Tesis: "Decomposition de tenseur"

1

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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2

Blanc, Katy. "Description de contenu vidéo : mouvements et élasticité temporelle". Thesis, Université Côte d'Azur (ComUE), 2018. http://www.theses.fr/2018AZUR4212/document.

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La reconnaissance en vidéo atteint de meilleures performances ces dernières années, notamment grâce à l'amélioration des réseaux de neurones profonds sur les images. Pourtant l'explosion des taux de reconnaissance en images ne s'est pas directement répercuté sur les taux en reconnaissance vidéo. Cela est dû à cette dimension supplémentaire qu'est le temps et dont il est encore difficile d'extraire une description robuste. Les réseaux de neurones récurrents introduisent une temporalité mais ils ont une mémoire limitée dans le temps. Les méthodes de description vidéo de l'état de l'art gèrent généralement le temps comme une dimension spatiale supplémentaire et la combinaison de plusieurs méthodes de description vidéo apportent les meilleures performances actuelles. Or la dimension temporelle possède une élasticité propre, différente des dimensions spatiales. En effet, la dimension temporelle peut être déformée localement : une dilatation partielle provoquera un ralentissement visuel de la vidéo sans en changer la compréhension, à l'inverse d'une dilatation spatiale sur une image qui modifierait les proportions des objets. On peut donc espérer améliorer encore la classification de contenu vidéo par la conception d'une description invariante aux changements de vitesse. Cette thèse porte sur la problématique d'une description robuste de vidéo en considérant l'élasticité de la dimension temporelle sous trois angles différents. Dans un premier temps, nous avons décrit localement et explicitement les informations de mouvements. Des singularités sont détectées sur le flot optique, puis traquées et agrégées dans une chaîne pour décrire des portions de vidéos. Nous avons utilisé cette description sur du contenu sportif. Puis nous avons extrait des descriptions globales implicites grâce aux décompositions tensorielles. Les tenseurs permettent de considérer une vidéo comme un tableau de données multi-dimensionnelles. Les descriptions extraites sont évaluées dans une tache de classification. Pour finir, nous avons étudié les méthodes de normalisation de la dimension temporelle. Nous avons utilisé les méthodes de déformations temporelles dynamiques des séquences. Nous avons montré que cette normalisation aide à une meilleure classification
Video recognition gain in performance during the last years, especially due to the improvement in the deep learning performances on images. However the jump in recognition rate on images does not directly impact the recognition rate on videos. This limitation is certainly due to this added dimension, the time, on which a robust description is still hard to extract. The recurrent neural networks introduce temporality but they have a limited memory. State of the art methods for video description usually handle time as a spatial dimension and the combination of video description methods reach the current best accuracies. However the temporal dimension has its own elasticity, different from the spatial dimensions. Indeed, the temporal dimension of a video can be locally deformed: a partial dilatation produces a visual slow down during the video, without changing the understanding, in contrast with a spatial dilatation on an image which will modify the proportions of the shown objects. We can thus expect to improve the video content classification by creating an invariant description to these speed changes. This thesis focus on the question of a robust video description considering the elasticity of the temporal dimension under three different angles. First, we have locally and explicitly described the motion content. Singularities are detected in the optical flow, then tracked along the time axis and organized in chain to describe video part. We have used this description on sport content. Then we have extracted global and implicit description thanks to tensor decompositions. Tensor enables to consider a video as a multi-dimensional data table. The extracted description are evaluated in a classification task. Finally, we have studied speed normalization method thanks to Dynamical Time Warping methods on series. We have showed that this normalization improve the classification rates

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3

André, Rémi. "Algorithmes de diagonalisation conjointe par similitude pour la décomposition canonique polyadique de tenseurs : applications en séparation de sources". Thesis, Toulon, 2018. http://www.theses.fr/2018TOUL0011/document.

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Cette thèse présente de nouveaux algorithmes de diagonalisation conjointe par similitude. Cesalgorithmes permettent, entre autres, de résoudre le problème de décomposition canonique polyadiquede tenseurs. Cette décomposition est particulièrement utilisée dans les problèmes deséparation de sources. L’utilisation de la diagonalisation conjointe par similitude permet de paliercertains problèmes dont les autres types de méthode de décomposition canonique polyadiquesouffrent, tels que le taux de convergence, la sensibilité à la surestimation du nombre de facteurset la sensibilité aux facteurs corrélés. Les algorithmes de diagonalisation conjointe par similitudetraitant des données complexes donnent soit de bons résultats lorsque le niveau de bruit est faible,soit sont plus robustes au bruit mais ont un coût calcul élevé. Nous proposons donc en premierlieu des algorithmes de diagonalisation conjointe par similitude traitant les données réelles etcomplexes de la même manière. Par ailleurs, dans plusieurs applications, les matrices facteursde la décomposition canonique polyadique contiennent des éléments exclusivement non-négatifs.Prendre en compte cette contrainte de non-négativité permet de rendre les algorithmes de décompositioncanonique polyadique plus robustes à la surestimation du nombre de facteurs ou lorsqueces derniers ont un haut degré de corrélation. Nous proposons donc aussi des algorithmes dediagonalisation conjointe par similitude exploitant cette contrainte. Les simulations numériquesproposées montrent que le premier type d’algorithmes développés améliore l’estimation des paramètresinconnus et diminue le coût de calcul. Les simulations numériques montrent aussi queles algorithmes avec contrainte de non-négativité améliorent l’estimation des matrices facteurslorsque leurs colonnes ont un haut degré de corrélation. Enfin, nos résultats sont validés à traversdeux applications de séparation de sources en télécommunications numériques et en spectroscopiede fluorescence
This thesis introduces new joint eigenvalue decomposition algorithms. These algorithms allowamongst others to solve the canonical polyadic decomposition problem. This decomposition iswidely used for blind source separation. Using the joint eigenvalue decomposition to solve thecanonical polyadic decomposition problem allows to avoid some problems whose the others canonicalpolyadic decomposition algorithms generally suffer, such as the convergence rate, theoverfactoring sensibility and the correlated factors sensibility. The joint eigenvalue decompositionalgorithms dealing with complex data give either good results when the noise power is low, orthey are robust to the noise power but have a high numerical cost. Therefore, we first proposealgorithms equally dealing with real and complex. Moreover, in some applications, factor matricesof the canonical polyadic decomposition contain only nonnegative values. Taking this constraintinto account makes the algorithms more robust to the overfactoring and to the correlated factors.Therefore, we also offer joint eigenvalue decomposition algorithms taking advantage of thisnonnegativity constraint. Suggested numerical simulations show that the first developed algorithmsimprove the estimation accuracy and reduce the numerical cost in the case of complexdata. Our numerical simulations also highlight the fact that our nonnegative joint eigenvaluedecomposition algorithms improve the factor matrices estimation when their columns have ahigh correlation degree. Eventually, we successfully applied our algorithms to two blind sourceseparation problems : one concerning numerical telecommunications and the other concerningfluorescence spectroscopy

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4

Nguyen, Dinh Quoc Dang. "Representation of few-group homogenized cross sections by polynomials and tensor decomposition". Electronic Thesis or Diss., université Paris-Saclay, 2024. http://www.theses.fr/2024UPASP142.

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Cette thèse se concentre sur l'étude de la modélisation mathématique des sections efficaces homogénéisées à peu de groupes, un élément essentiel du schéma à deux étapes, qui est largement utilisé dans les simulations de réacteurs nucléaires. À mesure que les demandes industrielles nécessitent de plus en plus des maillages spatiaux et énergétiques fins pour améliorer la précision des calculs cœur, la taille de la bibliothèque des sections efficaces peut devenir excessive, entravant ainsi les performances des calculs cœur. Il est donc essentiel de développer une représentation qui minimise l'utilisation de la mémoire tout en permettant une interpolation des données efficace.Deux approches, la représentation polynomiale et la décomposition "Canonical Polyadic" des tenseurs, sont présentées et appliquées aux données de sections efficaces homogénéisées à peu de groupes. Les données sont préparées à l'aide d'APOLLO3 sur la géométrie de deux assemblages dans le benchmark X2 VVER-1000. Le taux de compression et la précision sont évalués et discutés pour chaque approche afin de déterminer leur applicabilité au schéma standard en deux étapes.De plus, des implémentations sur GPUs des deux approches sont testées pour évaluer la scalabilité des algorithmes en fonction du nombre de threads impliqués. Ces implémentations sont encapsulées dans une bibliothèque appelée Merlin, destinée à la recherche future et aux applications industrielles utilisant ces approches.Les deux approches, en particulier la méthode de décomposition des tenseurs, montrent des résultats prometteurs en termes de compression des données et de précision de reconstruction. L'intégration de ces méthodes dans le schéma standard en deux étapes permettrait non seulement de réduire considérablement l'utilisation de la mémoire pour le stockage des sections efficaces, mais aussi de diminuer significativement l'effort de calcul requis pour l'interpolation des sections efficaces lors des calculs cœur, réduisant donc le temps de calcul global pour les simulations de réacteurs industriels
This thesis focuses on studying the mathematical modeling of few-group homogenized cross sections, a critical element in the two-step scheme widely used in nuclear reactor simulations. As industrial demands increasingly require finer spatial and energy meshes to improve the accuracy of core calculations, the size of the cross section library can become excessive, hampering the performance of core calculations. Therefore, it is essential to develop a representation that minimizes memory usage while still enabling efficient data interpolation.Two approaches, polynomial representation and Canonical Polyadic decomposition of tensors, are presented and applied to few-group homogenized cross section data. The data is prepared using APOLLO3 on the geometry of two assemblies in the X2 VVER-1000 benchmark. The compression rate and accuracy are evaluated and discussed for each approach to determine their applicability to the standard two-step scheme.Additionally, GPU implementations of both approaches are tested to assess the scalability of the algorithms based on the number of threads involved. These implementations are encapsulated in a library called Merlin, intended for future research and industrial applications that involve these approaches.Both approaches, particularly the method of tensor decomposition, demonstrate promising results in terms of data compression and reconstruction accuracy. Integrating these methods into the standard two-step scheme would not only substantially reduce memory usage for storing cross sections, but also significantly decrease the computational effort required for interpolating cross sections during core calculations, thereby reducing overall calculation time for industrial reactor simulations

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5

Akhavanbahabadi, Saeed. "Analyse des Crises d’Epilepsie à l’Aide de Mesures de Profondeur". Thesis, Université Grenoble Alpes (ComUE), 2019. http://www.theses.fr/2019GREAT063.

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Le doctorant n'a pas délivré de résumé en français
Absence epilepsy syndrome is accompanied with sudden appearance of seizures in different regions of the brain. The sudden generalization of absence seizures to every region of the brain shows the existence of a mechanism which can quickly synchronizes the activities of the majority of neurons in the brain. The presence of such a mechanism challenges our information about the integrative properties of neurons and the functional connectivity of brain networks. For this reason, many researchers have tried to recognize the main origin of absence seizures. Recent studies have suggested a theory regarding the origin of absence seizures which states that somatosensory cortex drives the thalamus during the first cycles of absence seizures, while thereafter, cortex and thalamus mutually drive each other and continue absence seizures.This theory motivated the neuroscientists in Grenoble Institute of Neurosciences (GIN) to record data from different layers of somatosensory cortex of Genetic Absence Epilepsy Rats from Strasbourg (GAERS), which is a well-validate animal model for absence epilepsy, to explore the main starting region of absence seizures locally. An electrode with E = 16 sensors was vertically implanted in somatosensory cortex of GAERS, and potentials were recorded. In this study, we aim to localize the onset layers of somatosensory cortex during absence seizures and investigate the temporal evolution and dynamics of absence seizures using the recorded data. It is worth mentioning that all previous studies have investigated absence seizures using the data recorded from different regions of the brain, while this is the first study that performs the local exploration of absence seizures using the data recorded from different layers of somatosensory cortex, i.e., the main starting region of absence seizures.Using factor analysis, source separation, and blind deconvolution methods in different scenarios, we show that 1) the top and bottom layers of somatosensory cortex activate more than the other layers during absence seizures, 2) there is a background epileptic activity during absence seizures, 3) there are few activities or states which randomly activate with the background epileptic activity to generate the absence seizures, and 4) one of these states is dominant, and the others are unstable

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6

André, Rémi. "Algorithmes de diagonalisation conjointe par similitude pour la décomposition canonique polyadique de tenseurs : applications en séparation de sources". Electronic Thesis or Diss., Toulon, 2018. http://www.theses.fr/2018TOUL0011.

