The first part of this dissertation focuses on matrix-covariate regression models. While they have been studied in many existing works, classical statistical and computational methods for the analysis of the regression coefficient estimation are highly affected by high dimensional matrix-valued covariates. To address these issues, we proposes a framework of matrix-covariate regression models based on a low-rank constraint and an additional regularization for structured signals, with considerations of models of both continuous and binary responses. In the second part, we examine a Mixture Multilayer Stochastic Block Model (MMLSBM), where layers can be grouped into sets of similar networks. Each group of networks is endowed with a unique Stochastic Block Model. The objective is to partition the multilayer network into clusters of similar layers and identify communities within those layers. We present an alternative approach called the Alternating Minimization Algorithm (ALMA), which aims to simultaneously recover the layer partition and estimate the matrices of connection probabilities for the distinct layers. In the last part, we demonstrates the effectiveness of the projected gradient descent algorithm. Firstly, its local convergence rate is independent of the condition number. Secondly, under conditions where the objective function is rank-2r restricted L-smooth and μ-strongly convex, with L/μ < 3, projected gradient descent with appropriate step size converges linearly to the solution. Moreover, a perturbed version of this algorithm effectively navigates away from saddle points, converging to an approximate solution or a second-order local minimizer across a wide range of step sizes. Furthermore, we establish that there are no spurious local minimizes in estimating asymmetric low-rank matrices when the objective function satisfies L/μ < 3.

Nous nous intéressons, dans cette thèse, à l'étude des modèles de mélange et des modèles linéaires généralisés, avec une application aux données de co-infection entre les arbovirus et les parasites du paludisme. Après une première partie consacrée à l'étude de la co-infection par un modèle logistique multinomial, nous proposons dans une deuxième partie l'étude des mélanges de modèles linéaires généralisés. La méthode proposée pour estimer les paramètres du mélange est une combinaison d'une méthode des moments et d'une méthode spectrale. Nous proposons à la fin une dernière partie consacrée aux mélanges de valeurs extrêmes en présence de censure. La méthode d'estimation proposée dans cette partie se fait en deux étapes basées sur la maximisation d'une vraisemblance. / We are interested, in this thesis, to the study of mixture models and generalized linear models, with an application to co-infection data between arboviruses and malaria parasites. After a first part dedicated to the study of co-infection using a multinomial logistic model, we propose in a second part to study the mixtures of generalized linear models. The proposed method to estimate the parameters of the mixture is a combination of a moment method and a spectral method. Finally, we propose a final section for studing extreme value mixtures under random censoring. The estimation method proposed in this section is done in two steps based on the maximization of a likelihood.