Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / 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 âmatchâ 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 Ãltimas dÃcadas, sistemas de comunicaÃÃo sem fio de mÃltiplas antenas (MIMO - Multiple Input Multiple Output) tÃm sido objetos de intensas pesquisas devido à promessa teÃrica do aumento proporcional da capacidade com o aumento do nÃmero de antenas. Esta propriedade excepcional supÃe 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 interaÃÃo entre as antenas e meio de propagaÃÃo (fÃsico). AtravÃs da compreensÃo dessas interaÃÃes, arranjos de antenas de recepÃÃo e transmissÃo podem ser projetados para melhor "casar" com o meio de propagaÃÃo, tal que a qualidade do link de comunicaÃÃo e capacidade possam ser melhoradas em um sistema MIMO. Neste trabalho, consideramos o uso de harmÃnicos esfÃricos e decomposiÃÃes tensoriais no problema de modelagem de canal MIMO e estimaÃÃo. O uso de harmÃnicos esfÃricos permite representar os padrÃes de radiaÃÃo de antenas em termos de coeficientes de uma expansÃo, assim desacoplando as respostas dos arranjos de antenas (transmissoras e receptoras) do meio de propagaÃÃo fÃsica. Traduzindo modelos simples de canais baseados em propagaÃÃo, com informaÃÃes de polarizaÃÃo, para o domÃnio dos harmÃnicos esfÃricos, estudamos como os parÃmetros de propagaÃÃo si e configuraÃÃes especÃficas de antenas afetam o desempenho do sistema MIMO em termos de capacidade e de correlaÃÃo. A segunda parte deste trabalho aborda o problema de estimar canais direcionais MIMO no domÃnio dos harmÃnicos esfÃricos usando decomposiÃÃes por tensores. Considerando tanto cenos de espalhamento simples e de duplo espalhamento, fazemos uso das decomposiÃÃes PARAFAC e PARATUCK2, respectivamente, para estimar os modos esfÃricos propagantes, a partir das quais as direÃÃes de chegada (DoA) e as direÃÃes de saÃda (DoD) podem ser extraÃdas. Finalmente, propomos e comparamos dois mÃtodos de otimizaÃÃo dos coeficientes da expansÃo em harmÃnicos esfÃricos de arranjos de antenas para respostas de canais MIMO prÃ-especificados .

The center-focus and isochronicity problems are two classic problem in the qualitative theory of ordinary differential equations (ODEs). Although such problems have been studied during more than hundred years a complete understanding of them is far from be reached. Recently the computational algebra tools have been contributing significantly with the development of such problems. The aim of this thesis is to contribute with the studies of the center-focus and isochronicity problem. Using computational algebra tools we find conditions for the existence of two simultaneous centers for a family of quintic systems possessing symmetry. The studies of the simultaneous existence of two centers in differential systems is known as the bi-center problem. We investigate conditions for the isochronicity of centers for families of cubic and quintic systems and we study its global behaviour in the Poincaré disk. Finally, we study the existence of invariant surfaces and first integrals in a family of 3-dimensional systems. Such family is known as the May-Leonard asymmetric system and it appears in modelling, for instance it is a model for the competition of three species. / Os problemas do foco-centro e da isocronicidade são dois problemas clássicos da teoria qualitativa das equações diferenciais ordinárias (EDOs). Apesar de tais problemas serem investigados a mais de cem anos ainda pouco se sabe sobre eles. Recentemente o uso e desenvolvimento de ferramentas algebro-computacionais tem contribuído significativamente em seu avanço. O objetivo desta tese é colaborar com o estudo do problema do foco-centro e da isocronicidade. Utilizando ferramentas algebro-computacionais encontramos condições para a existência simultânea de dois centros em famílias de sistemas diferenciais quínticos com simetria. O estudo sobre a existência simultânea de dois centros é também conhecido como problema do bi-centro. Investigamos condições para a isocronicidade de centros para famílias de sistemas cubicos e quínticos e estudamos o comportamento global de suas órbitas no disco de Poincaré. Finalmente, tratamos da existência de superfícies invariantes e integrais primeiras para uma familia de sistemas 3-dimensionais encontrado entre outras situações na modelagem da competição entre três espécies e conhecido como sistema de May-Leonard.