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Cette thèse présente de nouveaux algorithmes de diagonalisation conjointe par similitude. Cesalgorithmes permettent, entre autres, de résoudre le problème de décomposition canonique polyadiquede tenseurs. Cette décomposition est particulièrement utilisée dans les problèmes deséparation de sources. L’utilisation de la diagonalisation conjointe par similitude permet de paliercertains problèmes dont les autres types de méthode de décomposition canonique polyadiquesouffrent, tels que le taux de convergence, la sensibilité à la surestimation du nombre de facteurset la sensibilité aux facteurs corrélés. Les algorithmes de diagonalisation conjointe par similitudetraitant des données complexes donnent soit de bons résultats lorsque le niveau de bruit est faible,soit sont plus robustes au bruit mais ont un coût calcul élevé. Nous proposons donc en premierlieu des algorithmes de diagonalisation conjointe par similitude traitant les données réelles etcomplexes de la même manière. Par ailleurs, dans plusieurs applications, les matrices facteursde la décomposition canonique polyadique contiennent des éléments exclusivement non-négatifs.Prendre en compte cette contrainte de non-négativité permet de rendre les algorithmes de décompositioncanonique polyadique plus robustes à la surestimation du nombre de facteurs ou lorsqueces derniers ont un haut degré de corrélation. Nous proposons donc aussi des algorithmes dediagonalisation conjointe par similitude exploitant cette contrainte. Les simulations numériquesproposées montrent que le premier type d’algorithmes développés améliore l’estimation des paramètresinconnus et diminue le coût de calcul. Les simulations numériques montrent aussi queles algorithmes avec contrainte de non-négativité améliorent l’estimation des matrices facteurslorsque leurs colonnes ont un haut degré de corrélation. Enfin, nos résultats sont validés à traversdeux applications de séparation de sources en télécommunications numériques et en spectroscopiede fluorescence
This thesis introduces new joint eigenvalue decomposition algorithms. These algorithms allowamongst others to solve the canonical polyadic decomposition problem. This decomposition iswidely used for blind source separation. Using the joint eigenvalue decomposition to solve thecanonical polyadic decomposition problem allows to avoid some problems whose the others canonicalpolyadic decomposition algorithms generally suffer, such as the convergence rate, theoverfactoring sensibility and the correlated factors sensibility. The joint eigenvalue decompositionalgorithms dealing with complex data give either good results when the noise power is low, orthey are robust to the noise power but have a high numerical cost. Therefore, we first proposealgorithms equally dealing with real and complex. Moreover, in some applications, factor matricesof the canonical polyadic decomposition contain only nonnegative values. Taking this constraintinto account makes the algorithms more robust to the overfactoring and to the correlated factors.Therefore, we also offer joint eigenvalue decomposition algorithms taking advantage of thisnonnegativity constraint. Suggested numerical simulations show that the first developed algorithmsimprove the estimation accuracy and reduce the numerical cost in the case of complexdata. Our numerical simulations also highlight the fact that our nonnegative joint eigenvaluedecomposition algorithms improve the factor matrices estimation when their columns have ahigh correlation degree. Eventually, we successfully applied our algorithms to two blind sourceseparation problems : one concerning numerical telecommunications and the other concerningfluorescence spectroscopy

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7

Silva, Alex Pereira da. "Techniques tensorielles pour le traitement du signal : algorithmes pour la décomposition polyadique canonique". Thesis, Université Grenoble Alpes (ComUE), 2016. http://www.theses.fr/2016GREAT042/document.

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L’approximation tensorielle de rang faible joue ces dernières années un rôle importantdans plusieurs applications, telles que la séparation aveugle de source, les télécommunications, letraitement d’antennes, les neurosciences, la chimiométrie, et l’exploration de données. La décompositiontensorielle Canonique Polyadique est très attractive comparativement à des outils matriciels classiques,notamment pour l’identification de systèmes. Dans cette thèse, nous proposons (i) plusieursalgorithmes pour calculer quelques approximations de rang faible spécifique: approximation de rang-1 itérative et en un nombre fini d’opérations, l’approximation par déflation itérative, et la décompositiontensorielle orthogonale; (ii) une nouvelle stratégie pour résoudre des systèmes quadratiquesmultivariés, où ce problème peut être réduit à la meilleure approximation de rang-1 d’un tenseur; (iii)des résultats théoriques pour étudier les performances ou prouver la convergence de quelques algorithmes.Toutes les performances sont illustrées par des simulations informatiques
Low rank tensor decomposition has been playing for the last years an important rolein many applications such as blind source separation, telecommunications, sensor array processing,neuroscience, chemometrics, and data mining. The Canonical Polyadic tensor decomposition is veryattractive when compared to standard matrix-based tools, manly on system identification. In this thesis,we propose: (i) several algorithms to compute specific low rank-approximations: finite/iterativerank-1 approximations, iterative deflation approximations, and orthogonal tensor decompositions. (ii)A new strategy to solve multivariate quadratic systems, where this problem is reduced to a best rank-1 tensor approximation problem. (iii) Theoretical results to study and proof the performance or theconvergence of some algorithms. All performances are supported by numerical experiments
A aproximação tensorial de baixo posto desempenha nestes últimos anos um papel importanteem várias aplicações, tais como separação cega de fontes, telecomunicações, processamentode antenas, neurociênca, quimiometria e exploração de dados. A decomposição tensorial canônicaé bastante atrativa se comparada às técnicas matriciais clássicas, principalmente na identificação desistemas. Nesta tese, propõe-se (i) vários algoritmos para calcular alguns tipos de aproximação deposto: aproximação de posto-1 iterativa e em um número finito de operações, a aproximação pordeflação iterativa, e a decomposição tensorial ortogonal; (ii) uma nova estratégia para resolver sistemasquadráticos em várias variáveis, em que tal problema pode ser reduzido à melhor aproximaçãode posto-1 de um tensor; (iii) resultados teóricos visando estudar o desempenho ou demonstrar aconvergência de alguns algoritmos. Todas os desempenhos são ilustrados através de simulações computacionais

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8

Marmin, Arthur. "Rational models optimized exactly for solving signal processing problems". Electronic Thesis or Diss., université Paris-Saclay, 2020. http://www.theses.fr/2020UPASG017.

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Une vaste classe de problèmes d'optimisation non convexes est celle de l'optimisation rationnelle. Cette dernière apparaît naturellement dans de nombreux domaines tels que le traitement du signal ou le génie des procédés. Toutefois, trouver les optima globaux pour ces problèmes est difficile. Une approche récente, appelée la hiérarchie de Lasserre, fournit néanmoins une suite de problèmes convexes assurée de converger vers le minimum global. Cependant, cette approche représente un défi calculatoire du fait de la très grande dimension de ses relaxations. Dans cette thèse, nous abordons ce défi pour divers problèmes de traitement du signal.Dans un premier temps, nous formulons la reconstruction de signaux parcimonieux en un problème d'optimisation rationnelle. Nous montrons alors que ce dernier possède une structure que nous exploitons afin de réduire la complexité des relaxations associées. Nous pouvons ainsi résoudre plusieurs problèmes pratiques comme la restoration de signaux de chromatographie. Nous étendons également notre méthode à la restoration de signaux dans différents contextes en proposant plusieurs modèles de bruit et de signal. Dans une deuxième partie, nous étudions les relaxations convexes générées par nos problèmes et qui se présentent sous la forme de problèmes d'optimisation semi-définie positive de très grandes dimensions. Nous considérons plusieurs algorithmes basés sur les opérateurs proximaux pour les résoudre efficacement.La dernière partie de cette thèse est consacrée au lien entre les problèmes d'optimisation polynomiaux et la décomposition de tenseurs symétriques. En effet, ces derniers peuvent être tous deux vus comme une instance du problème des moments. Nous proposons ainsi une méthode de détection de rang et de décomposition pour les tenseurs symétriques basée sur les outils connus en optimisation polynomiale. Parallèlement, nous proposons une technique d'extraction robuste des solutions d'un problème d'optimisation poylnomiale basée sur les algorithmes de décomposition de tenseurs. Ces méthodes sont illustrées sur des problèmes de traitement du signal
A wide class of nonconvex optimization problem is represented by rational optimization problems. The latter appear naturally in many areas such as signal processing or chemical engineering. However, finding the global optima of such problems is intricate. A recent approach called Lasserre's hierarchy provides a sequence of convex problems that has the theoretical guarantee to converge to the global optima. Nevertheless, this approach is computationally challenging due to the high dimensions of the convex relaxations. In this thesis, we tackle this challenge for various signal processing problems.First, we formulate the reconstruction of sparse signals as a rational optimization problem. We show that the latter has a structure that we wan exploit in order to reduce the complexity of the associated relaxations. We thus solve several practical problems such as the reconstruction of chromatography signals. We also extend our method to the reconstruction of various types of signal corrupted by different noise models.In a second part, we study the convex relaxations generated by our problems which take the form of high-dimensional semi-definite programming problems. We consider several algorithms mainly based on proximal operators to solve those high-dimensional problems efficiently.The last part of this thesis is dedicated to the link between polynomial optimization and symmetric tensor decomposition. Indeed, they both can be seen as an instance of the moment problem. We thereby propose a detection method as well as a decomposition algorithm for symmetric tensors based on the tools used in polynomial optimization. In parallel, we suggest a robust extraction method for polynomial optimization based on tensor decomposition algorithms. Those methods are illustrated on signal processing problems

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9

Brandoni, Domitilla. "Tensor decompositions for Face Recognition". Master's thesis, Alma Mater Studiorum - Università di Bologna, 2018. http://amslaurea.unibo.it/16867/.

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Automatic Face Recognition has become increasingly important in the past few years due to its several applications in daily life, such as in social media platforms and security services. Numerical linear algebra tools such as the SVD (Singular Value Decomposition) have been extensively used to allow machines to automatically process images in the recognition and classification contexts. On the other hand, several factors such as expression, view angle and illumination can significantly affect the image, making the processing more complex. To cope with these additional features, multilinear algebra tools, such as high-order tensors are being explored. In this thesis we first analyze tensor calculus and tensor approximation via several dif- ferent decompositions that have been recently proposed, which include HOSVD (Higher-Order Singular Value Decomposition) and Tensor-Train formats. A new algorithm is proposed to perform data recognition for the latter format.

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10

Bender, Matias Rafael. "Algorithms for sparse polynomial systems : Gröbner bases and resultants". Electronic Thesis or Diss., Sorbonne université, 2019. http://www.theses.fr/2019SORUS029.

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La résolution de systèmes polynomiaux est l’un des problèmes les plus anciens et importants en mathématiques informatiques et a des applications dans plusieurs domaines des sciences et de l’ingénierie. C'est un problème intrinsèquement difficile avec une complexité au moins exponentielle du nombre de variables. Cependant, dans la plupart des cas, les systèmes polynomiaux issus d'applications ont une structure quelconque. Dans cette thèse, nous nous concentrons sur l'exploitation de la structure liée à la faible densité des supports des polynômes; c'est-à-dire que nous exploitons le fait que les polynômes n'ont que quelques monômes à coefficients non nuls. Notre objectif est de résoudre les systèmes plus rapidement que les estimations les plus défavorables, qui supposent que tous les termes sont présents. Nous disons qu'un système creux est non mixte si tous ses polynômes ont le même polytope de Newton, et mixte autrement. La plupart des travaux sur la résolution de systèmes creux concernent le cas non mixte, à l'exception des résultants creux et des méthodes d'homotopie. Nous développons des algorithmes pour des systèmes mixtes. Nous utilisons les résultantes creux et les bases de Groebner. Nous travaillons sur chaque théorie indépendamment, mais nous les combinons également: nous tirons parti des propriétés algébriques des systèmes associés à une résultante non nulle pour améliorer la complexité du calcul de leurs bases de Groebner; par exemple, nous exploitons l’exactitude du complexe de Koszul pour déduire un critère d’arrêt précoce et éviter tout les réductions à zéro. De plus, nous développons des algorithmes quasi-optimaux pour décomposer des formes binaires
Solving polynomial systems is one of the oldest and most important problems in computational mathematics and has many applications in several domains of science and engineering. It is an intrinsically hard problem with complexity at least single exponential in the number of variables. However, in most of the cases, the polynomial systems coming from applications have some kind of structure. In this thesis we focus on exploiting the structure related to the sparsity of the supports of the polynomials; that is, we exploit the fact that the polynomials only have a few monomials with non-zero coefficients. Our objective is to solve the systems faster than the worst case estimates that assume that all the terms are present. We say that a sparse system is unmixed if all its polynomials have the same Newton polytope, and mixed otherwise. Most of the work on solving sparse systems concern the unmixed case, with the exceptions of mixed sparse resultants and homotopy methods. In this thesis, we develop algorithms for mixed systems. We use two prominent tools in nonlinear algebra: sparse resultants and Groebner bases. We work on each theory independently, but we also combine them to introduce new algorithms: we take advantage of the algebraic properties of the systems associated to a non-vanishing resultant to improve the complexity of computing their Groebner bases; for example, we exploit the exactness of some strands of the associated Koszul complex to deduce an early stopping criterion for our Groebner bases algorithms and to avoid every redundant computation (reductions to zero). In addition, we introduce quasi-optimal algorithms to decompose binary forms

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11

Shi, Qiquan. "Low rank tensor decomposition for feature extraction and tensor recovery". HKBU Institutional Repository, 2018. https://repository.hkbu.edu.hk/etd_oa/549.

