density operator corresponding to one part of the system. In this thesis we focus on a bipartition scheme known as the particle cut in which the particles in the system are divided into two groups and we look at the rank of the re- duced density operator. In the context of fractional quantum Hall physics it is conjectured that the rank of the reduced density operator for a model Hamiltonian describing the system is equal to the number of quasi-hole states. Here we consider the Laughlin wave function as the model state for the system and try to put this conjecture on a firmer ground by trying to determine the rank of the reduced density operator and calculating the number of quasi-hole states. This is done by relating this conjecture to the mathematical properties of symmetric polynomials and proving a theorem that enables us to find the lowest total degree of symmetric polynomials that vanish under some specific transformation referred to as clustering transformation.
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Grout, Jason Nicholas. "The Minimum Rank Problem Over Finite Fields." Diss., CLICK HERE for online access, 2007. http://contentdm.lib.byu.edu/ETD/image/etd1995.pdf.
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Palzer, Wolfgang [Verfasser], Alexander [Akademischer Betreuer] Alldridge, and George [Akademischer Betreuer] Marinescu. "Fourier Analysis on Non-Compact Symmetric Superspaces of Rank One / Wolfgang Palzer. Gutachter: Alexander Alldridge ; George Marinescu." Köln : Universitäts- und Stadtbibliothek Köln, 2014. http://d-nb.info/1051088070/34.
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Yip, Martha. "Genus one partitions." Thesis, University of Waterloo, 2006. http://hdl.handle.net/10012/2933.
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We obtain a tight upper bound for the genus of a partition, and calculate the number of partitions of maximal genus. The generating series for genus zero and genus one rooted hypermonopoles is obtained in closed form by specializing the genus series for hypermaps. We discuss the connection between partitions and rooted hypermonopoles, and suggest how a generating series for genus one partitions may be obtained via the generating series for genus one rooted hypermonopoles. An involution on the poset of genus one partitions is constructed from the associated hypermonopole diagrams, showing that the poset is rank-symmetric. Also, a symmetric chain decomposition is constructed for the poset of genus one partitions, which consequently shows that it is strongly Sperner.
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Garreis, Sebastian [Verfasser], Michael [Akademischer Betreuer] Ulbrich, Matthias [Gutachter] Heinkenschloss, Christian [Gutachter] Clason, and Michael [Gutachter] Ulbrich. "Optimal Control under Uncertainty: Theory and Numerical Solution with Low-Rank Tensors / Sebastian Garreis ; Gutachter: Matthias Heinkenschloss, Christian Clason, Michael Ulbrich ; Betreuer: Michael Ulbrich." München : Universitätsbibliothek der TU München, 2019. http://d-nb.info/1179360737/34.
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Owens, Kayla Denise. "Properties of the Zero Forcing Number." BYU ScholarsArchive, 2009. https://scholarsarchive.byu.edu/etd/2216.
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The zero forcing number is a graph parameter first introduced as a tool for solving the minimum rank problem, which is: Given a simple, undirected graph G, and a field F, let S(F,G) denote the set of all symmetric matrices A=[a_{ij}] with entries in F such that a_{ij} doess not equal 0 if and only if ij is an edge in G. Find the minimum possible rank of a matrix in S(F,G). It is known that the zero forcing number Z(G) provides an upper bound for the maximum nullity of a graph. I investigate properties of the zero forcing number, including its behavior under various graph operations.
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Rabusseau, Guillaume. "A tensor perspective on weighted automata, low-rank regression and algebraic mixtures." Thesis, Aix-Marseille, 2016. http://www.theses.fr/2016AIXM4062.
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Ce manuscrit regroupe différents travaux explorant les interactions entre les tenseurs et l'apprentissage automatique. Le premier chapitre est consacré à l'extension des modèles de séries reconnaissables de chaînes et d'arbres aux graphes. Nous y montrons que les modèles d'automates pondérés de chaînes et d'arbres peuvent être interprétés d'une manière simple et unifiée à l'aide de réseaux de tenseurs, et que cette interprétation s'étend naturellement aux graphes ; nous étudions certaines propriétés de ce modèle et présentons des résultats préliminaires sur leur apprentissage. Le second chapitre porte sur la minimisation approximée d'automates pondérés d'arbres et propose une approche théoriquement fondée à la problématique suivante : étant donné un automate pondéré d'arbres à n états, comment trouver un automate à mThis thesis tackles several problems exploring connections between tensors and machine learning. In the first chapter, we propose an extension of the classical notion of recognizable function on strings and trees to graphs. We first show that the computations of weighted automata on strings and trees can be interpreted in a natural and unifying way using tensor networks, which naturally leads us to define a computational model on graphs: graph weighted models; we then study fundamental properties of this model and present preliminary learning results. The second chapter tackles a model reduction problem for weighted tree automata. We propose a principled approach to the following problem: given a weighted tree automaton with n states, how can we find an automaton with m
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Guerrero, Flores Danny Joel. "On Updating Preconditioners for the Iterative Solution of Linear Systems." Doctoral thesis, Universitat Politècnica de València, 2018. http://hdl.handle.net/10251/104923.
