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<p dir="ltr">Continuous optimization is a fundamental topic both in theoretical computer science and applications of machine learning. Meanwhile, an important idea in the development modern algorithms it the use of randomness to achieve empirical speedup and improved theoretical runtimes. Stochastic gradient descent (SGD) and matrix-multiplication time linear program solvers [1] are two important examples of such achievements. Matrix sketching and related ideas provide a theoretical framework for the behavior of random matrices and vectors that arise in these algorithms, thereby provide a natural way to better understand the behavior of such randomized algorithms. In this dissertation, we consider three general problems in this area.</p>

[ES] Durante las primeras décadas del siglo pasado se estudiaron las inversas generalizadas que hoy en día se conocen como inversas generalizadas clásicas. Entre ellas cabe mencionar la inversa de Moore-Penrose (1955) y la inversa de Drazin (1958). Mientras que la inversa de Moore-Penrose se definió originalmente para matrices complejas rectangulares, la inversa de Drazin fue tratada, en un primer momento, únicamente para matrices cuadradas. Más tarde, en 1980, Cline y Greville realizaron la extensión del caso cuadrado al caso rectangular, mediante la consideración de una matriz de ponderación rectangular. Diferentes propiedades, caracterizaciones y aplicaciones fueron obtenidas para estos tipos de inversas generalizadas hasta finales del siglo pasado. En la última década, han aparecido nuevas nociones de inversas generalizadas. La primera de ellas fue la inversa core, introducida en el año 2010 por los autores Baksalary y Trenkler. La misma tuvo una amplia repercusión en la comunidad matemática debido a la sencillez de su definición, a su aplicación en la resolución de algunos sistemas lineales con restricciones que surgen en la teoría de redes eléctricas y también por su conexión con la inversa de Bott- Duffin. Muchos trabajos de investigación han surgido a partir de la inversa core, incluyendo sus extensiones a conjuntos más generales como el álgebra de operadores lineales acotados sobre espacios de Hilbert y/o al ámbito de anillos abstractos. El objetivo principal de esta tesis doctoral es definir y estudiar en profundidad una nueva inversa generalizada para matrices rectangulares, llamada inversa inversa de grupo débil ponderada, la cual extiende al caso rectangular la inversa de grupo débil recientemente definida (para el caso cuadrado) por Wang y Chen. También se considera un amplio estudio de la inversa core-EP ponderada definida por Ferreyra, Levis y Thome en el año 2018, y que extiende al caso rectangular inversa core-EP introducida por Manjunatha-Prasad y Mohana en el año 2014. Para ambas inversas generalizadas se obtienen nuevas propiedades, representaciones, caracterizaciones como así también su relación con otras inversas conocidas en la literatura. Además, se presentan dos algoritmos que permiten realizar un cálculo efectivo de las mismas. / [CA] Durant les primeres dècades del segle passat es van estudiar les inverses generalitzades que hui dia es coneixen com a inverses generalitzades clàssiques. Entre elles cal esmentar la inversa de Moore-Penrose (1955) i la inversa de Drazin (1958). Mentre que la inversa de Moore-Penrose es va definir originalment per a matrius complexes rectangulars, la inversa de Drazin va ser tractada, en un primer moment, únicament per a matrius quadrades. Més tard, en 1980, Cline i Greville van realitzar l'extensió del cas quadrat al cas rectangular, mitjançant la consideració d'una matriu de ponderació rectangular. Diferents propietats, caracteritzacions i aplicacions van ser obtingudes per a aquests tipus d'inverses generalitzades fins a finals del segle passat. En l'última dècada, han aparegut noves nocions d'inverses generalitzades. La primera d'elles va ser la inversa core, introduïda l'any 2010 pels autors Baksalary i Trenkler. La mateixa va tindre una àmplia repercussió en la comunitat matemàtica a causa de la senzillesa de la seua definició, a la seua aplicació en la resolució d'alguns sistemes lineals amb restriccions que sorgeixen en la teoria de xarxes elèctriques i també per la seua connexió amb la inversa de Bott-Duffinn. Molts treballs de recerca han sorgit a partir de la inversa core, incloent les seues extensions a conjunts més generals com l'àlgebra d'operadors lineals delimitats sobre espais de Hilbert i/o a l'àmbit d'anells abstractes. L'objectiu principal d'aquesta tesi doctoral és definir i estudiar en profunditat una nova inversa generalitzada per a matrius rectangulars, anomenada inversa inversa de grup feble ponderada, la qual estén al cas rectangular la inversa de grup feble recentment definida (per al cas quadrat) per Wang i Chen. Tamb é es considera un ampli estudi de la inversa core-EP ponderada definida per Ferreyra, Levis i Thome l'any 2018, i que estén al cas rectangular inversa core-EP introduïda per Manjunatha-Prasad i Mohana l'any 2014. Per a totes dues inverses generalitzades s'obtenen noves propietats, representacions, caracteritzacions com així també la seua relació amb altres inverses conegudes en la literatura. A més, es presenten dos algorismes que permeten realitzar un càlcul efectiu d'aquestes. / [EN] Generalized inverses, known today as Classical Generalized Inverses, were studied during the first decades of the last century. Two important classical generalized inverses are the Moore-Penrose inverse (1955) and the Drazin inverse (1958). The Moore-Penrose inverse was originally defined for complex rectangular matrices. In turn, the Drazin inverse was studied, at first, only for square matrices. It was in 1980 when Cline and Greville extended the case of square matrices to the case of rectangular matrices by considering a weight rectangular matrix. Throughout the entire past century there appeared difierent properties, characterizations and applications of these types of generalized inverses. This last decade gave rise to new notions of generalized inverses. The first of these new notions is known as the core inverse. Core inverses were introduced in 2010 by Baksalary and Trenkler. Their work had a wide repercussion in the mathematical community due to the simplicity of its denition and its application in the solution of some linear systems with restrictions. The core inverse further gain in interest due to their connection to the Bott-Duffin inverse. There is a large body of work on the core inverse, including extensions to more general sets if such as the algebra of bounded linear operators on Hilbert spaces and/or abstract rings. The main goal of this thesis is to define and study in depth a new generalized inverse for rectangular matrices. This new inverse is called weighted weak group inverse (or weighted WG inverse). Weighted WG inverses extend weak group inverse, recently defined for the square case by Wang and Chen, to the rectangular case. We also consider an extensive study of the weighted core-EP inverse. The latter type of inverse was dened by Ferreyra, Levis, and Thome in 2018. This inverse extends the core-EP inverse introduced by Manjunatha- Prasad and Mohana in 2014 to the rectangular case. This thesis presents new properties, representations, characterizations, as well as their relation with other inverses known in the literature are obtained, for weighted WG inverses and weighted core-EP inverse. In addition, the thesis presents two algorithms that allow for an efiective computation weighted WG inverses and weighted core-EP inverse. / Orquera, V. (2022). La inversa core-EP y la inversa de grupo débil para matrices rectangulares [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/185227