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Feature extraction and tensor recovery problems are important yet challenging, particularly for multi-dimensional data with missing values and/or noise. Low-rank tensor decomposition approaches are widely used for solving these problems. This thesis focuses on three common tensor decompositions (CP, Tucker and t-SVD) and develops a set of decomposition-based approaches. The proposed methods aim to extract low-dimensional features from complete/incomplete data and recover tensors given partial and/or grossly corrupted observations.;Based on CP decomposition, semi-orthogonal multilinear principal component analysis (SO-MPCA) seeks a tensor-to-vector projection that maximizes the captured variance with the orthogonality constraint imposed in only one mode, and it further integrates the relaxed start strategy (SO-MPCA-RS) to achieve better feature extraction performance. To directly obtain the features from incomplete data, low-rank CP and Tucker decomposition with feature variance maximization (TDVM-CP and TDVM-Tucker) are proposed. TDVM methods explore the relationship among tensor samples via feature variance maximization, while estimating the missing entries via low-rank CP and Tucker approximation, leading to informative features extracted directly from partial observations. TDVM-CP extracts low-dimensional vector features viewing the weight vectors as features and TDVM-Tucker learns low-dimensional tensor features viewing the core tensors as features. TDVM methods can be generalized to other variants based on other tensor decompositions. On the other hand, this thesis solves the missing data problem by introducing low-rank matrix/tensor completion methods, and also contributes to automatic rank estimation. Rank-one matrix decomposition coupled with L1-norm regularization (L1MC) addresses the matrix rank estimation problem. With the correct estimated rank, L1MC refines its model without L1-norm regularization (L1MC-RF) and achieve optimal recovery results given enough observations. In addition, CP-based nuclear norm regularized orthogonal CP decomposition (TREL1) solves the challenging CP- and Tucker-rank estimation problems. The estimated rank can improve the tensor completion accuracy of existing decomposition-based methods. Furthermore, tensor singular value decomposition (t-SVD) combined with tensor nuclear norm (TNN) regularization (ARE_TNN) provides automatic tubal-rank estimation. With the accurate tubal-rank determination, ARE_TNN relaxes its model without the TNN constraint (TC-ARE) and results in optimal tensor completion under mild conditions. In addition, ARE_TNN refines its model by explicitly utilizing its determined tubal-rank a priori and then successfully recovers low-rank tensors based on incomplete and/or grossly corrupted observations (RTC-ARE: robust tensor completion/RTPCA-ARE: robust tensor principal component analysis).;Experiments and evaluations are presented and analyzed using synthetic data and real-world images/videos in machine learning, computer vision, and data mining applications. For feature extraction, the experimental results of face and gait recognition show that SO-MPCA-RS achieves the best overall performance compared with competing algorithms, and its relaxed start strategy is also effective for other CP-based PCA methods. In the applications of face recognition, object/action classification, and face/gait clustering, TDVM methods not only stably yield similar good results under various multi-block missing settings and different parameters in general, but also outperform the competing methods with significant improvements. For matrix/tensor rank estimation and recovery, L1MC-RF efficiently estimates the true rank and exactly recovers the incomplete images/videos under mild conditions, and outperforms the state-of-the-art algorithms on the whole. Furthermore, the empirical evaluations show that TREL1 correctly determines the CP-/Tucker- ranks well, given sufficient observed entries, which consistently improves the recovery performance of existing decomposition-based tensor completion. The t-SVD recovery methods TC-ARE, RTPCA-ARE, and RTC-ARE not only inherit the ability of ARE_TNN to achieve accurate rank estimation, but also achieve good performance in the tasks of (robust) image/video completion, video denoising, and background modeling. This outperforms the state-of-the-art methods in all cases we have tried so far with significant improvements.

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12

Brandoni, Domitilla <1994&gt. "Tensor-Train decomposition for image classification problems". Doctoral thesis, Alma Mater Studiorum - Università di Bologna, 2022. http://amsdottorato.unibo.it/10121/3/phd_thesis_DomitillaBrandoni_final.pdf.

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In these last years a great effort has been put in the development of new techniques for automatic object classification, also due to the consequences in many applications such as medical imaging or driverless cars. To this end, several mathematical models have been developed from logistic regression to neural networks. A crucial aspect of these so called classification algorithms is the use of algebraic tools to represent and approximate the input data. In this thesis, we examine two different models for image classification based on a particular tensor decomposition named Tensor-Train (TT) decomposition. The use of tensor approaches preserves the multidimensional structure of the data and the neighboring relations among pixels. Furthermore the Tensor-Train, differently from other tensor decompositions, does not suffer from the curse of dimensionality making it an extremely powerful strategy when dealing with high-dimensional data. It also allows data compression when combined with truncation strategies that reduce memory requirements without spoiling classification performance. The first model we propose is based on a direct decomposition of the database by means of the TT decomposition to find basis vectors used to classify a new object. The second model is a tensor dictionary learning model, based on the TT decomposition where the terms of the decomposition are estimated using a proximal alternating linearized minimization algorithm with a spectral stepsize.
Negli ultimi anni si è registrato un notevole sviluppo di nuove tecniche per il riconoscimento automatico di oggetti, anche dovuto alle possibili ricadute di tali avanzamenti nel campo medico o automobilistico. A tal fine sono stati sviluppati svariati modelli matematici dai metodi di regressione fino alle reti neurali. Un aspetto cruciale di questi cosiddetti algoritmi di classificazione è l'uso di aspetti algebrici per la rappresentazione e l'approssimazione dei dati in input. In questa tesi esamineremo due diversi modelli per la classificazione di immagini basati sulla decomposizione Tensor-Train (TT). In generale, l'uso di approcci tensoriali è fondamentale per preservare la struttura intrinsecamente multidimensionale dei dati. Inoltre l'occupazione di memoria per la decomposizione Tensor-Train non cresce esponenzialmente all'aumentare dei dati, a differenza di altre decomposizioni tensoriali. Questo la rende particolarmente adatta nel caso di dati di grandi dimensioni. Inoltre permette, attraverso l'uso di opportune strategie di troncamento, di limitare notevolmente l'occupazione di memoria senza ricadute negative sulle performance di classificazione. Il primo modello proposto in questa tesi è basato su una decomposizione diretta del database tramite la decomposizione TT. In questo modo viene determinata una base che verrà di seguito utilizzata nella classificazione di nuove immagini. Il secondo è invece un modello di dictionary learning tensoriale sempre basato sulla decomposizione TT in cui i termini della decomposizione sono determinati utilizzando un nuovo metodo di ottimizzazione alternato con l'utilizzo di passi spettrali.

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13

Kaya, Oguz. "High Performance Parallel Algorithms for Tensor Decompositions". Thesis, Lyon, 2017. http://www.theses.fr/2017LYSEN051/document.

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La factorisation des tenseurs est au coeur des méthodes d'analyse des données massives multidimensionnelles dans de nombreux domaines, dont les systèmes de recommandation, les graphes, les données médicales, le traitement du signal, la chimiométrie, et bien d'autres.Pour toutes ces applications, l'obtention rapide de la décomposition des tenseurs est cruciale pour pouvoir traiter manipuler efficacement les énormes volumes de données en jeu.L'objectif principal de cette thèse est la conception d'algorithmes pour la décomposition de tenseurs multidimensionnels creux, possédant de plusieurs centaines de millions à quelques milliards de coefficients non-nuls. De tels tenseurs sont omniprésents dans les applications citées plus haut.Nous poursuivons cet objectif via trois approches.En premier lieu, nous proposons des algorithmes parallèles à mémoire distribuée, comprenant des schémas de communication point-à-point optimisés, afin de réduire les coûts de communication. Ces algorithmes sont indépendants du partitionnement des éléments du tenseur et des matrices de faible rang. Cette propriété nous permet de proposer des stratégies de partitionnement visant à minimiser le coût de communication tout en préservant l'équilibrage de charge entre les ressources. Nous utilisons des techniques d'hypergraphes pour analyser les paramètres de calcul et de communication de ces algorithmes, ainsi que des outils de partitionnement d'hypergraphe pour déterminer des partitions à même d'offrir un meilleur passage à l'échelle. Deuxièmement, nous étudions la parallélisation sur plate-forme à mémoire partagée de ces algorithmes. Dans ce contexte, nous déterminons soigneusement les tâches de calcul et leur dépendances, et nous les exprimons en termes d'une structure de données idoine, et dont la manipulation permet de révéler le parallélisme intrinsèque du problème. Troisièmement, nous présentons un schéma de calcul en forme d'arbre binaire pour représenter les noyaux de calcul les plus coûteux des algorithmes, comme la multiplication du tenseur par un ensemble de vecteurs ou de matrices donnés. L'arbre binaire permet de factoriser certains résultats intermédiaires, et de les ré-utiliser au fil du calcul. Grâce à ce schéma, nous montrons comment réduire significativement le nombre et le coût des multiplications tenseur-vecteur et tenseur-matrice, rendant ainsi la décomposition du tenseur plus rapide à la fois pour la version séquentielle et la version parallèle des algorithmes.Enfin, le reste de la thèse décrit deux extensions sur des thèmes similaires. La première extension consiste à appliquer le schéma d'arbre binaire à la décomposition des tenseurs denses, avec une analyse précise de la complexité du problème et des méthodes pour trouver la structure arborescente qui minimise le coût total. La seconde extension consiste à adapter les techniques de partitionnement utilisées pour la décomposition des tenseurs creux à la factorisation des matrices non-négatives, problème largement étudié et pour lequel nous obtenons des algorithmes parallèles plus efficaces que les meilleurs actuellement connus.Tous les résultats théoriques de cette thèse sont accompagnés d'implémentations parallèles,aussi bien en mémoire partagée que distribuée. Tous les algorithmes proposés, avec leur réalisation sur plate-forme HPC, contribuent ainsi à faire de la décomposition de tenseurs un outil prometteur pour le traitement des masses de données actuelles et à venir
Tensor factorization has been increasingly used to analyze high-dimensional low-rank data ofmassive scale in numerous application domains, including recommender systems, graphanalytics, health-care data analysis, signal processing, chemometrics, and many others.In these applications, efficient computation of tensor decompositions is crucial to be able tohandle such datasets of high volume. The main focus of this thesis is on efficient decompositionof high dimensional sparse tensors, with hundreds of millions to billions of nonzero entries,which arise in many emerging big data applications. We achieve this through three majorapproaches.In the first approach, we provide distributed memory parallel algorithms with efficientpoint-to-point communication scheme for reducing the communication cost. These algorithmsare agnostic to the partitioning of tensor elements and low rank decomposition matrices, whichallow us to investigate effective partitioning strategies for minimizing communication cost whileestablishing computational load balance. We use hypergraph-based techniques to analyze computational and communication requirements in these algorithms, and employ hypergraphpartitioning tools to find suitable partitions that provide much better scalability.Second, we investigate effective shared memory parallelizations of these algorithms. Here, we carefully determine unit computational tasks and their dependencies, and express them using aproper data structure that exposes the parallelism underneath.Third, we introduce a tree-based computational scheme that carries out expensive operations(involving the multiplication of the tensor with a set of vectors or matrices, found at the core ofthese algorithms) faster by factoring out and storing common partial results and effectivelyre-using them. With this computational scheme, we asymptotically reduce the number oftensor-vector and -matrix multiplications for high dimensional tensors, and thereby rendercomputing tensor decompositions significantly cheaper both for sequential and parallelalgorithms.Finally, we diversify this main course of research with two extensions on similar themes.The first extension involves applying the tree-based computational framework to computingdense tensor decompositions, with an in-depth analysis of computational complexity andmethods to find optimal tree structures minimizing the computational cost. The second workfocuses on adapting effective communication and partitioning schemes of our parallel sparsetensor decomposition algorithms to the widely used non-negative matrix factorization problem,through which we obtain significantly better parallel scalability over the state of the artimplementations.We point out that all theoretical results in the thesis are nicely corroborated by parallelexperiments on both shared-memory and distributed-memory platforms. With these fastalgorithms as well as their tuned implementations for modern HPC architectures, we rendertensor and matrix decomposition algorithms amenable to use for analyzing massive scaledatasets

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14

Dinckal, Cigdem. "Decomposition Of Elastic Constant Tensor Into Orthogonal Parts". Phd thesis, METU, 2010. http://etd.lib.metu.edu.tr/upload/12612226/index.pdf.