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El tema principal de esta tesis es el desarrollo de técnicas de actualización de precondicionadores para resolver sistemas lineales de gran tamaño y dispersos Ax=b mediante el uso de métodos iterativos de Krylov. Se consideran dos tipos interesantes de problemas. En el primero se estudia la solución iterativa de sistemas lineales no singulares y antisimétricos, donde la matriz de coeficientes A tiene parte antisimétrica de rango bajo o puede aproximarse bien con una matriz antisimétrica de rango bajo. Sistemas como este surgen de la discretización de PDEs con ciertas condiciones de frontera de Neumann, la discretización de ecuaciones integrales y métodos de puntos interiores, por ejemplo, el problema de Bratu y la ecuación integral de Love. El segundo tipo de sistemas lineales considerados son problemas de mínimos cuadrados (LS) que se resuelven considerando la solución del sistema equivalente de ecuaciones normales. Concretamente, consideramos la solución de problemas LS modificados y de rango incompleto. Por problema LS modificado se entiende que el conjunto de ecuaciones lineales se actualiza con alguna información nueva, se agrega una nueva variable o, por el contrario, se elimina alguna información o variable del conjunto. En los problemas LS de rango deficiente, la matriz de coeficientes no tiene rango completo, lo que dificulta el cálculo de una factorización incompleta de las ecuaciones normales. Los problemas LS surgen en muchas aplicaciones a gran escala de la ciencia y la ingeniería como, por ejemplo, redes neuronales, programación lineal, sismología de exploración o procesamiento de imágenes.
Los precondicionadores directos para métodos iterativos usados habitualmente son las factorizaciones incompletas LU, o de Cholesky cuando la matriz es simétrica definida positiva. La principal contribución de esta tesis es el desarrollo de técnicas de actualización de precondicionadores. Básicamente, el método consiste en el cálculo de una descomposición incompleta para un sistema lineal aumentado equivalente, que se utiliza como precondicionador para el problema original.
El estudio teórico y los resultados numéricos presentados en esta tesis muestran el rendimiento de la técnica de precondicionamiento propuesta y su competitividad en comparación con otros métodos disponibles en la literatura para calcular precondicionadores para los problemas estudiados.The main topic of this thesis is updating preconditioners for solving large sparse linear systems Ax=b by using Krylov iterative methods. Two interesting types of problems are considered. In the first one is studied the iterative solution of non-singular, non-symmetric linear systems where the coefficient matrix A has a skew-symmetric part of low-rank or can be well approximated with a skew-symmetric low-rank matrix. Systems like this arise from the discretization of PDEs with certain Neumann boundary conditions, the discretization of integral equations as well as path following methods, for example, the Bratu problem and the Love's integral equation. The second type of linear systems considered are least squares (LS) problems that are solved by considering the solution of the equivalent normal equations system. More precisely, we consider the solution of modified and rank deficient LS problems. By modified LS problem, it is understood that the set of linear relations is updated with some new information, a new variable is added or, contrarily, some information or variable is removed from the set. Rank deficient LS problems are characterized by a coefficient matrix that has not full rank, which makes difficult the computation of an incomplete factorization of the normal equations. LS problems arise in many large-scale applications of the science and engineering as for instance neural networks, linear programming, exploration seismology or image processing. Usually, incomplete LU or incomplete Cholesky factorization are used as preconditioners for iterative methods. The main contribution of this thesis is the development of a technique for updating preconditioners by bordering. It consists in the computation of an approximate decomposition for an equivalent augmented linear system, that is used as preconditioner for the original problem. The theoretical study and the results of the numerical experiments presented in this thesis show the performance of the preconditioner technique proposed and its competitiveness compared with other methods available in the literature for computing preconditioners for the problems studied.
