Academic literature on the topic 'Rank-one tensors'

<p> 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</p><p>

SOKAL, B. Semi-blind receivers for multi-relaying mimo systems using rank-one tensor factorizations. 2017. 85 f. Dissertação (Mestrado em Engenharia de Teleinformática)-Centro de Tecnologia, Universidade Federal do Ceará, Fortaleza, 2017.<br>Submitted by Renato Vasconcelos (ppgeti@ufc.br) on 2017-09-15T20:38:29Z No. of bitstreams: 1 2017_dis_bsokal.pdf: 1689224 bytes, checksum: f0e2e9424de721f23bf0629ba55330c1 (MD5)<br>Rejected by Marlene Sousa (mmarlene@ufc.br), reason: Prezado Bruno: Existe uma orientação para que normalizemos as dissertações e teses da UFC, em suas paginas pré-textuais e lista de referencias, pelas regras da ABNT. Por esse motivo, sugerimos consultar o modelo de template, para ajudá-lo nesta tarefa, disponível em: http://www.biblioteca.ufc.br/educacao-de-usuarios/templates/ Vamos agora as correções sempre de acordo com o template: 1. 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Não se destacam subtítulos e nos artigos de revistas se destaca-se apenas o ´nome da revista. Att. Marlene Rocha 3366-9620 mmarlene@ufc.br on 2017-09-18T11:38:11Z (GMT)<br>Submitted by Renato Vasconcelos (ppgeti@ufc.br) on 2017-09-20T14:04:25Z No. of bitstreams: 1 2017_dis_bsokal.pdf: 1481998 bytes, checksum: bdf4f504f50622f6b0e2084361272481 (MD5)<br>Approved for entry into archive by Marlene Sousa (mmarlene@ufc.br) on 2017-09-21T17:44:15Z (GMT) No. of bitstreams: 1 2017_dis_bsokal.pdf: 1481998 bytes, checksum: bdf4f504f50622f6b0e2084361272481 (MD5)<br>Made available in DSpace on 2017-09-21T17:44:15Z (GMT). No. of bitstreams: 1 2017_dis_bsokal.pdf: 1481998 bytes, checksum: bdf4f504f50622f6b0e2084361272481 (MD5) Previous issue date: 2017-07-27<br>Cooperative communications have shown to be an alternative to combat the impairments of signal propagation in wireless communications, such as path loss and shadowing, creating a virtual array of antennas for the source. In this work, we start with a two-hop MIMO system using a a single relay. By adding a space-time filtering step at the receiver, we propose a rank-one tensor factorization model for the resulting signal. Exploiting this model, two semi-blind receivers for joint symbol and channel estimation are derived: i) an iterative receiver based on the trilinear alternating least squares (Tri-ALS) algorithm and ii) a closed-form receiver based on the truncated higher order SVD (T-HOSVD). For this system, we also propose a space-time coding tensor having a PARAFAC decomposition structure, which gives more flexibility to system design, while allowing an orthogonal coding. In the second part of this work, we present an extension of the rank-one factorization approach to a multi-relaying scenario and a closed-form semi-blind receiver based on coupled SVDs (C-SVD) is derived. The C-SVD receiver efficiently combines all the available cooperative links to enhance channel and symbol estimation performance, while enjoying a parallel implementation.<br>Comunicações cooperativas têm mostrado ser uma alternativa para combater os efeitos de propagação do sinal em comunicações sem-fio, como, por exemplo, a perda por percurso e sombreamento, criando um array virtual de antenas para a fonte transmissora. Neste trabalho, toma-se como ponto de partida um modelo de sistema MIMO de dois saltos com um único relay. Adicionando um estágio de filtragem no receptor, é proposta uma fatoração de rank-um para o sinal resultante. A partir deste modelo, dois receptores semi-cegos para estimação conjunta de símbolo e canal são propostos: i) um receptor iterativo baseado no algoritmo trilinear de mínimos quadrados alternados (Tri-ALS) e ii) um receptor de solução fechada baseado na SVD de ordem superior truncada (T-HOSVD). Para este sistema, é também proposto um tensor de codificação espacial-temporal com uma estrutura PARAFAC, o que permite maior flexibilidade de design do sistema, além de uma codificação ortogonal. Na segunda parte deste trabalho, é apresentada uma extensão da fatoração de rank-um para o cenário multi-relay e um receptor semi-cego de solução fechada baseado em SVD's acopladas (C-SVD) é desenvolvido. O receptor C-SVD combina de modo eficiente todos os links cooperativos disponíveis, melhorando o desempenho da estimação de símbolos e de canal, além de oferecer uma implementação paralelizável.