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All procedures in the literature for decomposing symmetric second rank (stress) tensor and symmetric fourth rank (elastic constant) tensor are elaborated and compared which have many engineering and scientific applications for anisotropic materials. The decomposition methods for symmetric second rank tensors are orthonormal tensor basis method, complex variable representation and spectral method. For symmetric fourth rank (elastic constant) tensor, there are four mainly decomposition methods namely as, orthonormal tensor basis, irreducible, harmonic decomposition and spectral. Those are applied to anisotropic materials possessing various symmetry classes which are isotropic, cubic, transversely isotropic, tetragonal, trigonal and orthorhombic. For isotropic materials, an expression for the elastic constant tensor different than the traditionally known form is given. Some misprints found in the literature are corrected. For comparison purposes, numerical examples of each decomposition process are presented for the materials possessing different symmetry classes. Some applications of these decomposition methods are given. Besides, norm and norm ratio concepts are introduced to measure and compare the anisotropy degree for various materials with the same or di¤
erent symmetries. For these materials,norm and norm ratios are calculated. It is suggested that the norm of a tensor may be used as a criterion for comparing the overall e¤
ect of the properties of anisotropic materials and the norm ratios may be used as a criterion to represent the anisotropy degree of the properties of materials. Finally, comparison of all methods are done in order to determine similarities and differences between them. As a result of this comparison process, it is proposed that the spectral method is a non-linear decomposition method which yields non-linear orthogonal decomposed parts. For symmetric second rank and fourth rank tensors, this case is a significant innovation in decomposition procedures in the literature.

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15

Smart, David P. (David Paul). "Tensor decomposition and parallelization of Markov Decision Processes". Thesis, Massachusetts Institute of Technology, 2016. http://hdl.handle.net/1721.1/105018.

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Thesis: S.M., Massachusetts Institute of Technology, Computation for Design and Optimization Program, 2016.
Cataloged from PDF version of thesis.
Includes bibliographical references (pages 85-81).
Markov Decision Processes (MDPs) with large state spaces arise frequently when applied to real world problems. Optimal solutions to such problems exist, but may not be computationally tractable, as the required processing scales exponentially with the number of states. Unsurprisingly, investigating methods for efficiently determining optimal or near-optimal policies has generated substantial interest and remains an active area of research. A recent paper introduced an MDP representation as a tensor composition of a set of smaller component MDPs, and suggested a method for solving an MDP by decomposition into its tensor components and solving the smaller problems in parallel, combining their solutions into one for the original problem. Such an approach promises an increase in solution efficiency, since each smaller problem could be solved exponentially faster than the original. This paper develops this MDP tensor decomposition and parallelization algorithm, and analyzes both its computational performance and the optimality of its resultant solutions.
by David P. Smart.
S.M.

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16

SAPIENZA, ANNA. "Tensor decomposition techniques for analysing time-varying networks". Doctoral thesis, Politecnico di Torino, 2017. http://hdl.handle.net/11583/2668112.

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The aim of this Ph.D thesis is the study of time-varying networks via theoretical and data-driven approaches. Networks are natural objects to represent a vast variety of systems in nature, e.g., communication networks (phone calls and e-mails), online social networks (Facebook, Twitter), infrastructural networks, etc. Considering the temporal dimension of networks helps to better understand and predict complex phenomena, by taking into account both the fact that links in the network are not continuously active over time and the potential relation between multiple dimensions, such as space and time. A fundamental challenge in this area is the definition of mathematical models and tools able to capture topological and dynamical aspects and to reproduce properties observed on the real dynamics of networks. Thus, the purpose of this thesis is threefold: 1) we will focus on the analysis of the complex mesoscale patterns, as community like structures and their evolution in time, that characterize time-varying networks; 2) we will study how these patterns impact dynamical processes that occur over the network; 3) we will sketch a generative model to study the interplay between topological and temporal patterns of time-varying networks and dynamical processes occurring over the network, e.g., disease spreading. To tackle these problems, we adopt and extend an approach at the intersection between multi-linear algebra and machine learning: the decomposition of time-varying networks represented as tensors (multi-dimensional arrays). In particular, we focus on the study of Non-negative Tensor Factorization (NTF) techniques to detect complex topological and temporal patterns in the network. We first extend the NTF framework to tackle the problem of detecting anomalies in time-varying networks. Then, we propose a technique to approximate and reconstruct time-varying networks affected by missing information, to both recover the missing values and to reproduce dynamical processes on top of the network. Finally, we focus on the analysis of the interplay between the discovered patterns and dynamical processes. To this aim, we use the NTF as an hint to devise a generative model of time-varying networks, in which we can control both the topological and temporal patterns, to identify which of them has a major impact on the dynamics.

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17

Ratto, Francesco <1995&gt. "Tensor Decomposition, multifactorial model and strategic Asset Allocation". Master's Degree Thesis, Università Ca' Foscari Venezia, 2019. http://hdl.handle.net/10579/15382.

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L'obiettivo della tesi è applicare un algoritmo utilizzato nel Machine Learning per decomporre tensori non negativi. L'applicazione dell'algoritmo sarà dedicata all'asset allocation e all'ottimizzazione di portafoglio, proponendo vai metodi per selezionare con vari modelli i pesi di portafoglio ottimi.

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18

Morgan, William Russell IV. "Investigations into Parallelizing Rank-One Tensor Decompositions". Thesis, University of Maryland, Baltimore County, 2018. http://pqdtopen.proquest.com/#viewpdf?dispub=10683240.

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Tensor Decompositions are a solved problem in terms of evaluating for a result. Performance, however, is not. There are several projects to parallelize tensor decompositions, using a variety of different methods. This work focuses on investigating other possible strategies for parallelization of rank-one tensor decompositions, measuring performance across a variety of tensor sizes, and reporting the best avenues to continue investigation

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19

Senese, Frederick A. "A tensor product decomposition of the many-electron Hamiltonian". Diss., Virginia Polytechnic Institute and State University, 1989. http://hdl.handle.net/10919/54413.

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A new direct full variational approach is described. The approach exploits a tensor (Kronecker) product construction of the many-electron Hamiltonian and has a number of computational advantages. Explicit assembly and storage of the Hamiltonian matrix is avoided by using the Kronecker product structure to form matrix-vector products directly from the molecular integrals. Computation-intensive integral transformations and formula tapes are unnecessary. The wavefunction is expanded in terms of spin-free primitive kets rather than Slater determinants or configuration state functions and is equivalent to a full configuration interaction expansion. The approach suggests compact storage schemes and algorithms which are naturally suited to parallel and pipelined machines. Sample calculations for small two- and four-electron systems are presented. The preliminary ground state potential energy surface of the hydrogen molecule dimer is computed by the tensor product method using a small basis set.
Ph. D.

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20

Zhu, Lierong. "Topological visualization of tensor fields using a generalized Helmholtz decomposition". Morgantown, W. Va. : [West Virginia University Libraries], 2010. http://hdl.handle.net/10450/10962.

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Thesis (M.S.)--West Virginia University, 2010.
Title from document title page. Document formatted into pages; contains viii, 75 p. : ill. (some col.). Includes abstract. Includes bibliographical references (p. 72-75).

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21

Bi, Bin y 闭彬. "Third-order tensor decomposition for search in social tagging systems". Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2010. http://hub.hku.hk/bib/B45160041.

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Reising, Justin. "Function Space Tensor Decomposition and its Application in Sports Analytics". Digital Commons @ East Tennessee State University, 2019. https://dc.etsu.edu/etd/3676.

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Recent advancements in sports information and technology systems have ushered in a new age of applications of both supervised and unsupervised analytical techniques in the sports domain. These automated systems capture large volumes of data points about competitors during live competition. As a result, multi-relational analyses are gaining popularity in the field of Sports Analytics. We review two case studies of dimensionality reduction with Principal Component Analysis and latent factor analysis with Non-Negative Matrix Factorization applied in sports. Also, we provide a review of a framework for extending these techniques for higher order data structures. The primary scope of this thesis is to further extend the concept of tensor decomposition through the use of function spaces. In doing so, we address the limitations of PCA to vector and matrix representations and the CP-Decomposition to tensor representations. Lastly, we provide an application in the context of professional stock car racing.

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23

Avila, Gastón. "Asymptotic staticity and tensor decompositions with fast decay conditions". Phd thesis, Universität Potsdam, 2011. http://opus.kobv.de/ubp/volltexte/2011/5404/.

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Corvino, Corvino and Schoen, Chruściel and Delay have shown the existence of a large class of asymptotically flat vacuum initial data for Einstein's field equations which are static or stationary in a neighborhood of space-like infinity, yet quite general in the interior. The proof relies on some abstract, non-constructive arguments which makes it difficult to calculate such data numerically by using similar arguments. A quasilinear elliptic system of equations is presented of which we expect that it can be used to construct vacuum initial data which are asymptotically flat, time-reflection symmetric, and asymptotic to static data up to a prescribed order at space-like infinity. A perturbation argument is used to show the existence of solutions. It is valid when the order at which the solutions approach staticity is restricted to a certain range. Difficulties appear when trying to improve this result to show the existence of solutions that are asymptotically static at higher order. The problems arise from the lack of surjectivity of a certain operator. Some tensor decompositions in asymptotically flat manifolds exhibit some of the difficulties encountered above. The Helmholtz decomposition, which plays a role in the preparation of initial data for the Maxwell equations, is discussed as a model problem. A method to circumvent the difficulties that arise when fast decay rates are required is discussed. This is done in a way that opens the possibility to perform numerical computations. The insights from the analysis of the Helmholtz decomposition are applied to the York decomposition, which is related to that part of the quasilinear system which gives rise to the difficulties. For this decomposition analogous results are obtained. It turns out, however, that in this case the presence of symmetries of the underlying metric leads to certain complications. The question, whether the results obtained so far can be used again to show by a perturbation argument the existence of vacuum initial data which approach static solutions at infinity at any given order, thus remains open. The answer requires further analysis and perhaps new methods.
Corvino, Corvino und Schoen als auch Chruściel und Delay haben die Existenz einer grossen Klasse asymptotisch flacher Anfangsdaten für Einsteins Vakuumfeldgleichungen gezeigt, die in einer Umgebung des raumartig Unendlichen statisch oder stationär aber im Inneren der Anfangshyperfläche sehr allgemein sind. Der Beweis beruht zum Teil auf abstrakten, nicht konstruktiven Argumenten, die Schwierigkeiten bereiten, wenn derartige Daten numerisch berechnet werden sollen. In der Arbeit wird ein quasilineares elliptisches Gleichungssystem vorgestellt, von dem wir annehmen, dass es geeignet ist, asymptotisch flache Vakuumanfangsdaten zu berechnen, die zeitreflektionssymmetrisch sind und im raumartig Unendlichen in einer vorgeschriebenen Ordnung asymptotisch zu statischen Daten sind. Mit einem Störungsargument wird ein Existenzsatz bewiesen, der gilt, solange die Ordnung, in welcher die Lösungen asymptotisch statische Lösungen approximieren, in einem gewissen eingeschränkten Bereich liegt. Versucht man, den Gültigkeitsbereich des Satzes zu erweitern, treten Schwierigkeiten auf. Diese hängen damit zusammen, dass ein gewisser Operator nicht mehr surjektiv ist. In einigen Tensorzerlegungen auf asymptotisch flachen Räumen treten ähnliche Probleme auf, wie die oben erwähnten. Die Helmholtzzerlegung, die bei der Bereitstellung von Anfangsdaten für die Maxwellgleichungen eine Rolle spielt, wird als ein Modellfall diskutiert. Es wird eine Methode angegeben, die es erlaubt, die Schwierigkeiten zu umgehen, die auftreten, wenn ein schnelles Abfallverhalten des gesuchten Vektorfeldes im raumartig Unendlichen gefordert wird. Diese Methode gestattet es, solche Felder auch numerisch zu berechnen. Die Einsichten aus der Analyse der Helmholtzzerlegung werden dann auf die Yorkzerlegung angewandt, die in den Teil des quasilinearen Systems eingeht, der Anlass zu den genannten Schwierigkeiten gibt. Für diese Zerlegung ergeben sich analoge Resultate. Es treten allerdings Schwierigkeiten auf, wenn die zu Grunde liegende Metrik Symmetrien aufweist. Die Frage, ob die Ergebnisse, die soweit erhalten wurden, in einem Störungsargument verwendet werden können um die Existenz von Vakuumdaten zu zeigen, die im räumlich Unendlichen in jeder Ordnung statische Daten approximieren, bleibt daher offen. Die Antwort erfordert eine weitergehende Untersuchung und möglicherweise auch neue Methoden.