Constraint programming (CP) is a technology in which a combinatorial problem is modelled as a conjunction of constraints on variables ranging over given initial domains, and optionally an objective function on the variables. Such a model is given to a general-purpose solver performing systematic search to find constraint-satisfying domain values for the variables, giving an optimal value to the objective function. A constraint predicate (also known as a global constraint) does two things: from the modelling perspective, it allows a modeller to express a commonly occurring combinatorial substructure, for example that a set of variables must take distinct values; from the solving perspective, it comes with a propagation algorithm, called a propagator, which removes some but not necessarily all impossible values from the current domains of its variables when invoked during search. Although modern CP solvers have many constraint predicates, often a predicate one would like to use is not available. In the past, the choices were either to reformulate the model or to write one's own propagator. In this dissertation, we contribute to the automatic design of propagators for new predicates. Integer time series are often subject to constraints on the aggregation of the features of all maximal occurrences of some pattern. For example, the minimum width of the peaks may be constrained. Automata allow many constraint predicates for variable sequences, and in particular many time-series predicates, to be described in a high-level way. Our first contribution is an algorithm for generating an automaton-based predicate description from a pattern, a feature, and an aggregator. It has previously been shown how to decompose an automaton-described constraint on a variable sequence into a conjunction of constraints whose predicates have existing propagators. This conjunction provides the propagation, but it is unknown how to propagate it efficiently. Our second contribution is a tool for deriving, in an off-line process, implied constraints for automaton-induced constraint decompositions to improve propagation. Further, when a constraint predicate functionally determines a result variable that is unchanged under reversal of a variable sequence, we provide as our third contribution an algorithm for deriving an implied constraint between the result variables for a variable sequence, a prefix thereof, and the corresponding suffix.

The Ph.D. Thesis ¿Operators on weighted spaces of holomorphic functions¿ presented here treats different areas of functional analysis such as spaces of holomorphic functions, infinite dimensional holomorphy and dynamics of operators. After a first chapter that introduces the notation, definitions and the basic results we will use throughout the thesis, the text is divided into two parts. A first one, consisting of Chapters 1 and 2, focused on a study of weighted (LB)-spaces of entire functions on Banach spaces, and a second one, corresponding to Chapters 3 and 4, where we consider differentiation and integration operators acting on different classes of weighted spaces of entire functions to study its dynamical behaviour. In what follows, we give a brief description of the different chapters: In Chapter 1, given a decreasing sequence of continuous radial weights on a Banach space X, we consider the weighted inductive limits of spaces of entire functions VH(X) and VH0(X). Weighted spaces of holomorphic functions appear naturally in the study of growth conditions of holomorphic functions and have been investigated by many authors since the work of Williams in 1967, Rubel and Shields in 1970 and Shields and Williams in 1971. We determine conditions on the family of weights to ensure that the corresponding weighted space is an algebra or has polynomial Schauder decompositions. We study Hörmander algebras of entire functions defined on a Banach space and we give a description of them in terms of sequence spaces. We also focus on algebra homomorphisms between these spaces and obtain a Banach-Stone type theorem for a particular decreasing family of weights. Finally, we study the spectra of these weighted algebras, endowing them with an analytic structure, and we prove that each function f ¿ VH(X) extends naturally to an analytic function defined on the spectrum. Given an algebra homomorphism, we also investigate how the mapping induced between the spectra acts on the corresponding analytic structures and we show how in this setting composition operators have a different behavior from that for holomorphic functions of bounded type. This research is related to recent work by Carando, García, Maestre and Sevilla-Peris. The results included in this chapter are published by Beltrán in [14]. Chapter 2 is devoted to study the predual of VH(X) in order to linearize this space of entire functions. We apply Mujica¿s completeness theorem for (LB)-spaces to find a predual and to prove that VH(X) is regular and complete. We also study conditions to ensure that the equality VH0(X) = VH(X) holds. At this point, we will see some differences between the finite and the infinite dimensional cases. Finally, we give conditions which ensure that a function f defined in a subset A of X, with values in another Banach space E, and admitting certain weak extensions in a space of holomorphic functions can be holomorphically extended in the corresponding space of vector-valued functions. Most of the results obtained have been published by the author in [13]. The rest of the thesis is devoted to study the dynamical behaviour of