El tema principal d'esta tesi és actualitzar precondicionadors per a resoldre sistemes lineals grans i buits Ax=b per mitjà de l'ús de mètodes iteratius de Krylov. Es consideren dos tipus interessants de problemes. En el primer s'estudia la solució iterativa de sistemes lineals no singulars i antisimètrics, on la matriu de coeficients A té una part antisimètrica de baix rang, o bé pot aproximar-se amb una matriu antisimètrica de baix rang. Sistemes com este sorgixen de la discretització de PDEs amb certes condicions de frontera de Neumann, la discretització d'equacions integrals i mètodes de punts interiors, per exemple, el problema de Bratu i l'equació integral de Love. El segon tipus de sistemes lineals considerats, són problemes de mínims quadrats (LS) que es resolen considerant la solució del sistema equivalent d'equacions normals. Concretament, considerem la solució de problemes de LS modificats i de rang incomplet. Per problema LS modificat, s'entén que el conjunt d'equacions lineals s'actualitza amb alguna informació nova, s'agrega una nova variable o, al contrari, s'elimina alguna informació o variable del conjunt. En els problemes LS de rang deficient, la matriu de coeficients no té rang complet, la qual cosa dificultata el calcul d'una factorització incompleta de les equacions normals. Els problemes LS sorgixen en moltes aplicacions a gran escala de la ciència i l'enginyeria com, per exemple, xarxes neuronals, programació lineal, sismologia d'exploració o processament d'imatges. Els precondicionadors directes per a mètodes iteratius utilitzats més a sovint són les factoritzacions incompletes tipus ILU, o la factorització incompleta de Cholesky quan la matriu és simètrica definida positiva. La principal contribució d'esta tesi és el desenvolupament de tècniques d'actualització de precondicionadors. Bàsicament, el mètode consistix en el càlcul d'una descomposició incompleta per a un sistema lineal augmentat equivalent, que s'utilitza com a precondicionador pel problema original. L'estudi teòric i els resultats numèrics presentats en esta tesi mostren el rendiment de la tècnica de precondicionament proposta i la seua competitivitat en comparació amb altres mètodes disponibles en la literatura per a calcular precondicionadors per als problemes considerats.
Guerrero Flores, DJ. (2018). On Updating Preconditioners for the Iterative Solution of Linear Systems [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/104923
TESIS
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Nelson, Curtis G. "Diagonal Entry Restrictions in Minimum Rank Matrices, and the Inverse Inertia and Eigenvalue Problems for Graphs." BYU ScholarsArchive, 2012. https://scholarsarchive.byu.edu/etd/3246.
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Let F be a field, let G be an undirected graph on n vertices, and let SF(G) be the set of all F-valued symmetric n x n matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. Let MRF(G) be defined as the set of matrices in SF(G) whose rank achieves the minimum of the ranks of matrices in SF(G). We develop techniques involving Z-hat, a process termed nil forcing, and induced subgraphs, that can determine when diagonal entries corresponding to specific vertices of G must be zero or nonzero for all matrices in MRF(G). We call these vertices nil or nonzero vertices, respectively. If a vertex is not a nil or nonzero vertex, we call it a neutral vertex. In addition, we completely classify the vertices of trees in terms of the classifications: nil, nonzero and neutral. Next we give an example of how nil vertices can help solve the inverse inertia problem. Lastly we give results about the inverse eigenvalue problem and solve a more complex variation of the problem (the λ, µ problem) for the path on 4 vertices. We also obtain a general result for the λ, µ problem concerning the number of λ’s and µ’s that can be equal.
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Rydh, David. "Families of cycles and the Chow scheme." Doctoral thesis, Stockholm : Matematik, Mathematics, Kungliga Tekniska högskolan, 2008. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-4813.
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Muduli, Pranaba Kishor. "Ferromagnetic thin films of Fe and Fe 3 Si on low-symmetric GaAs(113)A substrates." Doctoral thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät I, 2006. http://dx.doi.org/10.18452/15473.
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In dieser Arbeit werden das Wachstum mittels Molekularstrahlepitaxie und die Eigenschaften der Ferromagneten Fe und Fe_3Si auf niedrig-symmetirschen GaAs(113)A-Substraten studiert. Drei wichtige Aspekte werden untersucht: (i) Wachstum und strukturelle Charakterisierung, (ii) magnetische Eigenschaften und (iii) Magnetotransporteigenschaften der Fe und Fe_3Si Schichten auf GaAs(113)A-Substraten. Das Wachstum der Fe- und Fe_3Si-Schichten wurde bei einer Wachstumstemperatur von = bzw. 250 °C optimiert. Bei diesen Wachstumstemperaturen zeigen die Schichten eine hohe Kristallperfektion und glatte Grenz- und Oberflächen analog zu [001]-orientierten Schichten. Weiterhin wurde die Stabilität der Fe_(3+x)Si_(1-x) Phase über einen weiten Kompositionsbereich innerhalb der Fe_3Si-Stoichiometry demonstriert. Die Abhängigkeit der magnetischen