For a linear regression, the traditional technique deals with a case where the number of observations n more than the number of predictor variables p (n > p). In the case n < p, the classical method fails to estimate the coefficients. A solution of the problem is the case of correlated predictors is provided in this thesis. A new regularization and variable selection is proposed under the name of Sparse Ridge Fusion (SRF). In the case of highly correlated predictor, the simulated examples and a real data show that the SRF always outperforms the lasso, eleastic net, and the S-Lasso, and the results show that the SRF selects more predictor variables than the sample size n while the maximum selected variables by lasso is n size.

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Systems of polynomial equations appear both in mathematics, as well as in many applications in the sciences, economics and engineering. Solving these systems is at the heart of computational algebraic geometry, a field which is often associated with symbolic computations based on Gr¨obner bases. Over the last thirty years, increasing performance and versatility made numerical algebraic geometry emerge as an alternative. It enables us to solve problems which are infeasible with symbolic methods. The focus of this thesis is the rich interplay between algebraic geometry, numerical computation and optimization in various instances. As a first application of algebraic geometry, we investigate global optimization problems whose objective function and constraints are all described by multivariate polynomials. One of the most important, and also most common, features of real world data is sparsity. We explore the effects of sparsity in global optimization, when exhibited by constraints and objective functions. Exploiting this property can lead to dramatic improvements of computational performance of algorithms. As a second application of geometry we study a particularly structured class of polynomial programs which stems from the optimization of sequencial decision rules. In the framework of partially observable Markov decision rules, an agent manipulates a system in a sequence of events. It selects an action at every time step, which in turn influences the state of the system at the next time step, and depending on the state it receives an instantaneous reward. Optimizing the long term reward has a long-standing history in computer science, economics and statistics. The ability to incorporate nondeterministic effects makes the framework particularly well suited for real world applications. We initiate a novel, geometric perspective on the underlying optimization problem and explore algorithmic consequences. As a third application of geometry we present the usage of tropical geometry in order to numerically compute defining equations of unirational varieties from their parametrization. Tropical geometry is an emerging field in mathematics at the boundary of discrete geometry and algebraic geometry. The tropicalization of a variety is a polyhedral complex which encodes geometric information of the variety. Tropical implicitization means computing the tropicalization of a unirational variety from its parametrization. In the case of a hypersurface, this amounts to finding the Newton polytope of the implicit equation, without computing its coefficients. We use this as a preprocessing step for numerical computation. Contrary to the above uses of geometry in application, we also employ numerical computation in pure mathematics. When relying on numerical methods, problems can be solved that are infeasible with symbolic methods, but the computational results lack a certificate for correctness. This often hinders the application of numerical computation with the purpose of proving mathematical theorems. With this in mind, we develop interval arithmetic as a practical tool for certification in numerical algebraic geometry.:1. Introduction 2. Certifying zeros of polynomial systems using interval arithmetic 3. Algebraic methods in decision processes 4. Discriminants and tropical implicitization

Incomplete data represents a major challenge for a successful prediction and subsequent removal of multiples. In this paper, a new method will be represented that tackles this challenge in a two-step approach. During the first step, the recenly developed curvelet-based recovery by sparsity-promoting inversion (CRSI) is applied to the data, followed by a prediction of the primaries. During the second high-resolution step, the estimated primaries are used to improve the frequency content of the recovered data by combining the focal transform, defined in terms of the estimated primaries, with the curvelet transform. This focused curvelet transform leads to an improved recovery, which can subsequently be used as input for a second stage of multiple prediction and primary-multiple separation.