No / Spectrum sensing is used in cognitive radio systems to detect the availability of spectrum holes for secondary usage. The simplest and most famous spectrum sensing techniques are based either on energy detection or eigenspace analysis from Random Matrix Theory (RMT) such as using the Marchenko-Pastur law. These schemes suffer from uncertainty in estimating the noise variance which reduces their performance. In this paper we propose a new method to evaluate the noise variance that can eliminate the limitations of the aforementioned schemes. This method estimates the noise variance from a measurement set of noisy signals or noise-only signals. Extensive simulations show that the proposed method performs well in estimating the noise variance. Its performance greatly improves with increasing numbers of measurements and also with increasing numbers of samples taken per measurement.

No / Dynamic Spectrum Access (DSA) for secondary usage of underutilized radio spectrum is currently of great interest for radio regulatory authorities and for cellular network operators. However, the co-existence of multiple devices operating in the same bands, such as wireless microphones which also operate in TV bands, poses a challenge to DSA. Efficient and proactive spectrum sensing could prevent harmful interference between collocated devices, but existing blind spectrum sensing schemes such as energy detection and schemes based on Random Matrix Theory (RMT) have performance limitations. We propose a new blind spectrum sensing scheme for cognitive radio. The proposed scheme uses a new formula for the estimation of noise variance. The scheme has been evaluated through extensive simulations on wireless microphone signals and shows higher performance as compared to energy detection and RMT-based sensing schemes such as MME and EME. It also shows higher performance in terms of probability of detection (Pd).

