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Статті в журналах з теми "Decomposition de tenseur"
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Zheng, Yu-Bang, Ting-Zhu Huang, Xi-Le Zhao, Qibin Zhao, and Tai-Xiang Jiang. "Fully-Connected Tensor Network Decomposition and Its Application to Higher-Order Tensor Completion." Proceedings of the AAAI Conference on Artificial Intelligence 35, no. 12 (May 18, 2021): 11071–78. http://dx.doi.org/10.1609/aaai.v35i12.17321.
Повний текст джерелаАнотація:
The popular tensor train (TT) and tensor ring (TR) decompositions have achieved promising results in science and engineering. However, TT and TR decompositions only establish an operation between adjacent two factors and are highly sensitive to the permutation of tensor modes, leading to an inadequate and inflexible representation. In this paper, we propose a generalized tensor decomposition, which decomposes an Nth-order tensor into a set of Nth-order factors and establishes an operation between any two factors. Since it can be graphically interpreted as a fully-connected network, we named it fully-connected tensor network (FCTN) decomposition. The superiorities of the FCTN decomposition lie in the outstanding capability for characterizing adequately the intrinsic correlations between any two modes of tensors and the essential invariance for transposition. Furthermore, we employ the FCTN decomposition to one representative task, i.e., tensor completion, and develop an efficient solving algorithm based on proximal alternating minimization. Theoretically, we prove the convergence of the developed algorithm, i.e., the sequence obtained by it globally converges to a critical point. Experimental results substantiate that the proposed method compares favorably to the state-of-the-art methods based on other tensor decompositions.
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Luo, Dijun, Chris Ding, and Heng Huang. "Multi-Level Cluster Indicator Decompositions of Matrices and Tensors." Proceedings of the AAAI Conference on Artificial Intelligence 25, no. 1 (August 4, 2011): 423–28. http://dx.doi.org/10.1609/aaai.v25i1.7933.
Повний текст джерелаАнотація:
A main challenging problem for many machine learning and data mining applications is that the amount of data and features are very large, so that low-rank approximations of original data are often required for efficient computation. We propose new multi-level clustering based low-rank matrix approximations which are comparable and even more compact than Singular Value Decomposition (SVD). We utilize the cluster indicators of data clustering results to form the subspaces, hence our decomposition results are more interpretable. We further generalize our clustering based matrix decompositions to tensor decompositions that are useful in high-order data analysis. We also provide an upper bound for the approximation error of our tensor decomposition algorithm. In all experimental results, our methods significantly outperform traditional decomposition methods such as SVD and high-order SVD.
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Breiding, Paul, and Nick Vannieuwenhoven. "On the average condition number of tensor rank decompositions." IMA Journal of Numerical Analysis 40, no. 3 (June 20, 2019): 1908–36. http://dx.doi.org/10.1093/imanum/drz026.
Повний текст джерелаАнотація:
Abstract We compute the expected value of powers of the geometric condition number of random tensor rank decompositions. It is shown in particular that the expected value of the condition number of $n_1\times n_2 \times 2$ tensors with a random rank-$r$ decomposition, given by factor matrices with independent and identically distributed standard normal entries, is infinite. This entails that it is expected and probable that such a rank-$r$ decomposition is sensitive to perturbations of the tensor. Moreover, it provides concrete further evidence that tensor decomposition can be a challenging problem, also from the numerical point of view. On the other hand, we provide strong theoretical and empirical evidence that tensors of size $n_1~\times ~n_2~\times ~n_3$ with all $n_1,n_2,n_3 \geqslant 3$ have a finite average condition number. This suggests that there exists a gap in the expected sensitivity of tensors between those of format $n_1\times n_2 \times 2$ and other order-3 tensors. To establish these results we show that a natural weighted distance from a tensor rank decomposition to the locus of ill-posed decompositions with an infinite geometric condition number is bounded from below by the inverse of this condition number. That is, we prove one inequality towards a so-called condition number theorem for the tensor rank decomposition.
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Alhamadi, Khaled M., Fahmi Yaseen Qasem, and Meqdad Ahmed Ali. "Different types of decomposition for certain tensors in \(K^h-BR-F_n\) and \(K^h-BR\)-affinely connected space." University of Aden Journal of Natural and Applied Sciences 20, no. 2 (August 31, 2016): 355–63. http://dx.doi.org/10.47372/uajnas.2016.n2.a10.