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Silva, Alex Pereira da. "Tensor techniques in signal processing: algorithms for the canonical polyadic decomposition (PARAFAC)". reponame:Repositório Institucional da UFC, 2016. http://www.repositorio.ufc.br/handle/riufc/19361.

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SILVA, A. P. Tensor techniques in signal processing: algorithms for the canonical polyadic decomposition (PARAFAC). 2016. 124 f. Tese (Doutorado em Engenharia de Teleinformática) - Centro de Tecnologia, Universidade Federal do Ceará, Fortaleza, 2016.
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Low rank tensor decomposition has been playing for the last years an important role in many applications such as blind source separation, telecommunications, sensor array processing, neuroscience, chemometrics, and data mining. The Canonical Polyadic tensor decomposition is very attractive when compared to standard matrix-based tools, manly on system identification. In this thesis, we propose: (i) several algorithms to compute specific low rank-approximations: finite/iterative rank-1 approximations, iterative deflation approximations, and orthogonal tensor decompositions. (ii) A new strategy to solve multivariate quadratic systems, where this problem is reduced to a best rank-1 tensor approximation problem. (iii) Theoretical results to study and proof the performance or the convergence of some algorithms. All performances are supported by numerical experiments

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25

Lawlor, Matthew. "Tensor Decomposition by Modified BCM Neurons Finds Mixture Means Through Input Triplets". Thesis, Yale University, 2014. http://pqdtopen.proquest.com/#viewpdf?dispub=3580742.

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Grimm, Christopher Lee Jr. "A tensor-train-decomposition-based algorithm for high-dimensional pursuit-evasion games". Thesis, Massachusetts Institute of Technology, 2016. http://hdl.handle.net/1721.1/105615.

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Thesis: S.M., Massachusetts Institute of Technology, Department of Aeronautics and Astronautics, 2016.
Cataloged from PDF version of thesis.
Includes bibliographical references (pages 99-100).
The research presented in this thesis was inspired by an interest in determining feedback strategies for high-dimensional pursuit-evasion games. When a problem is high-dimensional or involves a state space that is defined by several variables, various methods used to solve pursuit-evasion games often require unrealistic computation time. This problem, called the curse of dimensionality, can be mitigated under certain circumstances by utilizing tensor-train (TT) decomposition. By using this intuition, a new algorithm for solving high dimensional pursuit-evasion problems called Best-Response Tensor-Train-decomposition-based Value Iteration (BR-TT-VI) was developed. BR-TT-VI builds on concepts from game theory, dynamic programming (DP), and tensor-train decomposition. By using TT decomposition, BR-TT-VI greatly reduces the effects of the curse of dimensionality. This work culminates in the application of BR-TT-VI to two different pursuit-evasion problems. First, a four-dimensional problem capable of being solved by traditional value iteration(VI) is tackled by the BR-TT-VI algorithm. This problem allows a direct comparison between VI and BR-TT-VI to demonstrate the reduced computational time of the new algorithm. Finally, BR-TT-VI is used to solve a six-dimensional problem involving two Dubins vehicles that is impractical to solve with VI.
by Christopher Lee Grimm Jr.
S.M.

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27

Pham, Thi. "Visualization and Classification of Neurological Status with Tensor Decomposition and Machine Learning". Thesis, Linköpings universitet, Institutionen för datavetenskap, 2019. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-158497.

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Recognition of physical and mental responses to stress is important for stress assessment and management as its negative effects in health can be prevented or reduced. Wearable technology, mainly using electroencephalogram (EEG), provides information such as tracking fitness activity, disease detection, and monitoring neurologicalstates of individuals. However, the recording of EEG signals from a wearable device is inconvenient, expensive, and uncomfortable during normal daily activities. This study introduces the application of tensor decomposition of non-EEG data for visualizing and classifying neurological statuses with application to human stress recognition. The multimodal dataset of non-EEG physiological signals publicly available from the PhysioNet database was used for testing the proposed method. To visualize the biosignals in a low dimensional feature space, the multi-way factorization technique known as the PARAFAC was applied for feature extraction. Results show visualizations that well separate the four groups of neurological statuses obtained from twenty healthy subjects. The extracted features were then used for pattern classification. Two statistical classifiers, which are the multinomial logit regression(MLR) and linear discriminant analysis (LDA), were implemented. The results show that the MLR and LDA can identify the four neurological statuses with accuracies of 95% and 98.8%, respectively. This study suggests the potential application of tensor decomposition for the analysis of physiological measurements collected from multiple sensors. Moreover, the proposed study contributes to the advancement of wearable technology for human stress monitoring. With tensor decomposition of complex multi-sensor or multi-channel data, simple classification techniques can be employed to achieve similar results obtained using sophisticated machine-learning techniques.

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28

Vasile, Flavian. "Uncovering the structure of hypergraphs through tensor decomposition an application to folksonomy analysis /". [Ames, Iowa : Iowa State University], 2008.

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29

Alora, John Irvin P. "Automated synthesis of low-rank stochastic dynamical systems using the tensor-train decomposition". Thesis, Massachusetts Institute of Technology, 2016. http://hdl.handle.net/1721.1/105006.

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Thesis: S.M., Massachusetts Institute of Technology, Department of Aeronautics and Astronautics, 2016.
This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.
Cataloged from student-submitted PDF version of thesis.
Includes bibliographical references (pages 79-83).
Cyber-physical systems are increasingly becoming integrated in various fields such as medicine, finance, robotics, and energy. In these systems and their applications, safety and correctness of operation is of primary concern, sparking a large amount of interest in the development of ways to verify system behavior. The tight coupling of physical constraints and computation that typically characterize cyber-physical systems make them extremely complex, resulting in unexpected failure modes. Furthermore, disturbances in the environment and uncertainties in the physical model require these systems to be robust. These are difficult constraints, requiring cyberphysical systems to be able to reason about their behavior and respond to events in real-time. Thus, the goal of automated synthesis is to construct a controller that provably implements a range of behaviors given by a specification of how the system should operate. Unfortunately, many approaches to automated synthesis are ad hoc and are limited to simple systems that admit specific structure (e.g. linear, affine systems). Not only that, but they are also designed without taking into account uncertainty. In order to tackle more general problems, several computational frameworks that allow for more general dynamics and uncertainty to be investigated. Furthermore, all of the existing computational algorithms suffer from the curse of dimensionality, the run time scales exponentially with increasing dimensionality of the state space. As a result, existing algorithms apply to systems with only a few degrees of freedom. In this thesis, we consider a stochastic optimal control problem with a special class of linear temporal logic specifications and propose a novel algorithm based on the tensor-train decomposition. We prove that the run time of the proposed algorithm scales linearly with the dimensionality of the state space and polynomially with the rank of the optimal cost-to-go function.
by John Irvin P. Alora.
S.M.

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30

Mosskull, Albin y Arfvidsson Kaj Munhoz. "Solving the Hamilton-Jacobi-Bellman Equation for Route Planning Problems Using Tensor Decomposition". Thesis, KTH, Skolan för elektroteknik och datavetenskap (EECS), 2020. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-289326.

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Optimizing routes for multiple autonomous vehiclesin complex traffic situations can lead to improved efficiency intraffic. Attempting to solve these optimization problems centrally,i.e. for all vehicles involved, often lead to algorithms that exhibitthe curse of dimensionality: that is, the computation time andmemory needed scale exponentially with the number of vehiclesresulting in infeasible calculations for moderate number ofvehicles. However, using a numerical framework called tensordecomposition one can calculate and store solutions for theseproblems in a more manageable way. In this project, we investi-gate different tensor decomposition methods and correspondingalgorithms for solving optimal control problems, by evaluatingtheir accuracy for a known solution. We also formulate complextraffic situations as optimal control problems and solve them.We do this by using the best tensor decomposition and carefullyadjusting different cost parameters. From these results it canbe concluded that the Sequential Alternating Least Squaresalgorithm used with canonical tensor decomposition performedthe best. By asserting a smooth cost function one can solve certainscenarios and acquire satisfactory solutions, but it requiresextensive testing to achieve such results, since numerical errorsoften can occur as a result of an ill-formed problem.
Att optimera färdvägen för flertalet au-tonoma fordon i komplexa trafiksituationer kan leda till effekti-vare trafik. Om man försöker lösa dessa optimeringsproblemcentralt, för alla fordon samtidigt, leder det ofta till algorit-mer som uppvisar The curse of dimensionality, vilket är då beräkningstiden och minnes-användandet växer exponentielltmed antalet fordon. Detta gör många problem olösbara för endasten måttlig mängd fordon. Däremot kan sådana problem hanterasgenom numeriska verktyg så som tensornedbrytning. I det här projektet undersöker vi olika metoder för tensornedbrytningoch motsvarandes algoritmer för att lösa optimala styrproblem,genom att jämföra dessa för ett problem med en känd lösning.Dessutom formulerar vi komplexa trafiksituationer som optimalastyrproblem för att sedan lösa dem. Detta gör vi genom attanvända den bästa tensornedbrytningen och genom att noggrantanpassa kostnadsparametrar. Från dessa resultat framgår det att Sequential Alternating Least Squaresalgoritmen, tillsammans medkanonisk tensornedbrytning, överträffade de andra algoritmersom testades. De komplexa trafiksituationerna kan lösas genomatt ansätta släta kostnadsfunktioner, men det kräver omfattandetestning för att uppnå sådana resultat då numeriska fel lätt kan uppstå som ett resultat av dålig problemformulering.
Kandidatexjobb i elektroteknik 2020, KTH, Stockholm

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31

Kang, Kingston. "ESTIMATING THE RESPIRATORY LUNG MOTION MODEL USING TENSOR DECOMPOSITION ON DISPLACEMENT VECTOR FIELD". VCU Scholars Compass, 2018. https://scholarscompass.vcu.edu/etd/5254.

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Modern big data often emerge as tensors. Standard statistical methods are inadequate to deal with datasets of large volume, high dimensionality, and complex structure. Therefore, it is important to develop algorithms such as low-rank tensor decomposition for data compression, dimensionality reduction, and approximation. With the advancement in technology, high-dimensional images are becoming ubiquitous in the medical field. In lung radiation therapy, the respiratory motion of the lung introduces variabilities during treatment as the tumor inside the lung is moving, which brings challenges to the precise delivery of radiation to the tumor. Several approaches to quantifying this uncertainty propose using a model to formulate the motion through a mathematical function over time. [Li et al., 2011] uses principal component analysis (PCA) to propose one such model using each image as a long vector. However, the images come in a multidimensional arrays, and vectorization breaks the spatial structure. Driven by the needs to develop low-rank tensor decomposition and provided the 4DCT and Displacement Vector Field (DVF), we introduce two tensor decompositions, Population Value Decomposition (PVD) and Population Tucker Decomposition (PTD), to estimate the respiratory lung motion with high levels of accuracy and data compression. The first algorithm is a generalization of PVD [Crainiceanu et al., 2011] to higher order tensor. The second algorithm generalizes the concept of PVD using Tucker decomposition. Both algorithms are tested on clinical and phantom DVFs. New metrics for measuring the model performance are developed in our research. Results of the two new algorithms are compared to the result of the PCA algorithm.

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32

Ximenes, Leandro Ronchini. "MIMO channel modeling and estimation: application of spherical harmonics and tensor decompositions". reponame:Repositório Institucional da UFC, 2011. http://www.repositorio.ufc.br/handle/riufc/12915.