Cette thèse est dédiée à améliorer la compréhension neuro-scientifique à l'aide d'imagerie par résonance magnétique de diffusion (IRMd). Nous nous concentrons sur la modélisation du signal de diffusion et l'estimation par IRMd des biomarqueurs liés à la microstructure, appelé «Microstructure Imaging». Cette thèse est organisée en trois parties. Dans partie I nous commençons par la base de l'IRMd et un aperçu de l'anisotropie en diffusion. Puis nous examinons la plupart des modèles de microstructure utilisant PGSE, en mettant l'accent sur leurs hypothèses et limites, suivi par une validation par l'histologie de la moelle épinière de leur estimation. La partie II présente nos contributions à l'imagerie en 3D et à l’estimation de microstructure. Nous proposons une régularisation laplacienne de la base fonctionnelle MAP, ce qui nous permet d'estimer de façon robuste les indices d'espace q liés au tissu. Nous appliquons cette approche aux données du Human Connectome Project, où nous l'utilisons comme prétraitement pour d'autres modèles de microstructure. Enfin, nous comparons les biomarqueurs dans une étude ex-vivo de rats Alzheimer à différents âges. La partie III présente nos contributions au représentation de l’espace qt - variant sur l'espace q 3D et le temps de diffusion. Nous présentons une approche initiale qui se concentre sur l'estimation du diamètre de l'axone depuis l'espace qt. Nous terminons avec notre approche finale, où nous proposons une nouvelle base fonctionnelle régularisée pour représenter de façon robuste le signal qt, appelé qt-IRMd. Ce qui permet l'estimation des indices d’espace q dépendants du temps, quantifiant la dépendance temporelle du signal IRMd. / This thesis is dedicated to furthering neuroscientific understanding of the human brain using diffusion-sensitized Magnetic Resonance Imaging (dMRI). Within dMRI, we focus on the estimation and interpretation of microstructure-related markers, often referred to as ``Microstructure Imaging''. This thesis is organized in three parts. Part I focuses on understanding the state-of-the-art in Microstructure Imaging. We start with the basic of diffusion MRI and a brief overview of diffusion anisotropy. We then review and compare most state-of-the-art microstructure models in PGSE-based Microstructure Imaging, emphasizing model assumptions and limitations, as well as validating them using spinal cord data with registered ground truth histology. In Part II we present our contributions to 3D q-space imaging and microstructure recovery. We propose closed-form Laplacian regularization for the recent MAP functional basis, allowing robust estimation of tissue-related q-space indices. We also apply this approach to Human Connectome Project data, where we use it as a preprocessing for other microstructure models. Finally, we compare tissue biomarkers in a ex-vivo study of Alzheimer rats at different ages. In Part III, we present our contributions to representing the qt-space - varying over 3D q-space and diffusion time. We present an initial approach that focuses on 3D axon diameter estimation from the qt-space. We end with our final approach, where we propose a novel, regularized functional basis to represent the qt-signal, which we call qt-dMRI. Our approach allows for the estimation of time-dependent q-space indices, which quantify the time-dependence of the diffusion signal.