Повний текст джерелаАнотація:
In this paper we defined \(K^h\)-birecurrent space which is characterized by the condition\(K_jkh|m|l^i=a_lm K_jkh^i\) , \(K_jkh^i≠0\), also we introduced some decompositions of Cartan's fourth and third curvature tensor and Berwald curvature tensor and its torsion tensor. The aim of this paper is devoted to the discussion of decomposition for different tensors in \(K^h\)-birecurrent space and \(K^h\)-birecurrent affinely connected space and the decomposition of curvature tensor Cartan's fourth and third in \(K^h\)-birecurrent space, also the decomposition of curvature tensor of Berwald in \(K^h\)-birecurrent affinely connected space, various results, formulas, theorems and different identities have been obtained.
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Hameduddin, Ismail, Charles Meneveau, Tamer A. Zaki, and Dennice F. Gayme. "Geometric decomposition of the conformation tensor in viscoelastic turbulence." Journal of Fluid Mechanics 842 (March 12, 2018): 395–427. http://dx.doi.org/10.1017/jfm.2018.118.
Повний текст джерелаАнотація:
This work introduces a mathematical approach to analysing the polymer dynamics in turbulent viscoelastic flows that uses a new geometric decomposition of the conformation tensor, along with associated scalar measures of the polymer fluctuations. The approach circumvents an inherent difficulty in traditional Reynolds decompositions of the conformation tensor: the fluctuating tensor fields are not positive definite and so do not retain the physical meaning of the tensor. The geometric decomposition of the conformation tensor yields both mean and fluctuating tensor fields that are positive definite. The fluctuating tensor in the present decomposition has a clear physical interpretation as a polymer deformation relative to the mean configuration. Scalar measures of this fluctuating conformation tensor are developed based on the non-Euclidean geometry of the set of positive definite tensors. Drag-reduced viscoelastic turbulent channel flow is then used an example case study. The conformation tensor field, obtained using direct numerical simulations, is analysed using the proposed framework.
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Obster, Dennis, and Naoki Sasakura. "Counting Tensor Rank Decompositions." Universe 7, no. 8 (August 15, 2021): 302. http://dx.doi.org/10.3390/universe7080302.
Повний текст джерелаАнотація:
Tensor rank decomposition is a useful tool for geometric interpretation of the tensors in the canonical tensor model (CTM) of quantum gravity. In order to understand the stability of this interpretation, it is important to be able to estimate how many tensor rank decompositions can approximate a given tensor. More precisely, finding an approximate symmetric tensor rank decomposition of a symmetric tensor Q with an error allowance Δ is to find vectors ϕi satisfying ∥Q−∑i=1Rϕi⊗ϕi⋯⊗ϕi∥2≤Δ. The volume of all such possible ϕi is an interesting quantity which measures the amount of possible decompositions for a tensor Q within an allowance. While it would be difficult to evaluate this quantity for each Q, we find an explicit formula for a similar quantity by integrating over all Q of unit norm. The expression as a function of Δ is given by the product of a hypergeometric function and a power function. By combining new numerical analysis and previous results, we conjecture a formula for the critical rank, yielding an estimate for the spacetime degrees of freedom of the CTM. We also extend the formula to generic decompositions of non-symmetric tensors in order to make our results more broadly applicable. Interestingly, the derivation depends on the existence (convergence) of the partition function of a matrix model which previously appeared in the context of the CTM.
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Adkins, William A. "Primary decomposition of torsionR[X]-modules." International Journal of Mathematics and Mathematical Sciences 17, no. 1 (1994): 41–46. http://dx.doi.org/10.1155/s0161171294000074.
Повний текст джерелаАнотація:
This paper is concerned with studying hereditary properties of primary decompositions of torsionR[X]-modulesMwhich are torsion free asR-modules. Specifically, if anR[X]-submodule ofMis pure as anR-submodule, then the primary decomposition ofMdetermines a primary decomposition of the submodule. This is a generalization of the classical fact from linear algebra that a diagonalizable linear transformation on a vector space restricts to a diagonalizable linear transformation of any invariant subspace. Additionally, primary decompositions are considered under direct sums and tensor product.
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Cai, Yunfeng, and Ping Li. "A Blind Block Term Decomposition of High Order Tensors." Proceedings of the AAAI Conference on Artificial Intelligence 35, no. 8 (May 18, 2021): 6868–76. http://dx.doi.org/10.1609/aaai.v35i8.16847.
