Modal Seminar, 2013-2014 (6 sessions)

Date/Time: A11 / 14:00

Title: Sharp Thresholds for High-Dimensional and Noisy Spartsity Recovery Using l1-Constrained Quadratic Programming (Lasso) (article de Martin J. Wainwright)

Abstract: The problem of consistently estimating the sparsity pattern of a vector \beta based on observations contaminated by noise arises in various contexts, including signal denoising, sparse approximation, compressed sensing, and model selection. We analyze the behavior of l1-constrained quadratic programming (QP), also referred to as the Lasso, for recovering the sparsity pattern. Our main result is to establish precise conditions on the problem dimension p, the number of nonzero elements k, in \beta^*, and the number of observations n that are necessary and sufficient for sparsity pattern recovery using the Lasso. We first analyze the case of observations made using deterministic design matrices and sub-Gaussian additive noise, and provide sufficient conditions for support recovery and l_\infty-error bounds, as well as results showing the necessity of incoherence and bounds on the minimum value. We then turn to the case of random designs, in which each row of the design is drawn from a N(0,Sigma) ensemble. For a broad class of Gaussian ensembles satisfying mutual incoherence conditions, we compute explicit values of thresholds 0<\theta_l(\Sigma)<\theta_u(\Sigma)<+\infty with the following properties: for any delta>0, if n>2(\theta_u+\delta)k log(p-k), then the Lasso succeeds in recovering the sparsity pattern with probability converging to one for large problems, whereas for n<2(\theta_l – \delta) k log(p-k), then the probability of successful recovery converges to zero. For the special case of the uniform Gaussian ensemble, we show that \theta_u=\theta_l=1, so that the precise threshold n=2k log(p-k) is exactly determined.

Room: A11

Title: Clustering dans des modèles de graphes aléatoires à variables latentes

Abstract: L’hétérogénéité dans les réseaux peut être modélisée en attribuant à chaque noeud une couleur ou une position dans un espace latent. Dans le cas des couleurs, les probabilités de connexion entre noeuds ne dépendent que des couleurs des noeuds, qui correspondent ainsi à des profils sociaux dans le réseau. Une question importante est en particulier celle de la classification non supervisée, visant à retrouver les couleurs à partir du réseau observé. Dans le cas des positions, on suppose que les arêtes sont d’autant plus probables que les sommets sont proches selon une métrique donnée. Le problème posé est de retrouver la structure en clusters de la densité des positions latentes à partir uniquement du réseau observé. On utilise pour cela les composantes connexes de sous-graphes biens choisis. La problématique commune à ces deux problèmes est le développement d’algorithmes rapides et consistents dans ces modèles, notamment pour traiter efficacement de grands graphes.

Room: plénière

Title: Classification supervisée de processus de Cox

Abstract: Dans cet exposé, nous discuterons du problème de la classification supervisée de processus de Cox. Les processus de Cox, qui sont en particulier des processus de comptage, ont vocation à modéliser, par exemple, le nombre de visites d’un patient à l’hôpital au cours du temps et leur étude présente de nombreuses applications en Biologie. Notre étude est basée sur un calcul explicite de la règle de Bayes (i.e. la règle de classification optimale). Nous proposons une stratégie de classification de type boosting : la règle de classification proposée minimise un critère empirique convexe régularisé. Nous présenterons une inégalité oracle pour évaluer la performance de notre règle de classification. D’autre part, nous montrerons que cette règle de classification converge vers la règle de Bayes à une vitesse qui s’adapte à la régularité inconnue de l’intensité du processus. Ce travail a été effectué en collaboration avec Gérard Biau et Benoît Cadre.

Room: plénière

Title: Régression multivariée régularisée avec variance inconnue et integration d’information a priori

Room: A11

Title: Statistical modeling and classification of Multivariate Functional Data: an application to ECG signals

Abstract: Nowadays data coming from a biomedical context are frequently functions produced by medical devices. This calls for the identification of suitable inferential techniques for managing their complexity. In the research works that will be presented, some inferential tools for dealing with classification and prediction in the context of multivariate functional data, i.e., data where each observation is a set of possibly correlated functions, are proposed. We aim at modeling the binary outcome of interest (the presence of cardiovascular ischaemic event), estimating the probability of each patient to be affected by acute myocardial infarction. We are also interest in the assignment of a curve to a specific disease/group, i.e., in carrying out semi-automatic diagnosis starting from purely statistical assessments. We then first use the 8-leads ECG trace of each patient and its first derivative as multivariate functional predictor in a generalized regression model, reducing their dimensionality via Multivariate Functional PCA, in order to identify a robust base of coefficients that enables a reliable prediction for a new patient entering the study. On the other hand, we summarize data using an extension of Depth Measure to the multivariate functional setting. In particular, to define a multivariate depth measure, we weight the contribution of each component according to the estimated correlation among data components themselves. Data depths are then used within the regression model aimed at predicting patient’s status. A generalization of the k-means algorithm to the multivariate functional case is also adopted in order to classify signals in different groups of diseases.

Room: A11

Title: Scaling up MCMC: a subsampling approach

Abstract: Markov chain Monte Carlo (MCMC) methods are often deemed far too computationally intensive to be of any practical use for large datasets. In this talk, I will describe a methodology that aims to scale up the Metropolis-Hastings (MH) algorithm in this context. We propose an approximate implementation of the accept/reject step of MH based on concentration inequalities, which only requires evaluating the likelihood of a random subset of the data, yet is guaranteed to coincide with the accept/reject step based on the full dataset with a probability superior to a user-specified tolerance level. This adaptive subsampling technique is an alternative to the recent approach developed in (Korattikara et al., to appear in ICML’14), and it allows to establish rigorously that the resulting approximate MH algorithm samples from a perturbed version of the target distribution of interest. Furthermore, the total variation distance between this perturbed target and the target of interest is controlled explicitely. I will demonstrate the benefits and limitations of this scheme on several examples.