Network analysis provides a rich framework to model complex phenomena, such as human brain connectivity. It has proven efficient to understand their natural properties and design predictive models. In this paper, we study the variability within groups of networks, i.e., the structure of connection similarities and differences across a set of networks. We propose a statistical framework to model these variations based on manifold-valued latent factors. Each network adjacency matrix is decomposed as a weighted sum of matrix patterns with rank one. Each pattern is described as a random perturbation of a dictionary element. As a hierarchical statistical model, it enables the analysis of heterogeneous populations of adjacency matrices using mixtures. Our framework can also be used to infer the weight of missing edges. We estimate the parameters of the model using an Expectation-Maximization-based algorithm. Experimenting on synthetic data, we show that the algorithm is able to accurately estimate the latent structure in both low and high dimensions. We apply our model on a large data set of functional brain connectivity matrices from the UK Biobank. Our results suggest that the proposed model accurately describes the complex variability in the data set with a small number of degrees of freedom.

Variational auto-encoders (VAEs) have proven to be a well suited tool for performing dimensionality reduction by extracting latent variables lying in a potentially much smaller dimensional space than the data. Their ability to capture meaningful information from the data can be easily apprehended when considering their capability to generate new realistic samples or perform potentially meaningful interpolations in a much smaller space. However, such generative models may perform poorly when trained on small data sets which are abundant in many real-life fields such as medicine. This may, among others, come from the lack of structure of the latent space, the geometry of which is often under-considered. We thus propose in this paper to see the latent space as a Riemannian manifold endowed with a parametrized metric learned at the same time as the encoder and decoder networks. This metric is then used in what we called the Riemannian Hamiltonian VAE which extends the Hamiltonian VAE introduced by arXiv:1805.11328 to better exploit the underlying geometry of the latent space. We argue that such latent space modelling provides useful information about its underlying structure leading to far more meaningful interpolations, more realistic data-generation and more reliable clustering.

Matrix data sets arise in network analysis for medical applications, where each network belongs to a subject and represents a measurable phenotype. These large dimensional data are often modeled using lower-dimensional latent variables, which explain most of the observed variability and can be used for predictive purposes. In this paper, we provide asymptotic convergence guarantees for the estimation of a hierarchical statistical model for matrix data sets. It captures the variability of matrices by modeling a truncation of their eigendecomposition. We show that this model is identifiable, and that consistent Maximum A Posteriori (MAP) estimation can be performed to estimate the distribution of eigenvalues and eigenvectors. The MAP estimator is shown to be asymptotically normal for a restricted version of the model.

Modélisation statistique et inférence pour les populations de réseaux de connectivité cérébrale et les données longitudinales: Le développement et la massification des bases de données d'imagerie médicale et de suivi clinique ouvrent de nouvelles perspectives pour la compréhension de phénomènes complexes comme le vieillissement ou les maladies neurodégénératives. En particulier, la connectivité cérébrale, c'est-à-dire l'étude des connexions et des interactions entre les régions du cerveau, peut maintenant être étudiée à l'échelle d'une population et non plus d'un individu isolé. Ce cadre offre la possibilité d'une meilleure prise en compte des spécificités individuelles dans le développement d'outils de suivi. Dans cette thèse, nous proposons dans un premier temps de nouvelles approches pour modéliser et comprendre la variabilité de la connectivité cérébrale au sein d'un groupe de sujets. Plus généralement, nous nous intéressons aux collections de réseaux où chaque réseau décrit des interactions entre les mêmes entités. Nous nous appuyons sur la propriété empirique de rang faible des matrices d'adjacence de ces réseaux pour rendre compte de leur distribution. Nous proposons deux approches, l'une variationnelle et l'autre statistique, pour rendre compte de l'hétérogénéité de ces matrices. En particulier, dans le second cas, nous montrons qu'un nombre restreint de paramètres suffit à donner une description fidèle et interprétable de la variabilité de la connectivité cérébrale. Nous montrons également la consistence et l'identifiabilité de notre approche sur le plan théorique. Dans un second temps, nous étudions un modèle longitudinal pour le suivi de la progression de la maladie de Parkinson. Dans ce modèle, la trajectoire de chaque patient est divisée en plusieurs morceaux pouvant correspondre aux différentes phases de la maladie ou d'un traitement. Nous nous montrons qu'il est possible d'estimer les trajectoires constituées de plusieurs morceaux, et de sélectionner le nombre de ruptures le mieux adapté pour décrire l'évolution moyenne de la population.

Variational auto-encoders (VAEs) have proven to be a well suited tool for performing dimensionality reduction by extracting latent variables lying in a potentially much smaller dimensional space than the data. Their ability to capture meaningful information from the data can be easily apprehended when considering their capability to generate new realistic samples or perform potentially meaningful interpolations in a much smaller space. However, such generative models may perform poorly when trained on small data sets which are abundant in many real-life fields such as medicine. This may, among others, come from the lack of structure of the latent space, the geometry of which is often under-considered. We thus propose in this paper to see the latent space as a Riemannian manifold endowed with a parametrized metric learned at the same time as the encoder and decoder networks. This metric is then used in what we called the Riemannian Hamiltonian VAE which extends the Hamiltonian VAE introduced by Caterini et al. (2018) to better exploit the underlying geometry of the latent space. We argue that such latent space modelling provides useful information about its underlying structure leading to far more meaningful interpolations, more realistic data-generation and more reliable clustering.