Повний текст джерелаАнотація:
Tensor decompositions have found many applications in signal processing, data mining, machine learning, etc. In particular, the block term decomposition (BTD), which is a generalization of CP decomposition and Tucker decomposition/HOSVD, has been successfully used for the compression and acceleration of neural networks. However, computing BTD is NP-hard, and optimization based methods usually suffer from slow convergence or even fail to converge, which limits the applications of BTD. This paper considers a “blind” block term decomposition (BBTD) of high order tensors, in which the block diagonal structure of the core tensor is unknown. Our contributions include: 1) We establish the necessary and sufficient conditions for the existence of BTD, characterize the condition when a BTD solves the BBTD problem, and show that the BBTD is unique under a “low rank” assumption. 2) We propose an algebraic method to compute the BBTD. This method transforms the problem of determining the block diagonal structure of the core tensor into a clustering problem of complex numbers, in polynomial time. And once the clustering problem is solved, the BBTD can be obtained via computing several matrix decompositions. Numerical results show that our method is able to compute the BBTD, even in the presence of noise to some extent, whereas optimization based methods (e.g., MINF and NLS in TENSORLAB) may fail to converge.
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Hauenstein, Jonathan D., Luke Oeding, Giorgio Ottaviani, and Andrew J. Sommese. "Homotopy techniques for tensor decomposition and perfect identifiability." Journal für die reine und angewandte Mathematik (Crelles Journal) 2019, no. 753 (August 1, 2019): 1–22. http://dx.doi.org/10.1515/crelle-2016-0067.
Повний текст джерелаАнотація:
AbstractLetTbe a general complex tensor of format{(n_{1},\dots,n_{d})}. When the fraction{\prod_{i}n_{i}/[1+\sum_{i}(n_{i}-1)]}is an integer, and a natural inequality (called balancedness) is satisfied, it is expected thatThas finitely many minimal decomposition as a sum of decomposable tensors. We show how homotopy techniques allow us to find all the decompositions ofT, starting from a given one. Computationally, this gives a guess regarding the total number of such decompositions. This guess matches exactly with all cases previously known, and predicts several unknown cases. Some surprising experiments yielded two new cases of generic identifiability: formats{(3,4,5)}and{(2,2,2,3)}which have a unique decomposition as the sum of six and four decomposable tensors, respectively. We conjecture that these two cases together with the classically known matrix pencils are the only cases where generic identifiability holds, i.e., the onlyidentifiablecases. Building on the computational experiments, we use algebraic geometry to prove these two new cases are indeed generically identifiable.
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Zhu, Ben-Chao, and Xiang-Song Chen. "Tensor gauge condition and tensor field decomposition." Modern Physics Letters A 30, no. 35 (October 28, 2015): 1550192. http://dx.doi.org/10.1142/s0217732315501928.
Повний текст джерелаАнотація:
We discuss various proposals of separating a tensor field into pure-gauge and gauge-invariant components. Such tensor field decomposition is intimately related to the effort of identifying the real gravitational degrees of freedom out of the metric tensor in Einstein’s general relativity. We show that as for a vector field, the tensor field decomposition has exact correspondence to and can be derived from the gauge-fixing approach. The complication for the tensor field, however, is that there are infinitely many complete gauge conditions in contrast to the uniqueness of Coulomb gauge for a vector field. The cause of such complication, as we reveal, is the emergence of a peculiar gauge-invariant pure-gauge construction for any gauge field of spin [Formula: see text]. We make an extensive exploration of the complete tensor gauge conditions and their corresponding tensor field decompositions, regarding mathematical structures, equations of motion for the fields and nonlinear properties. Apparently, no single choice is superior in all aspects, due to an awkward fact that no gauge-fixing can reduce a tensor field to be purely dynamical (i.e. transverse and traceless), as can the Coulomb gauge in a vector case.
Дисертації з теми "Decomposition de tenseur"
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Harmouch, Jouhayna. "Décomposition de petit rang, problèmes de complétion et applications : décomposition de matrices de Hankel et des tenseurs de rang faible." Thesis, Université Côte d'Azur (ComUE), 2018. http://www.theses.fr/2018AZUR4236/document.