the following three operators on weighted spaces of entire functions: the differentiation operator Df(z) = f (z), the integration operator Jf(z) = z 0 f(¿)d¿ and the Hardy operator Hf(z) = 1 z z 0 f(¿)d¿, z ¿ C. In Chapter 3 we focus on the dynamics of these operators on a wide class of weighted Banach spaces of entire functions defined by means of integrals and supremum norms: the weighted spaces of entire functions Bp,q(v), 1 ¿ p ¿ ¿, and 1 ¿ q ¿ ¿. For q = ¿ they are known as generalized weighted Bergman spaces of entire functions, denoted by Hv(C) and H0 v (C) if, in addition, p = ¿. We analyze when they are hypercyclic, chaotic, power bounded, mean ergodic or uniformly mean ergodic; thus complementing also work by Bonet and Ricker about mean ergodic multiplication operators. Moreover, for weights satisfying some conditions, we estimate the norm of the operators and study their spectrum. Special emphasis is made on exponential weights. The content of this chapter is published in [17] and [15]. For differential operators ¿(D) : Bp,q(v) ¿ Bp,q(v), whenever D : Bp,q(v) ¿ Bp,q(v) is continuous and ¿ is an entire function, we study hypercyclicity and chaos. The chapter ends with an example provided by A. Peris of a hypercyclic and uniformly mean ergodic operator. To our knowledge, this is the first example of an operator with these two properties. We thank him for giving us permission to include it in our thesis. The last chapter is devoted to the study of the dynamics of the differentiation and the integration operators on weighted inductive and projective limits of spaces of entire functions. We give sufficient conditions so that D and J are continuous on these spaces and we characterize when the differentiation operator is hypercyclic, topologically mixing or chaotic on projective limits. Finally, the dynamics of these operators is investigated in the Hörmander algebras Ap(C) and A0 p(C). The results concerning this topic are included by Bonet, Fernández and the author in [16]. / Beltrán Meneu, MJ. (2014). Operators on wighted spaces of holomorphic functions [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/36578 / TESIS / Premios Extraordinarios de tesis doctorales

Cette thèse étudie les causes et conséquences du processus de ségrégation résidentielle dans l’Afrique du Sud (AFS) post-Apartheid. Nous nous intéressons à plusieurs aspects encore débattus dans la littérature. Le premier concerne l’impact des préférences des individus pour la composition raciale de leur voisinage sur la ségrégation. Le second a trait à l’impact de la ségrégation résidentielle sur les niveaux de revenus des différents groupes raciaux. Le dernier quantifie les différentes causes de la ségrégation. Dans le premier chapitre, nous réconcilions la littérature théorique sur l’impact des préférences pour la composition raciale du voisinage avec les observations empiriques de niveaux décroissants de ségrégation aux US et en AFS. Nous soutenons l’idée que si les individus internalisent les apports économiques et sociaux de chaque nouvel arrivant dans leur voisinage alors des voisinages intégrés peuvent émerger. Cet effet est empiriquement plus fort que l’homophilie et le racisme. Dans le second chapitre, nous étudions l’impact de la ségrégation sur l’ensemble de la distribution des revenus. Nous montrons que la ségrégation a un effet positif sur les hauts revenus pour les Blancs tandis qu’elle a un effet négatif pour les Noirs au bas de la distribution. L’effet de la ségrégation est souvent plus important que l’effet de l’éducation. Enfin, dans le troisième chapitre, nous quantifions l’impact de chaque déterminant de la ségrégation. Nous trouvons que le manque d’accès aux services publics de base est le déterminant principal, alors que les différences de caractéristiques sociodémographiques ne comptent que pour une faible part pour les quartiers les plus ségrégués. / This thesis studies the causes and consequences of the residential segregation process in the post-Apartheid South Africa.Inside this general issue, we are interested in several aspects still debated in the literature on residential segregation. Thefirst concerns the impact of individuals’ preferences for the racial composition of their neighborhood on the segregationlevels. The second question deals with the impact of residential segregation on the income levels of each racial group. Thelast issue is related to quantifying the different causes of segregation.Three chapters constitute this thesis. In the first chapter, we reconcile the theoretical literature on the impact of preferencesfor the racial composition of the neighborhood with the empirical evidences of declining levels of segregation in theUnited-States and South Africa. We argue that if individuals internalize the economic and social life that a new entrantbrings with him, then integrated neighborhoods can emerge. This effect is empirically stronger than homophilly andracism. In the second chapter, we study the impact of residential segregation on the whole income distribution. We showthat residential segregation has a positif effect on top incomes for Whites, whereas it has a negatif effect for Blacks at thebottom of the distribution. The effect of residential segregation is even more important than the effect of education inmost cases. In the third chapter, we quantify the impact of each determinant of segregation. We find that the lackof access to basic public services is the main determinant, whereas differences in sociodemographics only account for asmall part in the most segregated areas.