On s’intéresse dans cette thèse à la PGD (Proper Generalized Decomposition), l’une des méthodes de réduction de modèles qui consiste à chercher, a priori, la solution d’une équation aux dérivées partielles sous forme de variables séparées. Ce travail est formé de cinq chapitres dans lesquels on vise à étendre la PGD aux espaces fractionnaires et aux espaces des fonctions à variation bornée, et à donner des interprétations théoriques de cette méthode pour une classe de problèmes elliptiques et paraboliques. Dans le premier chapitre, on fait un bref aperçu sur la littérature puis on présente les notions et outils mathématiques utilisés dans le corps de la thèse. Dans le second chapitre, la convergence des suites des directions alternées (AM) pour une classe de problèmes variationnels elliptiques est étudiée. Sous une condition de non-orthogonalité uniforme entre les itérés et le terme source, on montre que ces suites sont en général bornées et compactes. Alors, si en particulier la suite (AM) converge faiblement alors elle converge fortement et la limite serait la solution du problème de minimisation alternée. Dans le troisième chapitre, on introduit la notion des dérivées fractionnaires au sens de Riemann-Liouville puis on considère un problème variationnel qui est une généralisation d’ordre fractionnaire de l’équation de Poisson. En se basant sur la nature quadratique et la décomposabilité de l’énergie associée, on démontre que la suite PGD progressive converge fortement vers la solution faible de ce problème. Dans le quatrième chapitre, on profite de la structure tensorielle des espaces BV par rapport à la topologie faible étoile pour définir les suites PGD dans ce type d’espaces. La convergence de telle suite reste une question ouverte. Le dernier chapitre est consacré à l’équation de la chaleur d-dimensionnelle, où on discrétise en temps puis à chaque pas de temps on cherche la solution de l’équation elliptique en utilisant la PGD. On montre alors que la fonction affine par morceaux en temps obtenue à partir des solutions construites en utilisant la PGD converge vers la solution faible de l’équation<br>In this thesis, we are interested in the PGD (Proper Generalized Decomposition), one of the reduced order models which consists in searching, a priori, the solution of a partial differential equation in a separated form. This work is composed of five chapters in which we aim to extend the PGD to the fractional spaces and the spaces of functions of bounded variation and to give theoretical interpretations of this method for a class of elliptic and parabolic problems. In the first chapter, we give a brief review of the litterature and then we introduce the mathematical notions and tools used in this work. In the second chapter, the convergence of rank-one alternating minimisation AM algorithms for a class of variational linear elliptic equations is studied. We show that rank-one AM sequences are in general bounded in the ambient Hilbert space and are compact if a uniform non-orthogonality condition between iterates and the reaction term is fulfilled. In particular, if a rank-one (AM) sequence is weakly convergent then it converges strongly and the common limit is a solution of the alternating minimization problem. In the third chapter, we introduce the notion of fractional derivatives in the sense of Riemann-Liouville and then we consider a variational problem which is a generalization of fractional order of the Poisson equation. Basing on the quadratic nature and the decomposability of the associated energy, we prove that the progressive PGD sequence converges strongly towards the weak solution of this problem. In the fourth chapter, we benefit from tensorial structure of the spaces BV with respect to the weak-star topology to define the PGD sequences in this type of spaces. The convergence of this sequence remains an open question. The last chapter is devoted to the d-dimensional heat equation, we discretize in time and then at each time step one seeks the solution of the elliptic equation using the PGD. Then, we show that the piecewise linear function in time obtained from the solutions constructed using the PGD converges to the weak solution of the equation

The topic of this doctoral thesis is at the intersection between Real Algebraic Geometry, Optimization Theory and Multilinear Algebra. In particular, a relevant part of this thesis is dedicated to studying metric invariants of real algebraic varieties, with a particular interest in varieties in tensor spaces. In many applications, tensors arise as a useful way to store and organize experimental data. For example, it is widely known that tensor techniques are extremely useful in Algebraic Statistics. A strong relationship between classical algebraic geometry and multilinear algebra is established by the notion of tensor rank. Geometrically speaking, the problem of computing the rank of a tensor translates to a membership problem to a certain secant variety of a Segre product of projective spaces. In the last fifteen years, a new line of research in tensor theory has been undertaken and is commonly known as Spectral Theory of Tensors. One of the foundational motivations of this theory comes from the need, e.g., in some constrained optimization problems, to approximate a given tensor to its closest tensor of fixed lower rank, with respect to the Frobenius norm, also known as Bombieri norm. This is the so-called best rank- k approximation problem for real tensors. In this context, an important role is played by the singular vector tuples and the singular values of a tensor, which generalize the notions of eigenvector and eigenvalue of a matrix. Their symmetric counterpart is represented by the E-eigenvalues and the E-eigenvectors of a symmetric tensor. Of particular interest is the E-characteristic polynomial of a symmetric tensor, which has among its roots the E-eigenvalues of a symmetric tensor. For symmetric matrices, it coincides with the classical characteristic polynomial. We interpret the E-characteristic polynomial as an algebraic relation satisfied by the Frobenius distance between an assigned symmetric tensor and the dual affine cone of a Veronese variety. We show that the E-characteristic polynomial is monic only in the symmetric matrix case. We provide a rational formula for the product of the singular values of a partially symmetric tensor of hypercubic format. The formula generalizes the fact that the determinant of a symmetric matrix is equal to the product of its eigenvalues. This is the only case where no denominator occurs in the formula. Computing the distance from a variety of low-rank tensors is an important instance of a more general problem: computing the distance from a real algebraic variety X in a Euclidean space (V,q). We introduce a polynomial, called Euclidean Distance polynomial of X, which, for any data point u in V, has among its roots the distance ε from u to X. The ε^2-degree of the ED polynomial is the known Euclidean Distance degree of X. When X is transversal to the isotropic quadric Q={q=0}, we show that the ED polynomial of X is monic and we describe its lowest term completely.