The bloom of economics and technology has had an enormous impact on society. Along with these developments, human activities nowadays produce massive amounts of data that can be easily collected for relatively low cost with the aid of new technologies. Many examples can be mentioned here including data from web term-document data, sensor arrays, gene expression, finance data, imaging and hyperspectral analysis. Because of the enormous amount of data from various different and new sources, more and more challenging scientific problems appear. These problems have changed the types of problems which mathematical scientists work. / In traditional statistics, the dimension of the data, p say, is low, with many observations, n say. In this case, classical rules such as the Central Limit Theorem are often applied to obtain some understanding from data. A new challenge to statisticians today is dealing with a different setting, when the data dimension is very large and the number of observations is small. The mathematical assumption now could be p > n, or even p goes to infinity and n fixed in many cases, for example, there are few patients with many genes. In these cases, classical methods fail to produce a good understanding of the nature of the problem. Hence, new methods need to be found to solve these problems. Mathematical explanations are also needed to generalize these cases. / The research preferred in this thesis includes two problems: Variable selection and Classification, in the case where the dimension is very large. The work on variable selection problems, in particular the Adaptive Lasso was completed by June 2007 and the research on classification has been carried out through out 2008 and 2009. The research on the Dantzig selector and the Lasso were finished in July 2009. Therefore, this thesis is divided into two parts. In the first part of the thesis we study the Adaptive Lasso, the Lasso and the Dantzig selector. In particular, in Chapter 2 we present some results for the Adaptive Lasso. Chapter 3 will provides two examples that show that neither the Dantzig selector or the Lasso is definitely better than the other. The second part of the thesis is organized as follows. In Chapter 5, we shall construct the model setting. In Chapter 6, we summarize the results of the scaled centroid-based classifier. We also prove some results on the scaled centroid-based classifier. Because there are similarities between the Support Vector Machine (SVM) and Distance Weighted Discrimination (DWD) classifiers, Chapter 8 introduces a class of distance-based classifiers that could be considered a generalization of the SVM and DWD classifiers. Chapters 9 and 10 are about the SVM and DWD classifiers. Chapter 11 demonstrates the performance of these classifiers on simulated data sets and some cancer data sets.

The bloom of economics and technology has had an enormous impact on society. Along with these developments, human activities nowadays produce massive amounts of data that can be easily collected for relatively low cost with the aid of new technologies. Many examples can be mentioned here including data from web term-document data, sensor arrays, gene expression, finance data, imaging and hyperspectral analysis. Because of the enormous amount of data from various different and new sources, more and more challenging scientific problems appear. These problems have changed the types of problems which mathematical scientists work. / In traditional statistics, the dimension of the data, p say, is low, with many observations, n say. In this case, classical rules such as the Central Limit Theorem are often applied to obtain some understanding from data. A new challenge to statisticians today is dealing with a different setting, when the data dimension is very large and the number of observations is small. The mathematical assumption now could be p > n, or even p goes to infinity and n fixed in many cases, for example, there are few patients with many genes. In these cases, classical methods fail to produce a good understanding of the nature of the problem. Hence, new methods need to be found to solve these problems. Mathematical explanations are also needed to generalize these cases. / The research preferred in this thesis includes two problems: Variable selection and Classification, in the case where the dimension is very large. The work on variable selection problems, in particular the Adaptive Lasso was completed by June 2007 and the research on classification has been carried out through out 2008 and 2009. The research on the Dantzig selector and the Lasso were finished in July 2009. Therefore, this thesis is divided into two parts. In the first part of the thesis we study the Adaptive Lasso, the Lasso and the Dantzig selector. In particular, in Chapter 2 we present some results for the Adaptive Lasso. Chapter 3 will provides two examples that show that neither the Dantzig selector or the Lasso is definitely better than the other. The second part of the thesis is organized as follows. In Chapter 5, we shall construct the model setting. In Chapter 6, we summarize the results of the scaled centroid-based classifier. We also prove some results on the scaled centroid-based classifier. Because there are similarities between the Support Vector Machine (SVM) and Distance Weighted Discrimination (DWD) classifiers, Chapter 8 introduces a class of distance-based classifiers that could be considered a generalization of the SVM and DWD classifiers. Chapters 9 and 10 are about the SVM and DWD classifiers. Chapter 11 demonstrates the performance of these classifiers on simulated data sets and some cancer data sets.