Les milieux humides remplissent plusieurs fonctions écologiques d’importance et contribuent à la biodiversité de la faune et de la flore. Même s’il existe une reconnaissance croissante sur l’importante de protéger ces milieux, il n’en demeure pas moins que leur intégrité est encore menacée par la pression des activités humaines. L’inventaire et le suivi systématique des milieux humides constituent une nécessité et la télédétection est le seul moyen réaliste d’atteindre ce but. L’objectif de cette thèse consiste à contribuer et à améliorer la caractérisation des milieux humides en utilisant des données satellites acquises par des radars polarimétriques en bande L (ALOS-PALSAR) et C (RADARSAT-2). Cette thèse se fonde sur deux hypothèses (chap. 1). La première hypothèse stipule que les classes de physionomies végétales, basées sur la structure des végétaux, sont plus appropriées que les classes d’espèces végétales car mieux adaptées au contenu informationnel des images radar polarimétriques. La seconde hypothèse stipule que les algorithmes de décompositions polarimétriques permettent une extraction optimale de l’information polarimétrique comparativement à une approche multipolarisée basée sur les canaux de polarisation HH, HV et VV (chap. 3). En particulier, l’apport de la décomposition incohérente de Touzi pour l’inventaire et le suivi de milieux humides est examiné en détail. Cette décomposition permet de caractériser le type de diffusion, la phase, l’orientation, la symétrie, le degré de polarisation et la puissance rétrodiffusée d’une cible à l’aide d’une série de paramètres extraits d’une analyse des vecteurs et des valeurs propres de la matrice de cohérence. La région du lac Saint-Pierre a été sélectionnée comme site d’étude étant donné la grande diversité de ses milieux humides qui y couvrent plus de 20 000 ha. L’un des défis posés par cette thèse consiste au fait qu’il n’existe pas de système standard énumérant l’ensemble possible des classes physionomiques ni d’indications précises quant à leurs caractéristiques et dimensions. Une grande attention a donc été portée à la création de ces classes par recoupement de sources de données diverses et plus de 50 espèces végétales ont été regroupées en 9 classes physionomiques (chap. 7, 8 et 9). Plusieurs analyses sont proposées pour valider les hypothèses de cette thèse (chap. 9). Des analyses de sensibilité par diffusiogramme sont utilisées pour étudier les caractéristiques et la dispersion des physionomies végétales dans différents espaces constitués de paramètres polarimétriques ou canaux de polarisation (chap. 10 et 12). Des séries temporelles d’images RADARSAT-2 sont utilisées pour approfondir la compréhension de l’évolution saisonnière des physionomies végétales (chap. 12). L’algorithme de la divergence transformée est utilisé pour quantifier la séparabilité entre les classes physionomiques et pour identifier le ou les paramètres ayant le plus contribué(s) à leur séparabilité (chap. 11 et 13). Des classifications sont aussi proposées et les résultats comparés à une carte existante des milieux humide du lac Saint-Pierre (14). Finalement, une analyse du potentiel des paramètres polarimétrique en bande C et L est proposé pour le suivi de l’hydrologie des tourbières (chap. 15 et 16). Les analyses de sensibilité montrent que les paramètres de la 1re composante, relatifs à la portion dominante (polarisée) du signal, sont suffisants pour une caractérisation générale des physionomies végétales. Les paramètres des 2e et 3e composantes sont cependant nécessaires pour obtenir de meilleures séparabilités entre les classes (chap. 11 et 13) et une meilleure discrimination entre milieux humides et milieux secs (chap. 14). Cette thèse montre qu’il est préférable de considérer individuellement les paramètres des 1re, 2e et 3e composantes plutôt que leur somme pondérée par leurs valeurs propres respectives (chap. 10 et 12). Cette thèse examine également la complémentarité entre les paramètres de structure et ceux relatifs à la puissance rétrodiffusée, souvent ignorée et normalisée par la plupart des décompositions polarimétriques. La dimension temporelle (saisonnière) est essentielle pour la caractérisation et la classification des physionomies végétales (chap. 12, 13 et 14). Des images acquises au printemps (avril et mai) sont nécessaires pour discriminer les milieux secs des milieux humides alors que des images acquises en été (juillet et août) sont nécessaires pour raffiner la classification des physionomies végétales. Un arbre hiérarchique de classification développé dans cette thèse constitue une synthèse des connaissances acquises (chap. 14). À l’aide d’un nombre relativement réduit de paramètres polarimétriques et de règles de décisions simples, il