Cette thèse s'inscrit dans la théorie des matrices aléatoires, à l'intersection avec la théorie des probabilités libres et des algèbres d'opérateurs. Elle s'insère dans une démarche générale qui a fait ses preuves ces dernières décennies : importer les techniques et les concepts de la théorie des probabilités non commutatives pour l'étude du spectre de grandes matrices aléatoires. On s'intéresse ici à des généralisations du théorème de liberté asymptotique de Voiculescu. Dans les Chapitres 1 et 2, nous montrons des résultats de liberté asymptotique forte pour des matrices gaussiennes, unitaires aléatoires et déterministes. Dans les Chapitres 3 et 4, nous introduisons la notion de fausse liberté asymptotique pour des matrices déterministes et certaines matrices hermitiennes à entrées sous diagonales indépendantes, interpolant les modèles de matrices de Wigner et de Lévy. / The thesis fits into the random matrix theory, in intersection with free probability and operator algebra. It is part of a general approach which is common since the last decades: using tools and concepts of non commutative probability in order to get general results about the spectrum of large random matrices. Where are interested here in generalization of Voiculescu's asymptotic freeness theorem. In Chapter 1 and 2, we show some results of strong asymptotic freeness for gaussian, random unitary and deterministic matrices. In Chapter 3 and 4, we introduce the notion of asymptotic false freeness for deterministic matrices and certain random matrices, Hermitian with independent sub-diagonal entries, interpolating Wigner and Lévy models.

Dans cette thèse, nous considérons le problème de la localisation de source dans les grands réseaux de capteurs, quand le nombre d'antennes du réseau et le nombre d'échantillons du signal observé sont grands et du même ordre de grandeur. Nous considérons le cas où les signaux source émis sont déterministes, et nous développons un algorithme de localisation amélioré, basé sur la méthode MUSIC. Pour ce faire, nous montrons de nouveaux résultats concernant la localisation des valeurs propres des grandes matrices aléatoires gaussiennes complexes de type information plus bruit / In this thesis, we consider the problem of source localization in large sensor networks, when the number of antennas of the network and the number of samples of the observed signal are large and of the same order of magnitude. We also consider the case where the source signals are deterministic, and we develop an improved algorithm for source localization, based on the MUSIC method. For this, we fist show new results concerning the position of the eigen values of large information plus noise complex gaussian random matrices

Três ensembles, respectivamente relacionados com as distribuições Gaussiana, Lognormal e de Levy, são abordados neste trabalho primordialmente do ponto de vista da termodinâmica de seus espectros. Novas expressões para as grandezas termodinâmicas sao encontradas para os ensembles de Stieltjes e de Bertuola-Pato, e a conexão destes com os ensembles Gaussianos e estabelecida. Esta tese também se compromete com a continuação do desenvolvimento e aprimorarão do ensemble generalizado de Bertuola-Pato, estendendo alguns resultados para os ensembles simplifico e unitário generalizados, alem do ortogonal generalizado já introduzido anteriormente por A. C. Bertuola e M. P. Pato. / Three ensembles, related to the Gaussian, the Lognormal and the L´evy distributions respectively, have been studied in this work and were investigated most of all in what concerns their spectral thermodynamics. New expressions for the thermodynamics quantities were found for the Stieltjes and the Bertuola-Pato ensembles, and the connection with the gaussian ensembles is established. This work concerned with the development continuity and with the improvement of Bertuola-Pato generalized ensemble, extending some of the results to the simplectic and unitary generalized ensembles, besides the orthogonal generalized ensemble introduced before by A. C. Bertuola and M. P. Pato.