Les travaux de cette thèse ont été initiés par des problèmes d’apprentissage de données radiomiques. La Radiomique est une discipline médicale qui vise l’analyse à grande échelle de données issues d’imageries médicales traditionnelles, pour aider au diagnostic et au traitement des cancers. L’hypothèse principale de cette discipline est qu’en extrayant une grande quantité d’informations des images, on peut caractériser de bien meilleure façon que l’œil humain les spécificités de cette pathologie. Pour y parvenir, les données radiomiques sont généralement constituées de plusieurs types d’images et/ou de plusieurs types de caractéristiques (images, cliniques, génomiques). Cette thèse aborde ce problème sous l’angle de l’apprentissage automatique et a pour objectif de proposer une solution générique, adaptée à tous problèmes d’apprentissage du même type. Nous identifions ainsi en Radiomique deux problématiques d’apprentissage: (i) l’apprentissage de données en grande dimension et avec peu d’instances (high dimension, low sample size, a.k.a.HDLSS) et (ii) l’apprentissage multi-vues. Les solutions proposées dans ce manuscrit exploitent des représentations de dissimilarités obtenues à l’aide des Forêts Aléatoires. L’utilisation d’une représentation par dissimilarité permet de contourner les difficultés inhérentes à l’apprentissage en grande dimension et facilite l’analyse conjointe des descriptions multiples (les vues). Les contributions de cette thèse portent sur l’utilisation de la mesure de dissimilarité embarquée dans les méthodes de Forêts Aléatoires pour l’apprentissage multi-vue de données HDLSS. En particulier, nous présentons trois résultats: (i) la démonstration et l’analyse de l’efficacité de cette mesure pour l’apprentissage multi-vue de données HDLSS; (ii) une nouvelle méthode pour mesurer les dissimilarités à partir de Forêts Aléatoires, plus adaptée à ce type de problème d’apprentissage; et (iii) une nouvelle façon d’exploiter l’hétérogénèité des vues, à l’aide d’un mécanisme de combinaison dynamique. Ces résultats ont été obtenus sur des données radiomiques mais aussi sur des problèmes multi-vue classiques. / The work of this thesis was initiated by a Radiomic learning problem. Radiomics is a medical discipline that aims at the large-scale analysis of data from traditional medical imaging to assist in the diagnosis and treatment of cancer. The main hypothesis of this discipline is that by extracting a large amount of information from the images, we can characterize the specificities of this pathology in a much better way than the human eye. To achieve this, Radiomics data are generally based on several types of images and/or several types of features (from images, clinical, genomic). This thesis approaches this problem from the perspective of Machine Learning (ML) and aims to propose a generic solution, adapted to any similar learning problem. To do this, we identify two types of ML problems behind Radiomics: (i) learning from high dimension, low sample size (HDLSS) and (ii) multiview learning. The solutions proposed in this manuscript exploit dissimilarity representations obtained using the Random Forest method. The use of dissimilarity representations makes it possible to overcome the well-known difficulties of learning high dimensional data, and to facilitate the joint analysis of the multiple descriptions, i.e. the views.The contributions of this thesis focus on the use of the dissimilarity easurement embedded in the Random Forest method for HDLSS multi-view learning. In particular, we present three main results: (i) the demonstration and analysis of the effectiveness of this measure for HDLSS multi-view learning; (ii) a new method for measuring dissimilarities from Random Forests, better adapted to this type of learning problem; and (iii) a new way to exploit the heterogeneity of views, using a dynamic combination mechanism. These results have been obtained on radiomic data but also on classical multi-view learning problems.