Anisotropie innerhalb der Schichtebene von der Schichtdicke weist zwei Bereiche auf: einen Beresich mit dominanter uniaxialer Anisotropie für Fe-Schichten = 70 MLs. Weiterhin wird eine magnetische Anisotropie senkrecht zur Schichtebene in sehr dünnen Schichten gefunden. Der Grenzflächenbeitrag sowohl der uniaxialen als auch der senkrechten Anisotropiekonstanten, die aus der Dickenabhängigkeit bestimmt wurden, sind unabhängig von der [113]-Orientierung und eine inhärente Eigenschaft der Fe/GaAs-Grenzfläche. Die anisotrope Bindungskonfiguration zwischen den Fe und den As- oder Ga-Atomen an der Grenzfläche wird als Ursache für die uniaxiale magnetische Anisotropie betrachtet. Die magnetische Anisotropie der Fe_3Si-Schichten auf GaAs(113)A-Substraten zeigt ein komplexe Abhängigkeit von der Wachstumsbedingungen und der Komposition der Schichten. In den Magnetotransportuntersuchungen tritt sowohl in Fe(113)- als auch in Fe_3Si(113)-Schichten eine antisymmetrische Komponente (ASC) im planaren Hall-Effekt (PHE) auf. Ein phänomenologisches Modell, dass auf der Kristallsymmetrie basiert, liefert ein gute Beschreibung sowohl der ASC im PHE als auch des symmetrischen, anisotropen Magnetowiderstandes. Das Modell zeigt, dass die beobachtete ASC als Hall-Effekt zweiter Ordnung beschreiben werden kann.In this work, the molecular-beam epitaxial growth and properties of ferromagnets, namely Fe and Fe_3Si are studied on low-symmetric GaAs(113)A substrates. Three important aspects are investigated: (i) growth and structural characterization, (ii) magnetic properties, and (iii) magnetotransport properties of Fe and Fe_3Si films on GaAs(113)A substrates. The growth of Fe and Fe_3Si films is optimized at growth temperatures of 0 and 250 degree Celsius, respectively, where the layers exhibit high crystal quality and a smooth interface/surface similar to the [001]-oriented films. The stability of Fe_(3+x)Si_(1-x) phase over a range of composition around the Fe_3Si stoichiometry is also demonstrated. The evolution of the in-plane magnetic anisotropy with film thickness exhibits two regions: a uniaxial magnetic anisotropy (UMA) for Fe film thicknesses = 70 MLs. The existence of an out-of-plane perpendicular magnetic anisotropy is also detected in ultrathin Fe films. The interfacial contribution of both the uniaxial and the perpendicular anisotropy constants, derived from the thickness-dependent study, are found to be independent of the [113] orientation and are hence an inherent property of the Fe/GaAs interface. The origin of the UMA is attributed to anisotropic bonding between Fe and As or Ga at the interface, similarly to Fe/GaAs(001). The magnetic anisotropy in Fe_3Si on GaAs(113)A exhibits a complex dependence on the growth conditions and composition. Magnetotransport measurements of both Fe(113) and Fe_3Si(113) films shows the striking appearance of an antisymmetric component (ASC) in the planar Hall effect (PHE). A phenomenological model based on the symmetry of the crystal provides a good explanation to both the ASC in the PHE as well as the symmetric anisotropic magnetoresistance. The model shows that the observed ASC component can be ascribed to a second-order Hall effect.
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Medeiros, Rainelly Cunha de. "Degenerations of classical square matrices and their determinantal structure." Universidade Federal da Paraíba, 2017. http://tede.biblioteca.ufpb.br:8080/handle/tede/9318.
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In thisthesis,westudycertaindegenerations/specializationsofthegenericsquare matrix overa eld k of characteristiczeroalongitsmainrelatedstructures,suchthe determinantofthematrix,theidealgeneratedbyitspartialderivatives,thepolarmap de ned bythesederivatives,theHessianmatrixandtheidealofsubmaximalminorsof the matrix.Thedegenerationtypesofthegenericsquarematrixconsideredhereare: (1) degenerationby\cloning"(repeating)avariable;(2)replacingasubsetofentriesby zeros, inastrategiclayout;(3)furtherdegenerationsoftheabovetypesstartingfrom certain specializationsofthegenericsquarematrix,suchasthegenericsymmetric matrix andthegenericsquareHankelmatrix.Thefocusinallthesedegenerations is intheinvariantsdescribedabove,highlightingonthehomaloidalbehaviorofthe determinantofthematrix.Forthis,weemploytoolscomingfromcommutativealgebra, with emphasisonidealtheoryandsyzygytheory.
Nesta tese,estudamoscertasdegenera c~oes/especializa c~oesdamatrizquadradagen erica sobre umcorpo k de caracter sticazero,aolongodesuasprincipaisestruturasrela- cionadas, taiscomoodeterminantedamatriz,oidealgeradoporsuasderivadasparci- ais, omapapolarde nidoporessasderivadas,amatrizHessianaeoidealdosmenores subm aximosdamatriz.Ostiposdedegenera c~aodamatrizquadradagen ericacon- siderados aquis~ao:(1)degenera c~aopor\clonagem"(repeti c~ao)deumavari avel;(2) substitui c~aodeumsubconjuntodeentradasporzeros,emumadisposi c~aoestrat egica; (3) outrasdegenera c~oesdostiposacimapartindodecertasespecializa c~oesdamatriz quadrada gen erica,taiscomoamatrizgen ericasim etricaeamatrizquadradagen erica de Hankel.Ofocoemtodasessasdegenera c~oes enosinvariantesdescritosacima, com destaqueparaocomportamentohomaloidaldodeterminantedamatriz.Paratal, empregamos ferramentasprovenientesda algebracomutativa,com^enfasenateoriade ideais enateoriadesiz gias.