Повний текст джерелаАнотація:
On étudie la décomposition de matrice de Hankel comme une somme des matrices de Hankel de rang faible en corrélation avec la décomposition de son symbole σ comme une somme des séries exponentielles polynomiales. On présente un nouvel algorithme qui calcule la décomposition d’un opérateur de Hankel de petit rang et sa décomposition de son symbole en exploitant les propriétés de l’algèbre quotient de Gorenstein . La base de est calculée à partir la décomposition en valeurs singuliers d’une sous-matrice de matrice de Hankel . Les fréquences et les poids se déduisent des vecteurs propres généralisés des sous matrices de Hankel déplacés de . On présente une formule pour calculer les poids en fonction des vecteurs propres généralisés au lieu de résoudre un système de Vandermonde. Cette nouvelle méthode est une généralisation de Pencil méthode déjà utilisée pour résoudre un problème de décomposition de type de Prony. On analyse son comportement numérique en présence des moments contaminés et on décrit une technique de redimensionnement qui améliore la qualité numérique des fréquences d’une grande amplitude. On présente une nouvelle technique de Newton qui converge localement vers la matrice de Hankel de rang faible la plus proche au matrice initiale et on montre son effet à corriger les erreurs sur les moments. On étudie la décomposition d’un tenseur multi-symétrique T comme une somme des puissances de produit des formes linéaires en corrélation avec la décomposition de son dual comme une somme pondérée des évaluations. On utilise les propriétés de l’algèbre de Gorenstein associée pour calculer la décomposition de son dual qui est définie à partir d’une série formelle τ. On utilise la décomposition d’un opérateur de Hankel de rang faible associé au symbole τ comme une somme des opérateurs indécomposables de rang faible. La base d’ est choisie de façon que la multiplication par certains variables soit possible. On calcule les coordonnées des points et leurs poids correspondants à partir la structure propre des matrices de multiplication. Ce nouvel algorithme qu’on propose marche bien pour les matrices de Hankel de rang faible. On propose une approche théorique de la méthode dans un espace de dimension n. On donne un exemple numérique de la décomposition d’un tenseur multilinéaire de rang 3 en dimension 3 et un autre exemple de la décomposition d’un tenseur multi-symétrique de rang 3 en dimension 3. On étudie le problème de complétion de matrice de Hankel comme un problème de minimisation. On utilise la relaxation du problème basé sur la minimisation de la norme nucléaire de la matrice de Hankel. On adapte le SVT algorithme pour le cas d’une matrice de Hankel et on calcule l’opérateur linéaire qui décrit les contraintes du problème de minimisation de norme nucléaire. On montre l’utilité du problème de décomposition à dissocier un modèle statistique ou biologiqueWe study the decomposition of a multivariate Hankel matrix as a sum of Hankel matrices of small rank in correlation with the decomposition of its symbol σ as a sum of polynomialexponential series. We present a new algorithm to compute the low rank decomposition of the Hankel operator and the decomposition of its symbol exploiting the properties of the associated Artinian Gorenstein quotient algebra . A basis of is computed from the Singular Value Decomposition of a sub-matrix of the Hankel matrix . The frequencies and the weights are deduced from the generalized eigenvectors of pencils of shifted sub-matrices of Explicit formula for the weights in terms of the eigenvectors avoid us to solve a Vandermonde system. This new method is a multivariate generalization of the so-called Pencil method for solving Pronytype decomposition problems. We analyse its numerical behaviour in the presence of noisy input moments, and describe a rescaling technique which improves the numerical quality of the reconstruction for frequencies of high amplitudes. We also present a new Newton iteration, which converges locally to the closest multivariate Hankel matrix of low rank and show its impact for correcting errors on input moments. We study the decomposition of a multi-symmetric tensor T as a sum of powers of product of linear forms in correlation with the decomposition of its dual as a weighted sum of evaluations. We use the properties of the associated Artinian Gorenstein Algebra to compute the decomposition of its dual which is defined via a formal power series τ. We use the low rank decomposition of the Hankel operator associated to the symbol τ into a sum of indecomposable operators of low rank. A basis of is chosen such that the multiplication by some variables is possible. We compute the sub-coordinates of the evaluation points and their weights using the eigen-structure of multiplication matrices. The new algorithm that we propose works for small rank. We give a theoretical generalized approach of the method in n dimensional space. We show a numerical example of the decomposition of a multi-linear tensor of rank 3 in 3 dimensional space. We show a numerical example of the decomposition of a multi-symmetric tensor of rank 3 in 3 dimensional space. We study the completion problem of the low rank Hankel matrix as a minimization problem. We use the relaxation of it as a minimization problem of the nuclear norm of Hankel matrix. We adapt the SVT algorithm to the case of Hankel matrix and we compute the linear operator which describes the constraints of the problem and its adjoint. We try to show the utility of the decomposition algorithm in some applications such that the LDA model and the ODF model
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Blanc, Katy. "Description de contenu vidéo : mouvements et élasticité temporelle." Thesis, Université Côte d'Azur (ComUE), 2018. http://www.theses.fr/2018AZUR4212/document.