est possible d’identifier, entre autres, trois classes de bas marais et de discriminer avec succès les hauts marais herbacés des autres classes physionomiques sans avoir recours à des sources de données auxiliaires. Les résultats obtenus sont comparables à ceux provenant d’une classification supervisée utilisant deux images Landsat-5 avec une exactitude globale de 77.3% et 79.0% respectivement. Diverses classifications utilisant la machine à vecteurs de support (SVM) permettent de reproduire les résultats obtenus avec l’arbre hiérarchique de classification. L’exploitation d’une plus forte dimensionalitée par le SVM, avec une précision globale maximale de 79.1%, ne permet cependant pas d’obtenir des résultats significativement meilleurs. Finalement, la phase de la décomposition de Touzi apparaît être le seul paramètre (en bande L) sensible aux variations du niveau d’eau sous la surface des tourbières ouvertes (chap. 16). Ce paramètre offre donc un grand potentiel pour le suivi de l’hydrologie des tourbières comparativement à la différence de phase entre les canaux HH et VV. Cette thèse démontre que les paramètres de la décomposition de Touzi permettent une meilleure caractérisation, de meilleures séparabilités et de meilleures classifications des physionomies végétales des milieux humides que les canaux de polarisation HH, HV et VV. Le regroupement des espèces végétales en classes physionomiques est un concept valable. Mais certaines espèces végétales partageant une physionomie similaire, mais occupant un milieu différent (haut vs bas marais), ont cependant présenté des différences significatives quant aux propriétés de leur rétrodiffusion. / Wetlands fill many important ecological functions and contribute to the biodiversity of fauna and flora. Although there is a growing recognition of the importance to protect these areas, it remains that their integrity is still threatened by the pressure of human activities. The inventory and the systematic monitoring of wetlands are a necessity and remote sensing is the only realistic way to achieve this goal. The primary objective of this thesis is to contribute and improve the wetland characterization using satellite polarimetric data acquired in L (ALOS-PALSAR) and C (RADARSAT-2) band. This thesis is based on two hypotheses (Ch. 1). The first hypothesis stipulate that classes of plant physiognomies, based on plant structure, are more appropriate than classes of plant species because they are best adapted to the information content of polarimetric radar data. The second hypothesis states that polarimetric decomposition algorithms allow an optimal extraction of polarimetric information compared to a multi-polarized approach based on the HH, HV and VV channels (Ch. 3). In particular, the contribution of the incoherent Touzi decomposition for the inventory and monitoring of wetlands is examined in detail. This decomposition allows the characterization of the scattering type, its phase, orientation, symmetry, degree of polarization and the backscattered power of a target with a series of parameters extracted from an analysis of the coherency matrix eigenvectors and eigenvalues. The lake Saint-Pierre region was chosen as the study site because of the great diversity of its wetlands that are covering more than 20 000 ha. One of the challenges posed by this thesis is that there is neither a standard system enumerating all the possible physiognomic classes nor an accurate description of their characteristics and dimensions. Special attention was given to the creation of these classes by combining several data sources and more than 50 plant species were grouped into nine physiognomic classes (Ch. 7, 8 and 9). Several analyzes are proposed to validate the hypotheses of this thesis (Ch. 9). Sensitivity analysis using scatter plots are performs to study the characteristics and dispersion of plant physiognomic classes in various features space consisting of polarimetric parameters or polarization channels (Ch. 10 and 12). Time series of made of RADARSAT-2 images are used to deepen the understanding of the seasonal evolution of plant physiognomies (Ch. 12). The transformed divergence algorithm is used to quantify the separability between physiognomic classes and to identify the parameters (s) that contribute the most to their separability (Ch. 11 and 13). Classifications are also proposed and the results compared to an existing map of the lake Saint-Pierre wetlands (Ch. 14). Finally, an analysis of the potential of polarimetric parameters in C and L-band is proposed for the monitoring of peatlands hydrology (Ch. 15 and 16). Sensitivity analyses show that the parameters of the 1st component, relative to the dominant (polarized) part of the signal, are sufficient