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Lestandi, Lucas. "Approximations de rang faible et modèles d'ordre réduit appliqués à quelques problèmes de la mécanique des fluides." Thesis, Bordeaux, 2018. http://www.theses.fr/2018BORD0186/document.
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Les dernières décennies ont donné lieux à d'énormes progrès dans la simulation numérique des phénomènes physiques. D'une part grâce au raffinement des méthodes de discrétisation des équations aux dérivées partielles. Et d'autre part grâce à l'explosion de la puissance de calcul disponible. Pourtant, de nombreux problèmes soulevés en ingénierie tels que les simulations multi-physiques, les problèmes d'optimisation et de contrôle restent souvent hors de portée. Le dénominateur commun de ces problèmes est le fléau des dimensions. Un simple problème tridimensionnel requiert des centaines de millions de points de discrétisation auxquels il faut souvent ajouter des milliers de pas de temps pour capturer des dynamiques complexes. L'avènement des supercalculateurs permet de générer des simulations de plus en plus fines au prix de données gigantesques qui sont régulièrement de l'ordre du pétaoctet. Malgré tout, cela n'autorise pas une résolution ``exacte'' des problèmes requérant l'utilisation de plusieurs paramètres. L'une des voies envisagées pour résoudre ces difficultés est de proposer des représentations ne souffrant plus du fléau de la dimension. Ces représentations que l'on appelle séparées sont en fait un changement de paradigme. Elles vont convertir des objets tensoriels dont la croissance est exponentielle $n^d$ en fonction du nombre de dimensions $d$ en une représentation approchée dont la taille est linéaire en $d$. Pour le traitement des données tensorielles, une vaste littérature a émergé ces dernières années dans le domaine des mathématiques appliquées.Afin de faciliter leurs utilisations dans la communauté des mécaniciens et en particulier pour la simulation en mécanique des fluides, ce manuscrit présente dans un vocabulaire rigoureux mais accessible les formats de représentation des tenseurs et propose une étude détaillée des algorithmes de décomposition de données qui y sont associées. L'accent est porté sur l'utilisation de ces méthodes, aussi la bibliothèque de calcul texttt{pydecomp} développée est utilisée pour comparer l'efficacité de ces méthodes sur un ensemble de cas qui se veut représentatif. La seconde partie de ce manuscrit met en avant l'étude de l'écoulement dans une cavité entraînée à haut nombre de Reynolds. Cet écoulement propose une physique très riche (séquence de bifurcation de Hopf) qui doit être étudiée en amont de la construction de modèle réduit. Cette étude est enrichie par l'utilisation de la décomposition orthogonale aux valeurs propres (POD). Enfin une approche de construction ``physique'', qui diffère notablement des développements récents pour les modèles d'ordre réduit, est proposée. La connaissance détaillée de l'écoulement permet de construire un modèle réduit simple basé sur la mise à l'échelle des fréquences d'oscillation (time-scaling) et des techniques d'interpolation classiques (Lagrange,..)Numerical simulation has experienced tremendous improvements in the last decadesdriven by massive growth of computing power. Exascale computing has beenachieved this year and will allow solving ever more complex problems. But suchlarge systems produce colossal amounts of data which leads to its own difficulties.Moreover, many engineering problems such as multiphysics or optimisation andcontrol, require far more power that any computer architecture could achievewithin the current scientific computing paradigm. In this thesis, we proposeto shift the paradigm in order to break the curse of dimensionality byintroducing decomposition and building reduced order models (ROM) for complexfluid flows.This manuscript is organized into two parts. The first one proposes an extendedreview of data reduction techniques and intends to bridge between appliedmathematics community and the computational mechanics one. Thus, foundingbivariate separation is studied, including discussions on the equivalence ofproper orthogonal decomposition (POD, continuous framework) and singular valuedecomposition (SVD, discrete matrices). Then a wide review of tensor formats andtheir approximation is proposed. Such work has already been provided in theliterature but either on separate papers or into a purely applied mathematicsframework. Here, we offer to the data enthusiast scientist a comparison ofCanonical, Tucker, Hierarchical and Tensor train formats including theirapproximation algorithms. Their relative benefits are studied both theoreticallyand numerically thanks to the python library texttt{pydecomp} that wasdeveloped during this thesis. A careful analysis of the link between continuousand discrete methods is performed. Finally, we conclude that for mostapplications ST-HOSVD is best when the number of dimensions $d$ lower than fourand TT-SVD (or their POD equivalent) when $d$ grows larger.The second part is centered on a complex fluid dynamics flow, in particular thesingular lid driven cavity at high Reynolds number. This flow exhibits a seriesof Hopf bifurcation which are known to be hard to capture accurately which iswhy a detailed analysis was performed both with classical tools and POD. Oncethis flow has been characterized, emph{time-scaling}, a new ``physics based''interpolation ROM is presented on internal and external flows. This methodsgives encouraging results while excluding recent advanced developments in thearea such as EIM or Grassmann manifold interpolation
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Goulart, José Henrique De Morais. "Estimation de modèles tensoriels structurés et récupération de tenseurs de rang faible." Thesis, Université Côte d'Azur (ComUE), 2016. http://www.theses.fr/2016AZUR4147/document.