Повний текст джерелаАнотація:
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 classificationVideo 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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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.
Повний текст джерелаАнотація:
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 fluorescenceThis 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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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.
Повний текст джерелаАнотація:
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 industrielsThis 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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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.
Повний текст джерелаАнотація:
Le doctorant n'a pas délivré de résumé en françaisAbsence 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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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.
Повний текст джерелаАнотація:
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 fluorescenceThis 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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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.
Повний текст джерелаАнотація:
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 informatiquesLow 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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Marmin, Arthur. "Rational models optimized exactly for solving signal processing problems." Electronic Thesis or Diss., université Paris-Saclay, 2020. http://www.theses.fr/2020UPASG017.
Повний текст джерелаАнотація:
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 signalA 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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Brandoni, Domitilla. "Tensor decompositions for Face Recognition." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2018. http://amslaurea.unibo.it/16867/.
Повний текст джерелаАнотація:
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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Bender, Matias Rafael. "Algorithms for sparse polynomial systems : Gröbner bases and resultants." Electronic Thesis or Diss., Sorbonne université, 2019. http://www.theses.fr/2019SORUS029.
Повний текст джерелаАнотація:
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 binairesSolving 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
Книги з теми "Decomposition de tenseur"
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Cheng, Lei, Zhongtao Chen, and Yik-Chung Wu. Bayesian Tensor Decomposition for Signal Processing and Machine Learning. Cham: Springer International Publishing, 2023. http://dx.doi.org/10.1007/978-3-031-22438-6.
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Theocaris, Pericles S. On a general theory of anisotropy of matter: The spectral decomposition of the compliance tensor : application to crystallography. Athēnai: Grapheion Dēmosieumatōn tēs Akadēmias Athēnōn, 1999.
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Kondrat'ev, Gennadiy. Clifford Geometric Algebra. ru: INFRA-M Academic Publishing LLC., 2021. http://dx.doi.org/10.12737/1832489.
Повний текст джерелаАнотація:
The monograph is devoted to the fundamental aspects of geometric algebra and closely related issues. The category of Clifford algebras is considered as the conjugate category of vector spaces with a quadratic form. Possible constructions in this category and internal algebraic operations of an algebra with a geometric interpretation are studied. An application to the differential geometry of a Euclidean manifold based on a shape tensor is included.
We consider products, coproducts and tensor products in the category of associative algebras with application to the decomposition of Clifford algebras into simple components. Spinors are introduced. Methods of matrix representation of the Clifford algebra are studied.
It may be of interest to students, postgraduates and specialists in the field of mathematics, physics and cybernetics.
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Ninul, Anatolij Sergeevič. Tensor Trigonometry. Moscow, Russia: Fizmatlit Publisher, 2021.
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Ninul, Anatolij Sergeevič. Tenzornaja trigonometrija: Teorija i prilozenija / Theory and Applications /. Moscow, Russia: Mir Publisher, 2004.
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Structured Tensor Recovery and Decomposition. [New York, N.Y.?]: [publisher not identified], 2017.
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Maggiore, Michele. Helicity decomposition of metric perturbations. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198570899.003.0009.
Повний текст джерелаАнотація:
Decomposition of the perturbations over FRW into scalar, vector and tensor perturbations. Physical and unphysical degrees of freedom. Gauge-invariant metric perturbations, Bardeen variables. Gauge-invariant perturbations of the energy-momentum tensor
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Favier, Gérard. Matrix and Tensor Decompositions in Signal Processing. Wiley & Sons, Incorporated, John, 2021.
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Favier, Gérard. Matrix and Tensor Decompositions in Signal Processing. Wiley & Sons, Incorporated, John, 2021.
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Favier, Gérard. Matrix and Tensor Decompositions in Signal Processing. Wiley & Sons, Incorporated, John, 2021.
Знайти повний текст джерелаЧастини книг з теми "Decomposition de tenseur"
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Pajarola, Renato, Susanne K. Suter, Rafael Ballester-Ripoll, and Haiyan Yang. "Tensor Approximation for Multidimensional and Multivariate Data." In Mathematics and Visualization, 73–98. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-56215-1_4.