for a general characterization of plant physiognomies. The parameters of the second and third components are, however, needed for better class separability (Ch. 11 and 13) and a better discrimination between wetlands and uplands (Ch. 14). This thesis shows that it is preferable to consider individually the parameters of the 1st, 2nd and 3rd components rather than their weighted sum by their respective eigenvalues (Ch. 10 and 12). This thesis also examines the complementarity between the structural parameters and those related to the backscattered power, often ignored and normalized by most polarimetric decomposition. The temporal (seasonal) dimension is essential for the characterization and classification of plant physiognomies (Ch. 12, 13 and 14). Images acquired in spring (April and May) are needed to discriminate between upland and wetlands while images acquired in summer (July and August) are needed to refine the classifications of plant physiognomies. A hierarchical classification tree developed in this thesis represents a synthesis of the acquired knowledge (Chapter 14). Using a relatively small number of polarimetric parameters and simple decision rules, it is possible to identify, among other, three low marshes classes and to discriminate with success herbaceous high marshes from other physiognomic classes without using ancillary data source. The results obtained are comparable to those from a supervised classification using two Landsat-5 images with an overall accuracy of 77.3% and 79.0% respectively. Various classifications using the support vector machine (SVM) can reproduce the results obtained with the hierarchical classification tree. But the possible exploitation by the SVM of a higher dimensionality, with a maximum overall accuracy of 79.1%, does not allow however to achieve significantly better results. Finally, the phase of the Touzi decomposition appears to be the only parameter (in L-band) sensitive to changes in water level beneath the peat surface (Ch. 16). Therefore, this parameter offer a great potential for peatlands hydrology monitoring compared to the HH-VV phase difference. This thesis demonstrates that the Touzi decomposition parameters allow a better characterization, better separability and better classifications of wetlands plant physiognomic classes than HH, HV and VV polarization channels. The grouping of plant species into physiognomic classes is a valid concept. However, some plant species sharing a similar physiognomy, but occupying a different environment (high vs. low marshes), have presented significant differences in their scattering properties.

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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.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Métodos estocásticos oferecem uma poderosa ferramenta para a execução da compressão de dados e decomposições de matrizes. O método estocástico para decomposição de matrizes estudado utiliza amostragem aleatória para identificar um subespaço que captura a imagem de uma matriz de forma aproximada, preservando uma parte de sua informação essencial. Estas aproximações compactam a informação possibilitando a resolução de problemas práticos de maneira eficiente. Nesta dissertação é calculada uma decomposição em valores singulares (SVD) utilizando técnicas estocásticas. Esta SVD aleatória é empregada na tarefa de reconhecimento de faces. O reconhecimento de faces funciona de forma a projetar imagens de faces sobre um espaço de características que melhor descreve a variação de imagens de faces conhecidas. Estas características significantes são conhecidas como autofaces, pois são os autovetores de uma matriz associada a um conjunto de faces. Essa projeção caracteriza aproximadamente a face de um indivíduo por uma soma ponderada das autofaces características. Assim, a tarefa de reconhecimento de uma nova face consiste em comparar os pesos de sua projeção com os pesos da projeção de indivíduos conhecidos. A análise de componentes principais (PCA) é um método muito utilizado para determinar as autofaces características, este fornece as autofaces que representam maior variabilidade de informação de um conjunto de faces. Nesta dissertação verificamos a qualidade das autofaces obtidas pela SVD aleatória (que são os vetores singulares à esquerda de uma matriz contendo as imagens) por comparação de similaridade com as autofaces obtidas pela PCA. Para tanto, foram utilizados dois bancos de imagens, com tamanhos diferentes, e aplicadas diversas amostragens aleatórias sobre a matriz contendo as imagens. / Stochastic methods offer a powerful tool for performing data compression and decomposition of matrices. These methods use random sampling to identify a subspace that