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XIMENES, L. R. MIMO channel modeling and estimation: application of spherical harmonics and tensor decompositions. 2011. 120 f. Dissertação (Mestrado em Engenharia de Teleinformática) – Centro de Tecnologia, Universidade Federal do Ceará, Fortaleza, 2011.
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In the last two decades, multiple input multiple output (MIMO) wireless systems have been subject of intense research due to the theoretical promise of the proportional increase of the communications channel capacity as the number of antennas increases. This outstanding property supposes an efficient use of spatial diversity at both the transmitter and receiver. An important and not well explored path towards improving MIMO system performance using spatial diversity takes into account the interactions among the antennas and the (physical) propagation medium. By understanding these interactions, the transmit and receive antenna arrays can be designed to best “match” the propagation medium so that the link quality and capacity can be further improved in a MIMO system. In this work, we consider the use of spherical harmonics and tensor decompositions in the problem of MIMO channel modeling and estimation. The use of spherical harmonics allows to represent the radiation patterns of antennas in terms of coefficients of an expansion, thus decoupling the transmit and receive antenna array responses from the physical propagation medium. By translating simple propagation-motivated channel models with polarization information into the spherical harmonics domain, we study how propagation parameters themselves and antenna configurations affect MIMO performance in terms of capacity and correlation. A second part of this work addresses the problem of estimating directional MIMO channels in the spherical harmonics domain using tensor decompositions. Considering both single-scattering and double-scattering propagation scenarios, we make use of the parallel factor (PARAFAC) and PARATUCK-2 decompositions, respectively, to estimate the propagating spherical modes, from which the directions of arrival (DoA) and directions of departure (DoD) can be extracted. Finally, we propose and compare two methods for optimizing the coefficients of the spherical harmonics expansion of an antenna array for a prespecified MIMO channel response
Nas últimas décadas, sistemas de comunicação sem fio de múltiplas antenas (MIMO - Multiple Input Multiple Output) têm sido objetos de intensas pesquisas devido à promessa teórica do aumento proporcional da capacidade com o aumento do número de antenas. Esta propriedade excepcional supõe um uso eficiente da diversidade espacial no transmissor e receptor. Um caminho importante e não bem explorado no sentido de melhorar o desempenho de sistemas MIMO usando diversidade espacial leva em conta a interação entre as antenas e meio de propagação (físico). Através da compreensão dessas interações, arranjos de antenas de recepção e transmissão podem ser projetados para melhor "casar" com o meio de propagação, tal que a qualidade do link de comunicação e capacidade possam ser melhoradas em um sistema MIMO. Neste trabalho, consideramos o uso de harmônicos esféricos e decomposições tensoriais no problema de modelagem de canal MIMO e estimação. O uso de harmônicos esféricos permite representar os padrões de radiação de antenas em termos de coeficientes de uma expansão, assim desacoplando as respostas dos arranjos de antenas (transmissoras e receptoras) do meio de propagação física. Traduzindo modelos simples de canais baseados em propagação, com informações de polarização, para o domínio dos harmônicos esféricos, estudamos como os parâmetros de propagação si e configurações específicas de antenas afetam o desempenho do sistema MIMO em termos de capacidade e de correlação. A segunda parte deste trabalho aborda o problema de estimar canais direcionais MIMO no domínio dos harmônicos esféricos usando decomposições por tensores. Considerando tanto cenos de espalhamento simples e de duplo espalhamento, fazemos uso das decomposições PARAFAC e PARATUCK2, respectivamente, para estimar os modos esféricos propagantes, a partir das quais as direções de chegada (DoA) e as direções de saída (DoD) podem ser extraídas. Finalmente, propomos e comparamos dois métodos de otimização dos coeficientes da expansão em harmônicos esféricos de arranjos de antenas para respostas de canais MIMO pré-especificados

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33

Cavalcante, Ãtalo Vitor. "Tensor approach for channel estimation in MIMO multi-hop cooperative networks". Universidade Federal do CearÃ, 2014. http://www.teses.ufc.br/tde_busca/arquivo.php?codArquivo=12442.

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CoordenaÃÃo de AperfeÃoamento de Pessoal de NÃvel Superior
In this dissertation the problem of channel estimation in cooperative MIMO systems is investigated. More specifically, channel estimation techniques have been developed for a communication system assisted by relays with amplify-and-forward (AF) processing system in a three-hop scenario. The techniques developed use training sequences and enable, at the receiving node, the estimation of all the channels involved in the communication process. In an initial scenario, we consider a communication system with N transmit antennas and M receive antennas and between these nodes we have two relay groups with R1 and R2 antennas each. We propose protocols based on temporal multiplexing to coordinate the retransmission of the signals. At the end of the training phase, the receiving node estimates the channel matrices by combining the received data. By exploiting the multilinear (tensorial) structure of the sets of signals, we can model the received data through tensor models, such as PARAFAC and PARATUCK2 . This work proposes the combined use of these models and algebraic techniques to explore the spatial diversity. Secondly, we consider that the number of transmit and receive antennas at the relays may be different and that the data can travel in a bidirectional scheme (two-way). In order to validate the algorithms we use Monte-Carlo simulations in which we compare our proposed models with competing channel estimation algorithms, such as, the PARAFAC and Khatri-Rao factorization based algorithms in terms of NMSE and bit error rate.
Nesta dissertaÃÃo o problema de estimaÃÃo de canal em sistemas MIMO cooperativos Ã investigado. Mais especificamente, foram desenvolvidas tÃcnicas para estimaÃÃo de canal em um sistema de comunicaÃÃo assistida por relays com processamento do tipo amplifica-e-encaminha (do inglÃs, amplify-and-forward) em um cenÃrio de 3 saltos. As tÃcnicas desenvolvidas utilizam sequÃncia de treinamento e habilitam, no nÃ receptor, a estimaÃÃo de todos os canais envolvidos no processo de comunicaÃÃo. Em um cenÃrio inicial, consideramos um sistema de comunicaÃÃo com N antenas transmissoras e M antenas receptoras e entre esses nÃs temos dois grupos de relays com R1 e R2 antenas cada um. Foram desenvolvidos protocolos de transmissÃo baseado em multiplexaÃÃo temporal para coordenar as retransmissÃes dos sinais. Ao final da fase de treinamento, o nÃ receptor faz a estimaÃÃo das matrizes de canal atravÃs da combinaÃÃo dos dados recebidos. Explorando a estrutura multilinear (tensorial) dos diversos conjuntos de sinais, podemos modelar os dados recebidos atravÃs de modelos tensoriais, tais como: PARAFAC e PARATUCK2. Este trabalho propÃe a utilizaÃÃo combinada desses modelos e de tÃcnicas algÃbricas para explorar a diversidade espacial. Em um segundo momento, consideramos que o nÃmero de antenas transmissoras e receptoras dos relays podem ser diferentes e ainda que os dados podem trafegar em um esquema bidirecional (do inglÃs, two-way). Para fins de validaÃÃo dos algoritmos utilizamos simulaÃÃes de Monte-Carlo em que comparamos os modelos propostos com outros algoritmos de estimaÃÃo de canal, tais como os algoritmos baseados em PARAFAC e FatoraÃÃo de Khatri-Rao em termos de NMSE e taxa de erro de bit.

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34

Gomes, Paulo Ricardo Barboza. "Tensor Methods for Blind Spatial Signature Estimation". Universidade Federal do CearÃ, 2014. http://www.teses.ufc.br/tde_busca/arquivo.php?codArquivo=11635.

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FundaÃÃo Cearense de Apoio ao Desenvolvimento Cientifico e TecnolÃgico
In this dissertation the problem of spatial signature and direction of arrival estimation in Linear 2L-Shape and Planar arrays is investigated Methods based on tensor decompositions are proposed to treat the problem of estimating blind spatial signatures disregarding the use of training sequences and knowledge of the covariance structure of the sources By assuming that the power of the sources varies between successive time blocks decompositions for tensors of third and fourth orders obtained from spatial and spatio-temporal covariance of the received data in the array are proposed from which iterative algorithms are formulated to estimate spatial signatures of the sources Then greater spatial diversity is achieved by using the Spatial Smoothing in the 2L-Shape and Planar arrays In that case the estimation of the direction of arrival of the sources can not be obtained directly from the formulated algorithms The factorization of the Khatri-Rao product is then incorporated into these algorithms making it possible extracting estimates for the azimuth and elevation angles from matrices obtained using this method A distinguishing feature of the proposed tensor methods is their efficiency to treat the cases where the covariance matrix of the sources is non-diagonal and unknown which generally happens when working with sample data covariances computed from a reduced number of snapshots
Nesta dissertaÃÃo o problema de estimaÃÃo de assinaturas espaciais e consequentemente da direÃÃo de chegada dos sinais incidentes em arranjos Linear 2L-Shape e Planar Ã investigado MÃtodos baseados em decomposiÃÃes tensoriais sÃo propostos para tratar o problema de estimaÃÃo cega de assinaturas espaciais desconsiderando a utilizaÃÃo de sequÃncias de treinamento e o conhecimento da estrutura de covariÃncia das fontes Ao assumir que a potÃncia das fontes varia entre blocos de tempos sucessivos decomposiÃÃes para tensores de terceira e quarta ordem obtidas a partir da covariÃncia espacial e espaÃo-temporal dos dados recebidos no arranjo de sensores sÃo propostas a partir das quais algoritmos iterativos sÃo formulados para estimar a assinatura espacial das fontes em seguida uma maior diversidade espacial Ã alcanÃada utilizando a tÃcnica Spatial Smoothing na recepÃÃo de sinais nos arranjos 2L-Shape e Planar Nesse caso as estimaÃÃes da direÃÃo de chegada das fontes nÃo podem ser obtidas diretamente a partir dos algoritmos formulados de forma que a fatoraÃÃo do produto de Khatri-Rao Ã incorporada a estes algoritmos tornando possÃvel a obtenÃÃo de estimaÃÃes para os Ãngulos de azimute e elevaÃÃo a partir das matrizes obtidas utilizando este mÃtodo Uma caracterÃstica marcante dos mÃtodos tensoriais propostos estÃ presente na eficiÃncia obtida no tratamento de casos em que a matriz de covariÃncia das fontes Ã nÃo-diagonal e desconhecida o que geralmente ocorre quando se trabalha com covariÃncias de amostras reais calculadas a partir de um nÃmero reduzido de snapshots

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35

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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36

Riether, Fabian. "Agile quadrotor maneuvering using tensor-decomposition-based globally optimal control and onboard visual-inertial estimation". Thesis, Massachusetts Institute of Technology, 2016. http://hdl.handle.net/1721.1/106777.

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Thesis: S.M., Massachusetts Institute of Technology, Department of Mechanical Engineering, 2016.
Cataloged from PDF version of thesis.
Includes bibliographical references (pages 123-127).
Over the last few years, quadrotors have become increasingly popular amongst researchers and hobbyist. Although tremendous progress has been made towards making drones autonomous, advanced capabilities, such as aggressive maneuvering and visual perception, are still confined to either laboratory environments with motion capture systems or drone platforms with large size, weight, and power requirements. We identify two recent developments that may help address these shortcomings. On the one hand, new embedded high-performance computers equipped with powerful Graphics Processor Units (GPUs). These computers enable real-time onboard processing of vision data. On the other hand, recently introduced compressed continuous computation techniques for stochastic optimal control allow designing feedback control systems for agile maneuvering. In this thesis, we design, implement and demonstrate a micro unmanned aerial vehicle capable of executing certain agile maneuvers using only a forward-facing camera and an inertial measurement unit. Specifically, we develop a hardware platform equipped with an Nvidia Jetson embedded super-computer for vision processing. We develop a low-latency software suite, including onboard visual marker detection, visual-inertial estimation and control algorithms. The full-state estimation is set up with respect to a visual target, such as a window. A nonlinear globally-optimal controller is designed to execute the desired flight maneuver. The resulting optimization problem is solved using tensor-train-decomposition-based compressed continuous computation techniques. The platform's capabilities and the potential of these types of controllers are demonstrated in both simulation studies and in experiments.
by Fabian Riether.
S.M.

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37

Zniyed, Yassine. "Breaking the curse of dimensionality based on tensor train : models and algorithms". Thesis, Université Paris-Saclay (ComUE), 2019. http://www.theses.fr/2019SACLS330.