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Dans la première partie de cette thèse, on formule deux méthodes pour le calcul d'une décomposition polyadique canonique avec facteurs matriciels linéairement structurés (tels que des facteurs de Toeplitz ou en bande): un algorithme de moindres carrés alternés contraint (CALS) et une solution algébrique dans le cas où tous les facteurs sont circulants. Des versions exacte et approchée de la première méthode sont étudiées. La deuxième méthode fait appel à la transformée de Fourier multidimensionnelle du tenseur considéré, ce qui conduit à la résolution d'un système d'équations monomiales homogènes. Nos simulations montrent que la combinaison de ces approches fournit un estimateur statistiquement efficace, ce qui reste vrai pour d'autres combinaisons de CALS dans des scénarios impliquant des facteurs non-circulants. La seconde partie de la thèse porte sur la récupération de tenseurs de rang faible et, en particulier, sur le problème de reconstruction tensorielle (TC). On propose un algorithme efficace, noté SeMPIHT, qui emploie des projections séquentiellement optimales par mode comme opérateur de seuillage dur. Une borne de performance est dérivée sous des conditions d'isométrie restreinte habituelles, ce qui fournit des bornes d'échantillonnage sous-optimales. Cependant, nos simulations suggèrent que SeMPIHT obéit à des bornes optimales pour des mesures Gaussiennes. Des heuristiques de sélection du pas et d'augmentation graduelle du rang sont aussi élaborées dans le but d'améliorer sa performance. On propose aussi un schéma d'imputation pour TC basé sur un seuillage doux du coeur du modèle de Tucker et son utilité est illustrée avec des données réelles de trafic routierIn the first part of this thesis, we formulate two methods for computing a canonical polyadic decomposition having linearly structured matrix factors (such as, e.g., Toeplitz or banded factors): a general constrained alternating least squares (CALS) algorithm and an algebraic solution for the case where all factors are circulant. Exact and approximate versions of the former method are studied. The latter method relies on a multidimensional discrete-time Fourier transform of the target tensor, which leads to a system of homogeneous monomial equations whose resolution provides the desired circulant factors. Our simulations show that combining these approaches yields a statistically efficient estimator, which is also true for other combinations of CALS in scenarios involving non-circulant factors. The second part of the thesis concerns low-rank tensor recovery (LRTR) and, in particular, the tensor completion (TC) problem. We propose an efficient algorithm, called SeMPIHT, employing sequentially optimal modal projections as its hard thresholding operator. Then, a performance bound is derived under usual restricted isometry conditions, which however yield suboptimal sampling bounds. Yet, our simulations suggest SeMPIHT obeys optimal sampling bounds for Gaussian measurements. Step size selection and gradual rank increase heuristics are also elaborated in order to improve performance. We also devise an imputation scheme for TC based on soft thresholding of a Tucker model core and illustrate its utility in completing real-world road traffic data acquired by an intelligent transportation
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Sodomaco, Luca. "The Distance Function from the Variety of partially symmetric rank-one Tensors." Doctoral thesis, 2020. http://hdl.handle.net/2158/1220535.
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The topic of this doctoral thesis is at the intersection between Real Algebraic Geometry, Optimization Theory and Multilinear Algebra. In particular, a relevant part of this thesis is dedicated to studying metric invariants of real algebraic varieties, with a particular interest in varieties in tensor spaces.
In many applications, tensors arise as a useful way to store and organize experimental data. For example, it is widely known that tensor techniques are extremely useful in Algebraic Statistics. A strong relationship between classical algebraic geometry and multilinear algebra is established by the notion of tensor rank. Geometrically speaking, the problem of computing the rank of a tensor translates to a membership problem to a certain secant variety of a Segre product of projective spaces.