captures the range of a matrix in an approximate way, preserving a part of its essential information. These approaches compress the information enabling the resolution of practical problems efficiently. This work computes a singular value decomposition (SVD) of a matrix using stochastic techniques. This random SVD is employed in the task of face recognition. The face recognition is based on the projection of images of faces on a feature space that best describes the variation of known image faces. These features are known as eigenfaces because they are the eigenvectors of a matrix constructed from a set of faces. This projection characterizes an individual face by a weighted sum of eigenfaces. The task of recognizing a new face is to compare the weights of its projection with the projection of the weights of known individuals. The principal components analysis (PCA) is a widely used method for determining the eigenfaces. This provides the greatest variability eigenfaces representing information from a set of faces. In this dissertation we discuss the quality of eigenfaces obtained by a random SVD (which are the left singular vectors of a matrix containing the images) by comparing the similarity with eigenfaces obtained by PCA. We use two databases of images, with different sizes and various random sampling applied on the matrix containing the images.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Métodos estocásticos oferecem uma poderosa ferramenta para a execução da compressão de dados e decomposições de matrizes. O método estocástico para decomposição de matrizes estudado utiliza amostragem aleatória para identificar um subespaço que captura a imagem de uma matriz de forma aproximada, preservando uma parte de sua informação essencial. Estas aproximações compactam a informação possibilitando a resolução de problemas práticos de maneira eficiente. Nesta dissertação é calculada uma decomposição em valores singulares (SVD) utilizando técnicas estocásticas. Esta SVD aleatória é empregada na tarefa de reconhecimento de faces. O reconhecimento de faces funciona de forma a projetar imagens de faces sobre um espaço de características que melhor descreve a variação de imagens de faces conhecidas. Estas características significantes são conhecidas como autofaces, pois são os autovetores de uma matriz associada a um conjunto de faces. Essa projeção caracteriza aproximadamente a face de um indivíduo por uma soma ponderada das autofaces características. Assim, a tarefa de reconhecimento de uma nova face consiste em comparar os pesos de sua projeção com os pesos da projeção de indivíduos conhecidos. A análise de componentes principais (PCA) é um método muito utilizado para determinar as autofaces características, este fornece as autofaces que representam maior variabilidade de informação de um conjunto de faces. Nesta dissertação verificamos a qualidade das autofaces obtidas pela SVD aleatória (que são os vetores singulares à esquerda de uma matriz contendo as imagens) por comparação de similaridade com as autofaces obtidas pela PCA. Para tanto, foram utilizados dois bancos de imagens, com tamanhos diferentes, e aplicadas diversas amostragens aleatórias sobre a matriz contendo as imagens. / Stochastic methods offer a powerful tool for performing data compression and decomposition of matrices. These methods use random sampling to identify a subspace that captures the range of a matrix in an approximate way, preserving a part of its essential information. These approaches compress the information enabling the resolution of practical problems efficiently. This work computes a singular value decomposition (SVD) of a matrix using stochastic techniques. This random SVD is employed in the task of face recognition. The face recognition is based on the projection of images of faces on a feature space that best describes the variation of known image faces. These features are known as eigenfaces because they are the eigenvectors of a matrix constructed from a set of faces. This projection characterizes an individual face by a weighted sum of eigenfaces. The task of recognizing a new face is to compare the weights of its projection with the projection of the weights of known individuals. The principal components analysis (PCA) is a widely used method for determining the eigenfaces. This provides the greatest variability eigenfaces representing information from a set of faces. In this dissertation we discuss the quality of eigenfaces obtained by a random SVD (which are the left singular vectors of a matrix containing the images) by comparing the similarity with eigenfaces obtained by PCA. We use two databases of images, with different sizes and various random sampling applied on the matrix containing the images.