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51	Transport optimal de mesures positives : modèles, méthodes numériques, applications / Unbalanced Optimal Transport : Models, Numerical Methods, Applications Chizat, Lénaïc 10 November 2017 (has links) L'objet de cette thèse est d'étendre le cadre théorique et les méthodes numériques du transport optimal à des objets plus généraux que des mesures de probabilité. En premier lieu, nous définissons des modèles de transport optimal entre mesures positives suivant deux approches, interpolation et couplage de mesures, dont nous montrons l'équivalence. De ces modèles découle une généralisation des métriques de Wasserstein. Dans une seconde partie, nous développons des méthodes numériques pour résoudre les deux formulations et étudions en particulier une nouvelle famille d'algorithmes de "scaling", s'appliquant à une grande variété de problèmes. La troisième partie contient des illustrations ainsi que l'étude théorique et numérique, d'un flot de gradient de type Hele-Shaw dans l'espace des mesures. Pour les mesures à valeurs matricielles, nous proposons aussi un modèle de transport optimal qui permet un bon arbitrage entre fidélité géométrique et efficacité algorithmique. / This thesis generalizes optimal transport beyond the classical "balanced" setting of probability distributions. We define unbalanced optimal transport models between nonnegative measures, based either on the notion of interpolation or the notion of coupling of measures. We show relationships between these approaches. One of the outcomes of this framework is a generalization of the p-Wasserstein metrics. Secondly, we build numerical methods to solve interpolation and coupling-based models. We study, in particular, a new family of scaling algorithms that generalize Sinkhorn's algorithm. The third part deals with applications. It contains a theoretical and numerical study of a Hele-Shaw type gradient flow in the space of nonnegative measures. It also adresses the case of measures taking values in the cone of positive semi-definite matrices, for which we introduce a model that achieves a balance between geometrical accuracy and algorithmic efficiency. Transport optimal Analyse convexe Optimisation Mesures positives Géométrie de l'information Algorithme de Sinkhorn Traitement d'image Flots de gradient Convergence faible Traitement de champs de tenseurs Barycentres Entropie relative Espace métrique géodésique Optimal transport Convex analysis Optimization Nonnegative measures Information geometry Sinkhorn's algorithm Image processing Gradient flows Weak convergence Tensor field processing Barycentres Relative entropy Geodesic metric space 519
52	Extensions of nonnegative matrix factorization for exploratory data analysis / 探索的なデータ分析のための非負値行列因子分解の拡張 / タンサクテキナデータブンセキノタメノヒフチギョウレツインシブンカイノカクチョウ阿部寛康, Hiroyasu Abe 22 March 2017 (has links) 非負値行列因子分解(NMF)は，全要素が非負であるデータ行列に対する行列分解法である．本論文では，実在するデータ行列に頻繁に見られる特徴や解釈容易性の向上を考慮に入れ，探索的にデータ分析を行うためのNMFの拡張について論じている．具体的には，零過剰行列や外れ値を含む行列を扱うための確率分布やダイバージェンス，さらには分解結果である因子行列の数や因子行列への直交制約について述べている． / Nonnegative matrix factorization (NMF) is a matrix decomposition technique to analyze nonnegative data matrices, which are matrices of which all elements are nonnegative. In this thesis, we discuss extensions of NMF for exploratory data analysis considering common features of a real nonnegative data matrix and an easy interpretation. In particular, we discuss probability distributions and divergences for zero-inflated data matrix and data matrix with outliers, two-factor vs. three-factor, and orthogonal constraint to factor matrices. / 博士(文化情報学) / Doctor of Culture and Information Science / 同志社大学 / Doshisha University 非負値行列因子分解 Tweedie分布零過剰複合ポアソン分布直交制約 3因子分解 nonnegative matrix factorization Tweedie distribution zero-inflation compound Poisson distribution orthogonal constraint tri-factorization
53	Some Advanced Model Selection Topics for Nonparametric/Semiparametric Models with High-Dimensional Data Fang, Zaili 13 November 2012 (has links) Model and variable selection have attracted considerable attention in areas of application where datasets usually contain thousands of variables. Variable selection is a critical step to reduce the dimension of high dimensional data by eliminating irrelevant variables. The general objective of variable selection is not only to obtain a set of cost-effective predictors selected but also to improve prediction and prediction variance. We have made several contributions to this issue through a range of advanced topics: providing a graphical view of Bayesian Variable Selection (BVS), recovering sparsity in multivariate nonparametric models and proposing a testing procedure for evaluating nonlinear interaction effect in a semiparametric model. To address the first topic, we propose a new Bayesian variable selection approach via the graphical model and the Ising model, which we refer to the ``Bayesian Ising Graphical Model'' (BIGM). There are several advantages of our BIGM: it is easy to (1) employ the single-site updating and cluster updating algorithm, both of which are suitable for problems with small sample sizes and a larger number of variables, (2) extend this approach to nonparametric regression models, and (3) incorporate graphical prior information. In the second topic, we propose a Nonnegative Garrote on a Kernel machine (NGK) to recover sparsity of input variables in smoothing functions. We model the smoothing function by a least squares kernel machine and construct a nonnegative garrote on the kernel model as the function of the similarity matrix. An efficient coordinate descent/backfitting algorithm is developed. The third topic involves a specific genetic pathway dataset in which the pathways interact with the environmental variables. We propose a semiparametric method to model the pathway-environment interaction. We then employ a restricted likelihood ratio test and a score test to evaluate the main pathway effect and the pathway-environment interaction. / Ph. D. Variable Selection Smoothing Splines Sparsistency Semiparametric Model Pathway Analysis Additive Model Cluster Algorithm Gaussian Random Process Global-Local Shrinkage Graphical Model Ising Model Kernel Machine KM Model LASSO Long Tail Prior Mixture Normals Model Selection Multivariate Smoothing Function Nonnegative Garrote Nonparametric Model

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