Multivariate linear regression methods are widely used statistical tools in data analysis, and were developed when some response variables are studied simultaneously, in which our aim is to study the relationship between predictor variables and response variables through the regression coefficient matrix. The rapid improvements of information technology have brought us a large number of large-scale data, but also brought us great challenges in data processing. When dealing with high dimensional data, the classical least squares estimation is not applicable in multivariate linear regression analysis. In recent years, some approaches have been developed to deal with high-dimensional data problems, among which dimension reduction is one of the main approaches. In some literature, random projection methods were used to reduce dimension in large datasets. In Chapter 2, a new random projection method, with low-rank matrix approximation, is proposed to reduce the dimension of the parameter space in high-dimensional multivariate linear regression model. Some statistical properties of the proposed method are studied and explicit expressions are then derived for the accuracy loss of the method with Gaussian random projection and orthogonal random projection. These expressions are precise rather than being bounds up to constants. In multivariate regression analysis, reduced rank regression is also a dimension reduction method, which has become an important tool for achieving dimension reduction goals due to its simplicity, computational efficiency and good predictive performance. In practical situations, however, the performance of the reduced rank estimator is not satisfactory when the predictor variables are highly correlated or the ratio of signal to noise is small. To overcome this problem, in Chapter 3, we incorporate matrix projections into reduced rank regression method, and then develop reduced rank regression estimators based on random projection and orthogonal projection in high-dimensional multivariate linear regression models. We also propose a consistent estimator of the rank of the coefficient matrix and achieve prediction performance bounds for the proposed estimators based on mean squared errors. Envelope technology is also a popular method in recent years to reduce estimative and predictive variations in multivariate regression, including a class of methods to improve the efficiency without changing the traditional objectives. Variable selection is the process of selecting a subset of relevant features variables for use in model construction. The purpose of using this technology is to avoid the curse of dimensionality, simplify models to make them easier to interpret, shorten training time and reduce overfitting. In Chapter 4, we combine envelope models and a group variable selection method to propose an envelope-based sparse reduced rank regression estimator in high-dimensional multivariate linear regression models, and then establish its consistency, asymptotic normality and oracle property. Tensor data are in frequent use today in a variety of fields in science and engineering. Processing tensor data is a practical but challenging problem. Recently, the prevalence of tensor data has resulted in several envelope tensor versions. In Chapter 5, we incorporate envelope technique into tensor regression analysis and propose a partial tensor envelope model, which leads to a parsimonious version for tensor response regression when some predictors are of special interest, and then consistency and asymptotic normality of the coefficient estimators are proved. The proposed method achieves significant gains in efficiency compared to the standard tensor response regression model in terms of the estimation of the coefficients for the selected predictors. Finally, in Chapter 6, we summarize the work carried out in the thesis, and then suggest some problems of further research interest. / Dissertation / Doctor of Philosophy (PhD)

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Nous étudions dans cette thèse une représentation particulière de signaux fondée sur une méthode d’apprentissage statistique, qui consiste à modéliser des données comme combinaisons linéaires de quelques éléments d’un dictionnaire appris. Ceci peut être vu comme une extension du cadre classique des ondelettes, dont le but est de construire de tels dictionnaires (souvent des bases orthonormales) qui sont adaptés aux signaux naturels. Un succès important de cette approche a été sa capacité à modéliser des imagettes, et la performance des méthodes de débruitage d’images fondées sur elle. Nous traitons plusieurs questions ouvertes, qui sont reliées à ce cadre : Comment apprendre efficacement un dictionnaire ? Comment enrichir ce modèle en ajoutant une structure sous-jacente au dictionnaire ? Est-il possible d’améliorer les méthodes actuelles de traitement d’image fondées sur cette approche ? Comment doit-on apprendre le dictionnaire lorsque celui-ci est utilisé pour une tâche autre que la reconstruction de signaux ? Y a-t-il des applications intéressantes de cette méthode en vision par ordinateur ? Nous répondons à ces questions, avec un point de vue multidisciplinaire, en empruntant des outils d’apprentissage statistique, d’optimisation convexe et stochastique, de traitement des signaux et des images, de vison par ordinateur, mais aussi d'optimisation sur des graphes. / We study in this thesis a particular machine learning approach to represent signals that that consists of modelling data as linear combinations of a few elements from a learned dictionary. It can be viewed as an extension of the classical wavelet framework, whose goal is to design such dictionaries (often orthonormal basis) that are adapted to natural signals. An important success of dictionary learning methods has been their ability to model natural image patches and the performance of image denoising algorithms that it has yielded. We address several open questions related to this framework: How to efficiently optimize the dictionary? How can the model be enriched by adding a structure to the dictionary? Can current image processing tools based on this method be further improved? How should one learn the dictionary when it is used for a different task than signal reconstruction? How can it be used for solving computer vision problems? We answer these questions with a multidisciplinarity approach, using tools from statistical machine learning, convex and stochastic optimization, image and signal processing, computer vision, but also optimization on graphs.