Simetrias compõe parte integral da análise na Teoria das Matrizes Aleatórias (RMT). As simetrias de inversão temporal e rotacional são aspectos-chave do Ensemble Gaussiano Ortogonal (GOE), enquanto esta última é quebrada no Ensemble Gaussiano Simplético (GSE) e ambas são quebradas no Conjunto Unitário Gaussiano (GUE). Desde o final da década de 1990, o crescente interesse no campo dos sistemas quânticos PT-simétricos levou os pesquisadores a considerar o efeito, em matrizes aleatórias, dessa classe de simetrias, bem como simetrias pseudo-hermitianas. A principal questão a ser respondida pela pesquisa apresentada nesta tese é se a simetria PT ou, de forma mais geral, a pseudo-Hermiticidade implica alguma distribuição de probabilidade específica para os autovalores. Ou, em outras palavras, se há um aspecto comum transmitido por tal simetria que pode ser usada para modelar alguma classe particular de sistemas físicos. A abordagem inicial considerada consistiu na introdução de um conjunto pseudo-hermitiano, isospectral ao conjunto -Hermite, que apresentaria o tipo de quebra de realidade típico dos sistemas PT-simétricos. Nesse modelo, a primeira abordagem adotada foi a introdução de perturbações que quebraram a realidade dos espectros. Os resultados obtidos permitem concluir que a transformação em seu similar pseudo-hermitiano conduz a um sistema assintoticamente instável. Esse modelo foi extendido ao considerar um pseudo-hermitiano não positivo, que leva a uma quebra similar na realidade dos espectros. Este caso apresenta um comportamento mais próximo do típico dos sistemas PT-simétricos presentes na literatura. Um modelo denso geral baseado em projetores foi proposto, e duas realizações particulares deste modelo receberam atenção mais detalhada. O comportamento espectral também foi similar àquele típico da simetria PT para as duas realizações consideradas, e seus limites assintóticos foram conectados a conjuntos clássicos de teoria de matriz aleatória. Além disso, as propriedades de seus polinômios característicos médios foram obtidas e os limites assintóticos desses polinômios também foram considerados e relacionados a polinômios clássicos. O comportamento estatístico deste conjunto foi estudado e comparado com o destes polinômios. Impor a pseudo-Hermiticidade não parece implicar qualquer distribuição particular de autovalores, sendo a característica comum a quebra da realidade dos autovalores comumente encontrados na literatura de simetria PT. O resultado mais notável dos estudos apresentados nesta tese é o fato de que uma interação pseudo-hermitiana pode ser construída de tal forma que o comportamento espectral médio possa ser controlado calibrando-se o mecanismo de interação, bem como sua intensidade. / The role of symmetries is an integral part of the analysis in Random Matrix Theory (RMT). Time reversal and rotational symmetries are key aspects of the Gaussian Orthogonal Ensemble (GOE), whereas the latter is broken in the Gaussian Sympletic Ensemble (GSE) and both are broken in the Gaussian Unitary Ensemble (GUE). Since the late 1990s, growing interest in the field of PT symmetric quantum systems has led researchers to consider the effect, in random matrices, of this class of symmetries, as well as that of pseudo-Hermitian symmetries. The primary question to be answered by the research presented in this thesis is whether PT-symmetry or, more generally, pseudo-Hermiticity implies some specific probability distribution for the eigenvalues. Or, in other words, whether there is a common aspect imparted by such a symmetry which may be used to model some particular class of physical systems. The initial approach considered consisted of introducing an pseudo-Hermitian ensemble, isospectral to the -Hermite ensemble, which would present the type of reality-breaking typical of PT-symmetrical systems. In this model, the first approach taken was to introduce perturbation which broke the reality of the spectra. The results obtained allow the conclusion that the transformation into its pseudo-Hermitian similar leads into a system which is asymptotically unstable. An extension of this model was to consider a non-positive pseudo-Hermitian , which lead to similar breaking in the reality of the spectra. This case displays behavior closer to that typical of the PT-symmetric systems present in the literature. A general dense projector model was proposed, and two particular realizations of this model were given more detailed attention. The spectral behavior was also similar to that typical of PT-symmetry for the two realizations considered, and their asymptotic limits were shown to connect to classical ensembles of random matrix theory. Furthermore, the properties of their average characteristic polynomials were obtained and the asymptotic limits of these polynomials were also considered and were related to classical polynomials. The statistical behavior of this ensemble was studied and compared to that of these polynomials. Imposing the pseudo-Hermitian does seem not imply any particular eigenvalue distribution, the common feature being the breaking of the reality of the eigenvalues commonly found in PT-symmetry literature. The most notable result of the studies presented herein is the fact that a pseudo-Hermitian interaction may be constructed such that the average spectral behavior may be controlled by calibrating the mechanism of interaction as well as its intensity.

Le sujet principal de cette thèse est la physique des verres de spin. Les verres de spin ont été introduits au début des années 70 pour décrire alliages magnétiques diluées. Ils ont désormais été considerés pour comprendre le comportement de liquides sousrefroidis. Parmis les systèmes qui peuvent être décrits par le langage des systèmes desordonnés, on trouve les problèmes d'optimisation combinatoire. Dans la première partie de cette thèse, nous considérons les modèles de verre de spin avec intéraction de Kac pour investiguer la phase de basse température des liquides sous-refroidis. Dans les chapitres qui suivent, nous montrons comment certaines caractéristiques des modèles de verre de spin peuvent être obtenues à partir de résultats de la théorie des matrices aléatoires en connection avec la statistique des valeurs extrêmes. Dans la dernière partie de la thèse, nous considérons la connexion entre la théorie desverres de spin et la science computationnelle, et présentons un nouvel algorithme qui peut être appliqué à certains problèmes dans le domaine des finances.