In the last fifteen years, a new line of research in tensor theory has been undertaken and is commonly known as Spectral Theory of Tensors. One of the foundational motivations of this theory comes from the need, e.g., in some constrained optimization problems, to approximate a given tensor to its closest tensor of fixed lower rank, with respect to the Frobenius norm, also known as Bombieri norm. This is the so-called best rank- k approximation problem for real tensors. In this context, an important role is played by the singular vector tuples and the singular values of a tensor, which generalize the notions of eigenvector and eigenvalue of a matrix. Their symmetric counterpart is represented by the E-eigenvalues and the E-eigenvectors of a symmetric tensor.
Of particular interest is the E-characteristic polynomial of a symmetric tensor, which has among its roots the E-eigenvalues of a symmetric tensor. For symmetric matrices, it coincides with the classical characteristic polynomial. We interpret the E-characteristic polynomial as an algebraic relation satisfied by the Frobenius distance between an assigned symmetric tensor and the dual affine cone of a Veronese variety. We show that the E-characteristic polynomial is monic only in the symmetric matrix case. We provide a rational formula for the product of the singular values of a partially symmetric tensor of hypercubic format. The formula generalizes the fact that the determinant of a symmetric matrix is equal to the product of its eigenvalues. This is the only case where no denominator occurs in the formula.
Computing the distance from a variety of low-rank tensors is an important instance of a more general problem: computing the distance from a real algebraic variety X in a Euclidean space (V,q). We introduce a polynomial, called Euclidean Distance polynomial of X, which, for any data point u in V, has among its roots the distance ε from u to X. The ε^2-degree of the ED polynomial is the known Euclidean Distance degree of X. When X is transversal to the isotropic quadric Q={q=0}, we show that the ED polynomial of X is monic and we describe its lowest term completely.
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MACCIONI, MAURO. "Tensor rank and eigenvectors." Doctoral thesis, 2017. http://hdl.handle.net/2158/1077336.
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I investigate on the number t of real eigenvectors of a real symmetric tensor. In particular, given a homogeneous polynomial f of degree d in 3 variables, I prove that t is greater or equal than 2c+1, if d is odd, and t is greater or equal than max(3,2c+1), if d is even, where c is the number of ovals in the zero locus of f. About binary forms, I prove that t is greater or equal than the number of real roots of f. Moreover, the above inequalities are sharp for binary forms of any degree and for cubic and quartic ternary forms. Previously, I worked on the computation of the real ranks of real binary forms of degree four and five with assigned complex rank.
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CHIEN-YI, MA. "Symmetric Tensors in Ortho-symplectic Lie Superalgebra of Dimension (4,4)." 2005. http://www.cetd.com.tw/ec/thesisdetail.aspx?etdun=U0001-0407200517394100.
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MA, CHIEN-YI, and 馬鑑一. "Symmetric Tensors in Ortho-symplectic Lie Superalgebra of Dimension (4,4)." Thesis, 2005. http://ndltd.ncl.edu.tw/handle/43066528074507216240.
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碩士國立臺灣大學
數學研究所
93
Ortho-symplectic Lie superalgebra osp can be realized as differential operators and homogeneous polynomial space is closed under its action, that is, homogeneous polynomial space is an osp-module. Our thesis is to study whether or not homogeneous polynomial space can be reduced to a direct sum of irreducible osp-modules. Our conclusion is for any odd homogeneous polynomial space, the answer is yes. For even, the answer is no in the case of degree 2, and therefore invalid for any even homogeneous polynomial space since it must contain a submodule isomorphic to degree 2 homogeneous polynomial space. However, a complete decomposition of arbitrary even homogeneous polynomial space has not been reached yet.
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Ashraphijuo, Morteza. "Low-Rank Tensor Completion - Fundamental Limits and Efficient Algorithms." Thesis, 2020. https://doi.org/10.7916/d8-a3j9-zn71.
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This dissertation is motivated by the increasing applications of high-dimensional large-scale data sets in various fields and lack of theoretical understanding of the existing algorithms as well as lack of efficient algorithms in many cases. Hence, identifying the geometrical properties of data sets is essential for many data processing tasks, such as data retrieval and denoising.
In Part I, we derive the fundamental limits on the sampling rate required to study three important problems (i) low-rank data completion, (ii) rank estimation, and (iii) data clustering. In Chapter 2 we characterize the geometrical conditions on the sampling pattern, i.e., locations of the sampled entries, for finite and unique completability of a low-rank tensor, assuming that its rank vector is given or estimated. To this end, we propose a manifold analysis and study the independence of a set of polynomials defined based on the sampling pattern. Then, using the polynomial analysis, we derive a lower bound on the sampling rate such that it guarantees that the proposed conditions on the sampling patterns for finite and unique completability hold true with high probability. Then, in Chapter 3, we study the problem of rank estimation, where a data structure is partially sampled and we propose a geometrical analysis on the sampling pattern to estimate the true value of rank for various data structures by providing extremely tight lower and upper bounds on the rank value. And in Chapters 4 and 5, we make use of the developed tools to obtain a lower bound on the sampling rate to be able to correctly cluster a union of sampled matrices or tensors by identifying their corresponding unknown subspaces.