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Le traitement des données massives, communément connu sous l’appellation “Big Data”, constitue l’un des principaux défis scientifiques de la communauté STIC.Plusieurs domaines, à savoir économique, industriel ou scientifique, produisent des données hétérogènes acquises selon des protocoles technologiques multi-modales. Traiter indépendamment chaque ensemble de données mesurées est clairement une approche réductrice et insatisfaisante. En faisant cela, des “relations cachées” ou des inter-corrélations entre les données peuvent être totalement ignorées.Les représentations tensorielles ont reçu une attention particulière dans ce sens en raison de leur capacité à extraire de données hétérogènes et volumineuses une information physiquement interprétable confinée à un sous-espace de dimension réduite. Dans ce cas, les données peuvent être organisées selon un tableau à D dimensions, aussi appelé tenseur d’ordre D.Dans ce contexte, le but de ce travail et que certaines propriétés soient présentes : (i) avoir des algorithmes de factorisation stables (ne souffrant pas de probème de convergence), (ii) avoir un faible coût de stockage (c’est-à-dire que le nombre de paramètres libres doit être linéaire en D), et (iii) avoir un formalisme sous forme de graphe permettant une visualisation mentale simple mais rigoureuse des décompositions tensorielles de tenseurs d’ordre élevé, soit pour D > 3.Par conséquent, nous nous appuyons sur la décomposition en train de tenseurs (TT) pour élaborer de nouveaux algorithmes de factorisation TT, et des nouvelles équivalences en termes de modélisation tensorielle, permettant une nouvelle stratégie de réduction de dimensionnalité et d'optimisation de critère des moindres carrés couplés pour l'estimation des paramètres d'intérêts nommé JIRAFE.Ces travaux d'ordre méthodologique ont eu des applications dans le contexte de l'analyse spectrale multidimensionelle et des systèmes de télécommunications à relais
Massive and heterogeneous data processing and analysis have been clearly identified by the scientific community as key problems in several application areas. It was popularized under the generic terms of "data science" or "big data". Processing large volumes of data, extracting their hidden patterns, while preforming prediction and inference tasks has become crucial in economy, industry and science.Treating independently each set of measured data is clearly a reductiveapproach. By doing that, "hidden relationships" or inter-correlations between thedatasets may be totally missed. Tensor decompositions have received a particular attention recently due to their capability to handle a variety of mining tasks applied to massive datasets, being a pertinent framework taking into account the heterogeneity and multi-modality of the data. In this case, data can be arranged as a D-dimensional array, also referred to as a D-order tensor.In this context, the purpose of this work is that the following properties are present: (i) having a stable factorization algorithms (not suffering from convergence problems), (ii) having a low storage cost (i.e., the number of free parameters must be linear in D), and (iii) having a formalism in the form of a graph allowing a simple but rigorous mental visualization of tensor decompositions of tensors of high order, i.e., for D> 3.Therefore, we rely on the tensor train decomposition (TT) to develop new TT factorization algorithms, and new equivalences in terms of tensor modeling, allowing a new strategy of dimensionality reduction and criterion optimization of coupled least squares for the estimation of parameters named JIRAFE.This methodological work has had applications in the context of multidimensional spectral analysis and relay telecommunications systems

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38

Gorodetsky, Alex Arkady. "Continuous low-rank tensor decompositions, with applications to stochastic optimal control and data assimilation". Thesis, Massachusetts Institute of Technology, 2017. http://hdl.handle.net/1721.1/108918.

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Thesis: Ph. D., Massachusetts Institute of Technology, Department of Aeronautics and Astronautics, 2017.
Cataloged from PDF version of thesis.
Includes bibliographical references (pages 205-214).
Optimal decision making under uncertainty is critical for control and optimization of complex systems. However, many techniques for solving problems such as stochastic optimal control and data assimilation encounter the curse of dimensionality when too many state variables are involved. In this thesis, we propose a framework for computing with high-dimensional functions that mitigates this exponential growth in complexity for problems with separable structure. Our framework tightly integrates two emerging areas: tensor decompositions and continuous computation. Tensor decompositions are able to effectively compress and operate with low-rank multidimensional arrays. Continuous computation is a paradigm for computing with functions instead of arrays, and it is best realized by Chebfun, a MATLAB package for computing with functions of up to three dimensions. Continuous computation provides a natural framework for building numerical algorithms that effectively, naturally, and automatically adapt to problem structure. The first part of this thesis describes a compressed continuous computation framework centered around a continuous analogue to the (discrete) tensor-train decomposition called the function-train decomposition. Computation with the function-train requires continuous matrix factorizations and continuous numerical linear algebra. Continuous analogues are presented for performing cross approximation; rounding; multilinear algebra operations such as addition, multiplication, integration, and differentiation; and continuous, rank-revealing, alternating least squares. Advantages of the function-train over the tensor-train include the ability to adaptively approximate functions and the ability to compute with functions that are parameterized differently. For example, while elementwise multiplication between tensors of different sizes is undefined, functions in FT format can be readily multiplied together. Next, we develop compressed versions of value iteration, policy iteration, and multilevel algorithms for solving dynamic programming problems arising in stochastic optimal control. These techniques enable computing global solutions to a broader set of problems, for example those with non-affine control inputs, than previously possible. Examples are presented for motion planning with robotic systems that have up to seven states. Finally, we use the FT to extend integration-based Gaussian filtering to larger state spaces than previously considered. Examples are presented for dynamical systems with up to twenty states.
by Alex Arkady Gorodetsky.
Ph. D.

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Sousa, Igor FlÃvio SimÃes de. "Distributed processing in receivers based on tensor for cooperative communications systems". Universidade Federal do CearÃ, 2014. http://www.teses.ufc.br/tde_busca/arquivo.php?codArquivo=12792.

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CoordenaÃÃo de AperfeÃoamento de Pessoal de NÃvel Superior
Ericsson Brasil
In this dissertation, we present a distributed data estimation and detection approach for the uplink of a network that uses CDMA at transmitters (users). The analyzed network can be represented by an undirected and connected graph, where the nodes use a distributed estimation algorithm based on consensus averaging to perform joint channel and symbol estimation using a receiver based on tensor signal processing. The centralized receiver, developed for a central base station, and the distributed receiver, developed for micro base stations, have their performances compared in a heterogeneous network. Then, two tensor-based receivers are proposed to be used in a relay-assisted network. In this case, the proposed receiver makes use of collaborative signal processing among relays to recover sources information before forwarding to the base station using a Decode-and-Forward protocol. The first receiver is based on the uncoded transmission of the tensor data reconstructed by the relays from the estimation of their factors matrix. The second one considers a tensor encoding of symbols estimated at the relays before transmission to the base station. The different proposed receivers are compared by means of computer simulations in terms of convergence and bit error rate.
Nesta dissertaÃÃo, apresentamos uma abordagem distribuÃda para a estimaÃÃo e detecÃÃo de dados para uplink em uma rede que emprega CDMA nos transmissores (usuÃrios). A rede analisada pode ser representada por um grafo sem direÃÃo e conectado, em que os nÃs fazem uso de um algoritmo de estimaÃÃo distribuÃda baseado em consenso mÃdio para realizar a estimaÃÃo conjunta de sÃmbolos transmitidos e do canal, utilizando um receptor baseado em processamento tensorial. O receptor centralizado, operando em uma EstaÃÃo RÃdio Base central, e o receptor distribuÃdo, operando em Micro EstaÃÃes RÃdio Base, tÃm seus desempenhos comparados em uma rede heterogÃnea. Em seguida, considerando-se uma rede assistida por repetidores, dois receptores tensoriais sÃo propostos. Neste caso, fazemos uso de um processamento de sinais colaborativo entre os repetidores para a recuperaÃÃo da informaÃÃo transmitida pela fonte, antes de ser encaminhada para estaÃÃo rÃdio base fazendo uso do protocolo Decode-and-Forward. O primeiro receptor Ã baseado na transmissÃo nÃo codificada do tensor de dados reconstruÃdo pelos repetidores a partir da estimaÃÃo de suas matrizes fatores. O segundo considera uma codificaÃÃo tensorial dos sÃmbolos previamente estimados nos repetidores antes da transmissÃo para estaÃÃo rÃdio base. Os diferentes receptores propostos sÃo comparados atravÃs de simulaÃÃes computacionais em termos de convergÃncia e taxa de erro de bit.

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Avila, Gastón Alejandro [Verfasser] y Helmut [Akademischer Betreuer] Friedrich. "Asymptotic staticity and tensor decompositions with fast decay conditions / Gastón Avila Alejandro. Betreuer: Helmut Friedrich". Potsdam : Universitätsbibliothek der Universität Potsdam, 2011. http://d-nb.info/1016138105/34.

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41

Parrish, Robert M. "Rank reduction methods in electronic structure theory". Diss., Georgia Institute of Technology, 2015. http://hdl.handle.net/1853/53850.

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Quantum chemistry is plagued by the presence of high-rank quantities, stemming from the N-body nature of the electronic Schrödinger equation. These high-rank quantities present a significant mathematical and computational barrier to the computation of chemical observables, and also drastically complicate the pedagogical understanding of important interactions between particles in a molecular system. The application of physically-motivated rank reduction approaches can help address these to problems. This thesis details recent efforts to apply rank reduction techniques in both of these arenas. With regards to computational tractability, the representation of the 1/r Coulomb repulsion between electrons is a critical stage in the solution of the electronic Schrödinger equation. Typically, this interaction is encapsulated via the order-4 electron repulsion integral (ERI) tensor, which is a major bottleneck in terms of generation, manipulation, and storage. Many rank reduction techniques for the ERI tensor have been proposed to ameliorate this bottleneck, most notably including the order-3 density fitting (DF) and pseudospectral (PS) representations. Here we detail a new and uniquely powerful factorization - tensor hypercontraction (THC). THC decomposes the ERI tensor as a product of five order-2 matrices (the first wholly order-2 compression proposed for the ERI) and offers great flexibility for low-scaling algorithms for the manipulations of the ERI tensor underlying electronic structure theory. THC is shown to be physically-motivated, markedly accurate, and uniquely efficient for some of the most difficult operations encountered in modern quantum chemistry. On the front of chemical understanding of electronic structure theory, we present our recent work in developing robust two-body partitions for ab initio computations of intermolecular interactions. Noncovalent interactions are the critical and delicate forces which govern such important processes as drug-protein docking, enzyme function, crystal packing, and zeolite adsorption. These forces arise as weak residual interactions leftover after the binding of electrons and nuclei into molecule, and, as such, are extremely difficult to accurately quantify or systematically understand. Symmetry-adapted perturbation theory (SAPT) provides an excellent approach to rigorously compute the interaction energy in terms of the physically-motivated components of electrostatics, exchange, induction, and dispersion. For small intermolecular dimers, this breakdown provides great insight into the nature of noncovalent interactions. However, SAPT abstracts away considerable details about the N-body interactions between particles on the two monomers which give rise to the interaction energy components. In the work presented herein, we step back slightly and extract an effective 2-body interaction for each of the N-body SAPT terms, rather than immediately tracing all the way down to the order-0 interaction energy. This effective order-2 representation of the order-N SAPT interaction allows for the robust assignment of interaction energy contributions to pairs of atoms or functional groups (the A-SAPT or F-SAPT partitions), allowing one to discuss the interaction in terms of atom- or functional-group-pairwise interactions. These A-SAPT and F-SAPT partitions can provide deep insight into the origins of complicated noncovalent interactions, e.g., by clearly shedding light on the long-contested question of the nature of the substituent effect in substituted sandwich benzene dimers.

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42

Nguyen, Viet-Dung. "Contribution aux décompositions rapides des matrices et tenseurs". Thesis, Orléans, 2016. http://www.theses.fr/2016ORLE2085/document.