Повний текст джерелаАнотація:
AbstractTensor decomposition methods and multilinear algebra are powerful tools to cope with challenges around multidimensional and multivariate data in computer graphics, image processing and data visualization, in particular with respect to compact representation and processing of increasingly large-scale data sets. Initially proposed as an extension of the concept of matrix rank for 3 and more dimensions, tensor decomposition methods have found applications in a remarkably wide range of disciplines. We briefly review the main concepts of tensor decompositions and their application to multidimensional visual data. Furthermore, we will include a first outlook on porting these techniques to multivariate data such as vector and tensor fields.
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Landsberg, J. "Tensor decomposition." In Graduate Studies in Mathematics, 289–310. Providence, Rhode Island: American Mathematical Society, 2011. http://dx.doi.org/10.1090/gsm/128/12.
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Taguchi, Y.-h. "Tensor Decomposition." In Unsupervised and Semi-Supervised Learning, 47–78. Cham: Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-22456-1_3.
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Liu, Yipeng, Jiani Liu, Zhen Long, and Ce Zhu. "Tensor Decomposition." In Tensor Computation for Data Analysis, 19–57. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-74386-4_2.
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Taguchi, Y.-h. "Tensor Decomposition." In Unsupervised and Semi-Supervised Learning, 47–77. Cham: Springer International Publishing, 2024. http://dx.doi.org/10.1007/978-3-031-60982-4_3.
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Hao, N., L. Horesh, and M. E. Kilmer. "Nonnegative Tensor Decomposition." In Compressed Sensing & Sparse Filtering, 123–48. Berlin, Heidelberg: Springer Berlin Heidelberg, 2013. http://dx.doi.org/10.1007/978-3-642-38398-4_5.
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Taguchi, Y. H. "Tensor Decomposition in Genomics." In Machine Learning and IoT Applications for Health Informatics, 131–43. Boca Raton: CRC Press, 2024. http://dx.doi.org/10.1201/9781003424987-7.
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Zhu, Hu, Yushan Pan, Lizhen Deng, and Guoxia Xu. "Low-Rank Tensor Decomposition." In Infrared Small Target Detection, 41–82. Singapore: Springer Nature Singapore, 2024. http://dx.doi.org/10.1007/978-981-99-9799-2_4.
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Harada, Kenji, Hiroaki Matsueda, and Tsuyoshi Okubo. "Application of Tensor Network Formalism for Processing Tensor Data." In Advanced Mathematical Science for Mobility Society, 79–100. Singapore: Springer Nature Singapore, 2024. http://dx.doi.org/10.1007/978-981-99-9772-5_5.
Повний текст джерелаАнотація:
AbstractNext-generation mobility services require a huge amount of data with multiple attributes. This data is stored as a multi-dimensional array called a tensor. A tensor network is an effective tool for representing a large composite tensor. As an application of the tensor-network formalism to tensor data processing, we present three research results from statistical physics: tree tensor networks, tensor ring decomposition, and MERA.
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Devarajan, Karthik. "Matrix and Tensor Decompositions." In Problem Solving Handbook in Computational Biology and Bioinformatics, 291–318. Boston, MA: Springer US, 2010. http://dx.doi.org/10.1007/978-0-387-09760-2_14.
Повний текст джерелаТези доповідей конференцій з теми "Decomposition de tenseur"
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Joshi, Spriha, Philippe Dreesen, Pietro Bonizzi, Joël Karel, Ralf Peeters, and Martijn Boussé. "Novel Tensor-based Singular Spectrum Decomposition." In 2024 32nd European Signal Processing Conference (EUSIPCO), 1327–31. IEEE, 2024. http://dx.doi.org/10.23919/eusipco63174.2024.10715185.
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Matveev, Sergey Alexandrovich, and Aleksandr A. Kurilovich. "Utilization of Tensor Decompositions for Video-compression." In 33rd International Conference on Computer Graphics and Vision. Keldysh Institute of Applied Mathematics, 2023. http://dx.doi.org/10.20948/graphicon-2023-582-589.