In Part II, first in Chapter 6, motivated by the algebraic tools developed in Part I, we develop a data completion algorithm based on solving a set of polynomial equations using Newton's method, that is effective especially when the sampling rate is low. Then, in Chapter 7, we consider a data structure consisting of a union of nested low-rank matrix or tensor subspaces, and develop a structured alternating minimization-based approach for completing such data, that is capable of taking advantage of multiple rank constraints simultaneously to achieve faster convergence and higher recovery accuracy.
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Kimaczyńska, Anna. "The differential operators in the bundle of symmetric tensors on a Riemannian manifold." Phd diss., 2016. http://hdl.handle.net/11089/22071.
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Differential operators: the gradient grad and the divergence div
are defined and examined in the bundles of symmetric tensors on a Riemannian
manifold. For the second order operator div grad which appears to be elliptic
and a manifold with boundary a system of natural boundary conditions is
constructed and investigated. There are 2k+1 conditions in the bundle Sk
of symmetric tensors of degree k. This is in contrast to the bundle of skewsymmetric
forms where (for analogous differential operators) there are always
four such conditions independently of the degree of forms (i.e. independently
of k). All the 2k+1 conditions are investigated in detail. In particular, it is
proved that each of them is self-adjoint and elliptic. Such the ellipticity of a
given boundary condition has an essential significance for the existing of an
orthonormal basis in L2 consisting of smooth sections that are the eigenvalues
of the operator and satisfy the boundary condition. Some special cases, e.g.
k = 1 or the the cases that the boundary is umbilical or totally geodesic are
also discussed.
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Garrido, Garcia Miguel Angel. "Characterization of the Fluctuations in a Symmetric Ensemble of Rank-Based Interacting Particles." Thesis, 2021. https://doi.org/10.7916/d8-azx1-sn93.
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Within the context of rank-based interacting particle systems, we study the fluctuations in a symmetric ensemble around its stable distribution. This system is inspired by the classic Atlas model but represents its opposite pole since both the highest- and lowest-ranked particles will have non-zero drifts. In the first part of the dissertation, we derive a fine asymptotic analysis that includes a Law of Large Numbers. The lack of monotonicity of the ensemble requires that we develop alternative tools to those traditionally used in the analysis of the Atlas model. In the second part of the dissertation, we characterize the system’s fluctuations and show that, as the number of particles goes to infinity, they converge weakly to the mild solution of the Additive Stochastic Heat Equation on the real line with a symmetric initial condition. To establish this result, we use the technique proposed by Dembo and Tsai, 2017, where the Empirical Measure Process is used as a proxy for the ensemble’s fluctuations. We expect that a combination of our work, and the available knowledge about the Atlas model, could help draw a full picture of how a finite rank-based interacting particle system with a general drift structure fluctuates around its stationary distribution as the number of particles goes to infinity, a long-standing open question in the field.
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XIE, MING-HUA, and 謝明華. "The low rank modification of the symmetric eigenproblem and the implementation of its application in symmetric tridiagonal eigenproblem on CRAY X-MP." Thesis, 1991. http://ndltd.ncl.edu.tw/handle/36773048168702319731.
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Pohl, Anke D. [Verfasser]. "Symbolic dynamics for the geodesic flow on locally symmetric good orbifolds of rank one / vorgelegt von Anke D. Pohl." 2009. http://d-nb.info/993834043/34.
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Van, Zyl Corli. "CUSUM procedures based on sequential ranks / Corli van Zyl." Thesis, 2015. http://hdl.handle.net/10394/15733.
Full textAbstract:
The main objective of this dissertation is the development of CUSUM procedures
based on signed and unsigned sequential ranks. These CUSUMs can be
applied to detect changes in the location or dispersion of a process. The signed
and unsigned sequential rank CUSUMs are distribution-free and robust against the
effect of outliers in the data. The only assumption that these CUSUMs require is
that the in-control distribution is symmetric around a known location parameter.
These procedures specifically do not require the existence of any higher order moments.
Another advantage of these CUSUMs is that Monte Carlo simulation can
readily be applied to deliver valid estimates of control limits, irrespective of what
the underlying distribution may be.
Other objectives of this dissertation include a brief discussion of the results
and refinements of the CUSUM in the literature. We justify the use of a signed
sequential rank statistic. Also, we evaluate the relative efficiency of the suggested
procedure numerically and provide three real-world applications from the engineering
and financial industries.MSc (Risk Analysis), North-West University, Potchefstroom Campus, 2015