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De nos jours, les grandes masses de données se retrouvent dans de nombreux domaines relatifs aux applications multimédia, sociologiques, biomédicales, radio astronomiques, etc. On parle alors du phénomène ‘Big Data’ qui nécessite le développement d’outils appropriés pour la manipulation et l’analyse appropriée de telles masses de données. Ce travail de thèse est dédié au développement de méthodes efficaces pour la décomposition rapide et adaptative de tenseurs ou matrices de grandes tailles et ce pour l’analyse de données multidimensionnelles. Nous proposons en premier une méthode d’estimation de sous espaces qui s’appuie sur la technique dite ‘divide and conquer’ permettant une estimation distribuée ou parallèle des sous-espaces désirés. Après avoir démontré l’efficacité numérique de cette solution, nous introduisons différentes variantes de celle-ci pour la poursuite adaptative ou bloc des sous espaces principaux ou mineurs ainsi que des vecteurs propres de la matrice de covariance des données. Une application à la suppression d’interférences radiofréquences en radioastronomie a été traitée. La seconde partie du travail a été consacrée aux décompositions rapides de type PARAFAC ou Tucker de tenseurs multidimensionnels. Nous commençons par généraliser l’approche ‘divide and conquer’ précédente au contexte tensoriel et ce en vue de la décomposition PARAFAC parallélisable des tenseurs. Ensuite nous adaptons une technique d’optimisation de type ‘all-at-once’ pour la décomposition robuste (à la méconnaissance des ordres) de tenseurs parcimonieux et non négatifs. Finalement, nous considérons le cas de flux de données continu et proposons deux algorithmes adaptatifs pour la décomposition rapide (à complexité linéaire) de tenseurs en dimension 3. Malgré leurs faibles complexités, ces algorithmes ont des performances similaires (voire parfois supérieures) à celles des méthodes existantes de la littérature. Au final, ce travail aboutit à un ensemble d’outils algorithmiques et algébriques efficaces pour la manipulation et l’analyse de données multidimensionnelles de grandes tailles
Large volumes of data are being generated at any given time, especially from transactional databases, multimedia content, social media, and applications of sensor networks. When the size of datasets is beyond the ability of typical database software tools to capture, store, manage, and analyze, we face the phenomenon of big data for which new and smarter data analytic tools are required. Big data provides opportunities for new form of data analytics, resulting in substantial productivity. In this thesis, we will explore fast matrix and tensor decompositions as computational tools to process and analyze multidimensional massive-data. We first aim to study fast subspace estimation, a specific technique used in matrix decomposition. Traditional subspace estimation yields high performance but suffers from processing large-scale data. We thus propose distributed/parallel subspace estimation following a divide-and-conquer approach in both batch and adaptive settings. Based on this technique, we further consider its important variants such as principal component analysis, minor and principal subspace tracking and principal eigenvector tracking. We demonstrate the potential of our proposed algorithms by solving the challenging radio frequency interference (RFI) mitigation problem in radio astronomy. In the second part, we concentrate on fast tensor decomposition, a natural extension of the matrix one. We generalize the results for the matrix case to make PARAFAC tensor decomposition parallelizable in batch setting. Then we adapt all-at-once optimization approach to consider sparse non-negative PARAFAC and Tucker decomposition with unknown tensor rank. Finally, we propose two PARAFAC decomposition algorithms for a classof third-order tensors that have one dimension growing linearly with time. The proposed algorithms have linear complexity, good convergence rate and good estimation accuracy. The results in a standard setting show that the performance of our proposed algorithms is comparable or even superior to the state-of-the-art algorithms. We also introduce an adaptive nonnegative PARAFAC problem and refine the solution of adaptive PARAFAC to tackle it. The main contributions of this thesis, as new tools to allow fast handling large-scale multidimensional data, thus bring a step forward real-time applications

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43

Han, Xu. "Robust low-rank tensor approximations using group sparsity". Thesis, Rennes 1, 2019. http://www.theses.fr/2019REN1S001/document.

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Le développement de méthodes de décomposition de tableaux multi-dimensionnels suscite toujours autant d'attention, notamment d'un point de vue applicatif. La plupart des algorithmes, de décompositions tensorielles, existants requièrent une estimation du rang du tenseur et sont sensibles à une surestimation de ce dernier. Toutefois, une telle estimation peut être difficile par exemple pour des rapports signal à bruit faibles. D'un autre côté, estimer simultanément le rang et les matrices de facteurs du tenseur ou du tenseur cœur n'est pas tâche facile tant les problèmes de minimisation de rang sont généralement NP-difficiles. Plusieurs travaux existants proposent d'utiliser la norme nucléaire afin de servir d'enveloppe convexe de la fonction de rang. Cependant, la minimisation de la norme nucléaire engendre généralement un coût de calcul prohibitif pour l'analyse de données de grande taille. Dans cette thèse, nous nous sommes donc intéressés à l'approximation d'un tenseur bruité par un tenseur de rang faible. Plus précisément, nous avons étudié trois modèles de décomposition tensorielle, le modèle CPD (Canonical Polyadic Decomposition), le modèle BTD (Block Term Decomposition) et le modèle MTD (Multilinear Tensor Decomposition). Pour chacun de ces modèles, nous avons proposé une nouvelle méthode d'estimation de rang utilisant une métrique moins coûteuse exploitant la parcimonie de groupe. Ces méthodes de décomposition comportent toutes deux étapes : une étape d'estimation de rang, et une étape d'estimation des matrices de facteurs exploitant le rang estimé. Des simulations sur données simulées et sur données réelles montrent que nos méthodes présentent toutes une plus grande robustesse à la présence de bruit que les approches classiques
Last decades, tensor decompositions have gained in popularity in several application domains. Most of the existing tensor decomposition methods require an estimating of the tensor rank in a preprocessing step to guarantee an outstanding decomposition results. Unfortunately, learning the exact rank of the tensor can be difficult in some particular cases, such as for low signal to noise ratio values. The objective of this thesis is to compute the best low-rank tensor approximation by a joint estimation of the rank and the loading matrices from the noisy tensor. Based on the low-rank property and an over estimation of the loading matrices or the core tensor, this joint estimation problem is solved by promoting group sparsity of over-estimated loading matrices and/or the core tensor. More particularly, three new methods are proposed to achieve efficient low rank estimation for three different tensors decomposition models, namely Canonical Polyadic Decomposition (CPD), Block Term Decomposition (BTD) and Multilinear Tensor Decomposition (MTD). All the proposed methods consist of two steps: the first step is designed to estimate the rank, and the second step uses the estimated rank to compute accurately the loading matrices. Numerical simulations with noisy tensor and results on real data the show effectiveness of the proposed methods compared to the state-of-the-art methods

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Krishnaswamy, Sriram. "On Computationally Efficient Frameworks For Data Association In Multi-Target Tracking". The Ohio State University, 2019. http://rave.ohiolink.edu/etdc/view?acc_num=osu1574672274983947.

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Breiding, Paul [Verfasser], Peter [Akademischer Betreuer] Bürgisser, Peter [Gutachter] Bürgisser y Felipe [Gutachter] Cucker. "Numerical and statistical aspects of tensor decompositions / Paul Breiding ; Gutachter: Peter Bürgisser, Felipe Cucker ; Betreuer: Peter Bürgisser". Berlin : Technische Universität Berlin, 2017. http://d-nb.info/1156018579/34.

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Giraldi, Loïc. "Contributions aux méthodes de calcul basées sur l'approximation de tenseurs et applications en mécanique numérique". Phd thesis, Ecole centrale de nantes - ECN, 2012. http://tel.archives-ouvertes.fr/tel-00861986.

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Cette thèse apporte différentes contributions à la résolution de problèmes de grande dimension dans le domaine du calcul scientifique, en particulier pour la quantification d'incertitudes. On considère ici des problèmes variationnels formulés dans des espaces produit tensoriel. On propose tout d'abord une stratégie de préconditionnement efficace pour la résolution de systèmes linéaires par des méthodes itératives utilisant des approximations de tenseurs de faible rang. Le préconditionneur est recherché comme une approximation de faible rang de l'inverse. Un algorithme glouton permet le calcul de cette approximation en imposant éventuellement des propriétés de symétrie ou un caractère creux. Ce préconditionneur est validé sur des problèmes linéaires symétriques ou non symétriques. Des contributions sont également apportées dans le cadre des méthodes d'approximation directes de tenseurs qui consistent à rechercher la meilleure approximation de la solution d'une équation dans un ensemble de tenseurs de faibles rangs. Ces méthodes, parfois appelées "Proper Generalized Decomposition" (PGD), définissent l'optimalité au sens de normes adaptées permettant le calcul a priori de cette approximation. On propose en particulier une extension des algorithmes gloutons classiquement utilisés pour la construction d'approximations dans les ensembles de tenseurs de Tucker ou hiérarchiques de Tucker. Ceci passe par la construction de corrections successives de rang un et de stratégies de mise à jour dans ces ensembles de tenseurs. L'algorithme proposé peut être interprété comme une méthode de construction d'une suite croissante d'espaces réduits dans lesquels on recherche une projection, éventuellement approchée, de la solution. L'application à des problèmes symétriques et non symétriques montre l'efficacité de cet algorithme. Le préconditionneur proposé est appliqué également dans ce contexte et permet de définir une meilleure norme pour l'approximation de la solution. On propose finalement une application de ces méthodes dans le cadre de l'homogénéisation numérique de matériaux hétérogènes dont la géométrie est extraite d'images. On présente tout d'abord des traitements particuliers de la géométrie ainsi que des conditions aux limites pour mettre le problème sous une forme adaptée à l'utilisation des méthodes d'approximation de tenseurs. Une démarche d'approximation adaptative basée sur un estimateur d'erreur a posteriori est utilisée afin de garantir une précision donnée sur les quantités d'intérêt que sont les propriétés effectives. La méthodologie est en premier lieu développée pour l'estimation de propriétés thermiques du matériau, puis est étendue à l'élasticité linéaire.

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Arnroth, Lukas. "Speeding up PARAFAC : Approximation of tensor rank using the Tucker core". Thesis, Uppsala universitet, Statistiska institutionen, 2018. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-353287.

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In this paper, the approach of utilizing the core tensor from the Tucker decomposition, in place of theuncompressed tensor, for nding a valid tensor rank for the PARAFAC decomposition is considered.Validity of the proposed method is investigated in terms of error and time consumption. As thesolutions of the PARAFAC decomposition are unique, stability of the solutions through split-halfanalysis is investigated. Simulated and real data are considered. Although, no general validity ofthe method could be observed, the results for some datasets look promising with 10% compressionin all modes. It is also shown that increased compression does not necessarily imply less timeconsumption.

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Baier, Stephan [Verfasser] y Volker [Akademischer Betreuer] Tresp. "Learning representations for supervised information fusion using tensor decompositions and deep learning methods / Stephan Baier ; Betreuer: Volker Tresp". München : Universitätsbibliothek der Ludwig-Maximilians-Universität, 2019. http://d-nb.info/1185979220/34.

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Yang, Yinchong [Verfasser] y Volker [Akademischer Betreuer] Tresp. "Enhancing representation learning with tensor decompositions for knowledge graphs and high dimensional sequence modeling / Yinchong Yang ; Betreuer: Volker Tresp". München : Universitätsbibliothek der Ludwig-Maximilians-Universität, 2018. http://d-nb.info/1156851939/34.

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Kühn, Stefan. "Hierarchische Tensordarstellung". Doctoral thesis, Universitätsbibliothek Leipzig, 2012. http://nbn-resolving.de/urn:nbn:de:bsz:15-qucosa-98906.

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In der vorliegenden Arbeit wird ein neues Tensorformat vorgestellt und eingehend analysiert. Das hierarchische Format verwendet einen binären Baum, um den Tensorraum der Ordnung d mit einer geschachtelten Unterraumstruktur zu versehen. Der Speicheraufwand für diese Darstellung ist von der Größenordnung O(dnr + dr^3), wobei n den Speicheraufwand in den Ansatzräumen kennzeichnet und r ein Rangparameter ist, der durch die Dimensionen der geschachtelten Unterräume bestimmt wird. Das hierarchische Format umfasst verschiedene Standardformate zur Tensordarstellung wie das kanonische oder r-Term-Format und die Unterraum-/Tucker-Darstellung. Die in dieser Arbeit entwickelte zugehörige Arithmetik inklusive mehrerer Approximationsmethoden basiert auf stabilen Methoden der Linearen Algebra, insbesondere die Singulärwertzerlegung und die QR-Zerlegung sind von zentraler Bedeutung. Die rechnerische Komplexität ist hierbei O(dnr^2+dr^4). Die lineare Abhängigkeit von der Ordnung d des Tensorraumes ist hervorzuheben. Für die verschiedenen Approximationsmethoden, deren Eﬃzienz und Eﬀektivität für die Anwendbarkeit des neuen Formates entscheidend sind, werden qualitative und quantitative Fehlerabschätzungen gezeigt. Umfassende numerische Experimente mit einem Fokus auf den Approximationsmethoden bestätigen zum einen die theoretischen Resultate und belegen die Stärken der neuen Tensordarstellung, zeigen aber zum anderen auch weitere, eher überraschende positive Eigenschaften der mit FastHOSVD bezeichneten schnellsten Kürzungsmethode
In this dissertation we present and a new format for the representation of tensors and analyse its properties. The hierarchical format uses a binary tree in order to define a hierarchical structure of nested subspaces in the tensor space of order d. The strorage requirements are O(dnr+dr^3) where n is determined by the storage requirements in the ansatz spaces and r is a rank parameter determined by the dimensions of the nested subspaces. The hierarchichal representation contains the standard representation like canonical or r-term representation and subspace or Tucker representation. The arithmetical operations that have been developed in this work, including several approximation methods, are based on stable Linear Alebra methods, especially the singular value decomposition (SVD) and the QR decomposition are of importance. The computational complexity is O(dnr^2+dr^4). The linear dependence from the order d of the tensor space is important. The approximation methods are one of the key ingredients for the applicability of the new format and we present qualitative and quantitative error estimates. Numerical experiments approve the theoretical results and show some additional, but unexpected positive aspects of the fastest method called FastHOSVD

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