Повний текст джерелаАнотація:
In this work, we provide a study of video compression with the use of tensor train and Tucker decomposi- tions. We measure the quality of compression with classical PSNR and SSIM metrics. Our approach allows us to control the quality of compressed video through the analytical evaluation of tensor decomposition ranks using the target value of PSNR. We achieve this aim because the PSNR is naturally related to the value of relative error in the Frobenius norm, which can be controlled for both tensor train and Tucker decompositions. In case of tensor train decomposition, we evaluate the idea of adding additional virtual dimensions and show that this trick allows us to improve the quality of compression without adding non- negligible additional errors. We discuss the advantages and visible artifacts introduced by the tensor-based algorithms to video compression and compare our results with industrial standards.
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Alexeev, Alexey Kirillovich, Alexander Evgenyevich Bondarev, and Yu S. Pyatakova. "On the Visualization of the Ensemble of Parametric Numerical Solutions Using Tensor Decomposition." In 33rd International Conference on Computer Graphics and Vision. Keldysh Institute of Applied Mathematics, 2023. http://dx.doi.org/10.20948/graphicon-2023-292-301.
Повний текст джерелаАнотація:
An approximation of the tensor obtained at a discretization of the multidimensional function on a uniform grid is addressed from the viewpoint of the storage and treating the result of the parametric computations in the CFD problems. The tensor decompositions are considered for this purpose. The new algorithm of the calculation of the canonical decomposition using the gradient descent for an approximately decomposable goal functional. This algorithm applies an ensemble of points on the hyperplane orthogonal to the computed core of the canonical decomposition (”umbrella”) that enables its flexible application for the approximation of the tensors with a priori unknown rank. This algorithm is naturally transferred on such tensor decomposition as the tensor train. The results of the numerical tests are presented for model six-dimensional functions and for the ensemble of the numerical solutions for two-dimensional Euler equations. These equations describe the flow of the compressible gas with two crossing shock waves. Mach number and the the flow deflection angles are considered as the flow parameters.
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Cao, Tianxiao, Lu Sun, Canh Hao Nguyen та Hiroshi Mamitsuka. "Learning Low-Rank Tensor Cores with Probabilistic ℓ0-Regularized Rank Selection for Model Compression". У Thirty-Third International Joint Conference on Artificial Intelligence {IJCAI-24}. California: International Joint Conferences on Artificial Intelligence Organization, 2024. http://dx.doi.org/10.24963/ijcai.2024/418.
Повний текст джерелаАнотація:
Compressing deep neural networks is of great importance for real-world applications on resource-constrained devices. Tensor decomposition is one promising answer that retains the functionality and most of the expressive power of the original deep models by replacing the weights with their decomposed cores. Decomposition with optimal ranks can achieve a good compression-accuracy trade-off, but it is expensive to optimize due to its discrete and combinatorial nature. A common practice is to set all ranks equal and tune one hyperparameter, but it may significantly harm the flexibility and generalization. In this paper, we propose a novel automatic rank selection method for deep model compression that allows learning model weights and decomposition ranks simultaneously. We propose to penalize the ℓ0 (quasi-)norm of the slices of decomposed tensor cores during model training. To avoid combinatorial optimization, we develop a probabilistic formulation and apply an approximate Bernoulli gate to each of the slices of tensor cores, which can be implemented in an end-to-end and scalable framework via gradient descent. It enables the automatic rank selection to be incorporated with arbitrary tensor decompositions and neural network layers such as linear layers, convolutional layers, and embedding layers. Comprehensive experiments on various tasks, including image classification, text sentiment classification, and neural machine translation, demonstrate the superior effectiveness of the proposed method over baselines.
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Ozdemir, Alp, Mark A. Iwen, and Selin Aviyente. "Multiscale tensor decomposition." In 2016 50th Asilomar Conference on Signals, Systems and Computers. IEEE, 2016. http://dx.doi.org/10.1109/acssc.2016.7869118.
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Kumar, Shrawan. "Tensor Product Decomposition." In Proceedings of the International Congress of Mathematicians 2010 (ICM 2010). Published by Hindustan Book Agency (HBA), India. WSPC Distribute for All Markets Except in India, 2011. http://dx.doi.org/10.1142/9789814324359_0094.
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Ghavam, Kamyar, and Reza Naghdabadi. "Corotational Analysis of Elastic-Plastic Hardening Materials Based on Different Kinematic Decompositions." In ASME 2006 Pressure Vessels and Piping/ICPVT-11 Conference. ASMEDC, 2006. http://dx.doi.org/10.1115/pvp2006-icpvt-11-93442.
Повний текст джерелаАнотація:
In this paper, two corotational modeling for elastic-plastic, mixed hardening materials at finite deformations are introduced. In these models, the additive decomposition of the strain rate tensor as well as the multiplicative decomposition of the deformation gradient tensor is used. For this purpose, corotational constitutive equations are derived for elastic-plastic hardening materials with the non-linear Armstrong-Frederick kinematic hardening and isotropic hardening models. As an application of the proposed constitutive modeling, the governing equations are solved numerically for the simple shear problem with different corotational rates and the stress components are plotted versus the shear displacement. The results for stress, using the additive and the multiplicative decompositions are compared with those obtained experimentally by Ishikawa [1]. This comparison shows a good agreement between the proposed theoretical models and the experimental data. As another example, the Prager kinematic hardening equation is used instead of the Armstrong-Frederick model. In this case the results for stress are compared with the theoretical results of Bruhns et al. [2].
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ZHANG, Jianfu, ZERUI TAO, LIQING ZHANG, and QIBIN ZHAO. "Tensor Decomposition Via Core Tensor Networks." In ICASSP 2021 - 2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). IEEE, 2021. http://dx.doi.org/10.1109/icassp39728.2021.9413637.
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Hinrich, Jesper L., and Morten Morup. "Probabilistic Tensor Train Decomposition." In 2019 27th European Signal Processing Conference (EUSIPCO). IEEE, 2019. http://dx.doi.org/10.23919/eusipco.2019.8903177.
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Du, Yishuai, Yimin Zheng, Kuang-chih Lee, and Shandian Zhe. "Probabilistic Streaming Tensor Decomposition." In 2018 IEEE International Conference on Data Mining (ICDM). IEEE, 2018. http://dx.doi.org/10.1109/icdm.2018.00025.
Повний текст джерелаЗвіти організацій з теми "Decomposition de tenseur"
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Tamara G. Kolda. Orthogonal tensor decompositions. Office of Scientific and Technical Information (OSTI), March 2000. http://dx.doi.org/10.2172/755101.
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Devine, Karen, and Grey Ballard. GentenMPI: Distributed Memory Sparse Tensor Decomposition. Office of Scientific and Technical Information (OSTI), August 2020. http://dx.doi.org/10.2172/1656940.
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Phipps, Eric, Nick Johnson, and Tamara Kolda. STREAMING GENERALIZED CANONICAL POLYADIC TENSOR DECOMPOSITIONS. Office of Scientific and Technical Information (OSTI), October 2021. http://dx.doi.org/10.2172/1832304.
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Anandkumar, Anima, Rong Ge, Daniel Hsu, Sham M. Kakade, and Matus Telgarsky. Tensor Decompositions for Learning Latent Variable Models. Fort Belvoir, VA: Defense Technical Information Center, December 2012. http://dx.doi.org/10.21236/ada604494.
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Lopez, Oscar, Richard Lehoucq, and Daniel Dunlavy. Zero-Truncated Poisson Tensor Decomposition for Sparse Count Data. Office of Scientific and Technical Information (OSTI), January 2022. http://dx.doi.org/10.2172/1841834.
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Dunlavy, Daniel M., Evrim Acar, and Tamara Gibson Kolda. An optimization approach for fitting canonical tensor decompositions. Office of Scientific and Technical Information (OSTI), February 2009. http://dx.doi.org/10.2172/978916.
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Ruiz, Trevor, and Charlotte R. Ellison. Spatiotemporally coherent tensor decompositions for the analysis of trajectory data. Engineer Research and Development Center (U.S.), July 2020. http://dx.doi.org/10.21079/11681/37355.
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Myers, Jeremy, and Daniel Dunlavy. Tensor Decompositions for Count Data that Leverage Stochastic and Deterministic Optimization. Office of Scientific and Technical Information (OSTI), August 2023. http://dx.doi.org/10.2172/2430475.
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Anandkumar, Animashree, Daniel Hsu, Majid Janzamin, and Sham Kakade. When are Overcomplete Representations Identifiable? Uniqueness of Tensor Decompositions Under Expansion Constraints. Fort Belvoir, VA: Defense Technical Information Center, June 2013. http://dx.doi.org/10.21236/ada604842.
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Myers, Jeremy, and Daniel Dunlavy. A Hybrid Method for Tensor Decompositions that Leverages Stochastic and Deterministic Optimization. Office of Scientific and Technical Information (OSTI), April 2022. http://dx.doi.org/10.2172/1865529.
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