We study a boundary value problem for an overdetermined elliptic system of nonlinear first order differential equations with linear boundary operators. Such a problem is solvable for a small set of data, and so we pass to its variational formulation which consists in minimising the discrepancy. The Euler-Lagrange equations for the variational problem are far-reaching analogues of the classical Laplace equation. Within the framework of Euler-Lagrange equations we specify an operator on the boundary whose zero set consists precisely of those boundary data for which the initial problem is solvable. The construction of such operator has much in common with that of the familiar Dirichlet to Neumann operator. In the case of linear problems we establish complete results.

<p>Bayesian nonparametric methods are useful for modeling data without having to define the complexity of the entire model <italic>a priori</italic>, but rather allowing for this complexity to be determined by the data. Two problems considered in this dissertation are the number of components in a mixture model, and the number of factors in a latent factor model, for which the Dirichlet process and the beta process are the two respective Bayesian nonparametric priors selected for handling these issues.</p> <p>The flexibility of Bayesian nonparametric priors arises from the prior's definition over an infinite dimensional parameter space. Therefore, there are theoretically an <italic>infinite</italic> number of latent components and an <italic>infinite</italic> number of latent factors. Nevertheless, draws from each respective prior will produce only a small number of components or factors that appear in a given data set. As mentioned, the number of these components and factors, and their corresponding parameter values, are left for the data to decide.</p> <p>This dissertation is split between novel practical applications and novel theoretical results for these priors. For the Dirichlet process, we investigate stick-breaking representations for the finite Dirichlet process and their application to novel sampling techniques, as well as a novel mixture modeling framework that incorporates multiple modalities within a data set. For the beta process, we present a new stick-breaking construction for the infinite-dimensional prior, and consider applications to image interpolation problems and dictionary learning for compressive sensing.</p> / Dissertation

<p>In most machine learning approaches, it is usually assumed that data are complete. When data are partially missing due to various reasons, for example, the failure of a subset of sensors, image corruption or inadequate medical measurements, many learning methods designed for complete data cannot be directly applied. In this dissertation we treat two kinds of problems with incomplete data using non-parametric Bayesian approaches: classification with incomplete features and analysis of low-rank matrices with missing entries.</p><p>Incomplete data in classification problems are handled by assuming input features to be generated from a mixture-of-experts model, with each individual expert (classifier) defined by a local Gaussian in feature space. With a linear classifier associated with each Gaussian component, nonlinear classification boundaries are achievable without the introduction of kernels. Within the proposed model, the number of components is theoretically ``infinite'' as defined by a Dirichlet process construction, with the actual number of mixture components (experts) needed inferred based upon the data under test. With a higher-level DP we further extend the classifier for analysis of multiple related tasks (multi-task learning), where model components may be shared across tasks. Available data could be augmented by this way of information transfer even when tasks are only similar in some local regions of feature space, which is particularly critical for cases with scarce incomplete training samples from each task. The proposed algorithms are implemented using efficient variational Bayesian inference and robust performance is demonstrated on synthetic data, benchmark data sets, and real data with natural missing values.</p><p>Another scenario of interest is to complete a data matrix with entries missing. The recovery of missing matrix entries is not possible without additional assumptions on the matrix under test, and here we employ the common assumption that the matrix is low-rank. Unlike methods with a preset fixed rank, we propose a non-parametric Bayesian alternative based on the singular value decomposition (SVD), where missing entries are handled naturally, and the number of underlying factors is imposed to be small and inferred in the light of observed entries. Although we assume missing at random, the proposed model is generalized to incorporate auxiliary information including missingness features. We also make a first attempt in the matrix-completion community to acquire new entries actively. By introducing a probit link function, we are able to handle counting matrices with the decomposed low-rank matrices latent. The basic model and its extensions are validated on</p><p>synthetic data, a movie-rating benchmark and a new data set presented for the first time.</p> / Dissertation

<p>The immune system is constantly faced with the daunting task of protecting the host from a large number of ever-evolving pathogens. In vertebrates, the immune response results from the interplay of two cellular systems: the innate immunity and the adaptive immunity. In the past decades, dendritic cells have emerged as major players in the modulation of the immune response, being one of the primary links between these two branches of the immune system.</p><p>Dendritic cells are pathogen-sensing cells that alert the rest of the immune system of the presence of infection. The signals sent by dendritic cells result in the recruitment of the appropriate cell types and molecules required for effectively clearing the infection. A question of utmost importance in our understanding of the immune response and our ability to manipulate it in the development of vaccines and therapies is: "How do dendritic cells translate the various cues they perceive from the environment into different signals that specifically activate the appropriate parts of the immune system that result in an immune response streamlined to clear the given pathogen?"</p><p>Here we have developed computational and statistical methods aimed to address specific aspects of this question. In particular, understanding how dendritic cells ultimately modulate the immune response requires an understanding of the subtleties of their maturation process in response to different environmental signals. Hence, the first part of this dissertation focuses on elucidating the changes in the transcriptional</p><p>program of dendritic cells in response to the detection of two common pathogen- associated molecules, LPS and CpG. We have developed a method based on Langevin and Dirichlet processes to model and cluster gene expression temporal data, and have used it to identify, on a large scale, genes that present unique and common transcriptional behaviors in response to these two stimuli. Additionally, we have also investigated a different, but related, aspect of dendritic cell modulation of the adaptive immune response. In the second part of this dissertation, we present a method to predict peptides that will bind to MHC molecules, a requirement for the activation of pathogen-specific T cells. Together, these studies contribute to the elucidation of important aspects of dendritic cell biology.</p> / Dissertation

Accompanying with the rapid growth of Internet, people around the world can easily distribute, browse, and share as much information as possible through the Internet. The enormous amount of information, however, causes the information overload problem that is beyond users¡¦ limited information processing ability. Therefore, recommender systems arise to help users to look for useful information when they cannot describe the requirements precisely. The filtering techniques in recommender systems can be divided into content-based filtering (CBF) and collaborative filtering (CF). Although CF is shown to be superior over CBF in literature, personalized document recommendation relies more on CBF simply because of its text content in nature. Nevertheless, document recommendation task provides a good chance to integrate both techniques into a hybrid one, and enhance the overall recommendation performance. The objective of this research is thus to propose a hybrid filtering approach for personalized document recommendation. Particularly, latent Dirichlet allocation to uncover latent semantic structure in documents is incorporated to help us to either obtain robust document similarity in CF, or explore user profiles in CBF. Two experiments are conducted accordingly. The results show that our proposed approach outperforms other counterparts on the recommendation performance, which justifies the feasibility of our proposed approach in real applications.

We are concerned with testing for differential expression and consider three different aspects of such testing procedures. First, we develop an exact ANOVA type model for discrete gene expression data, produced by technologies such as a Massively Parallel Signature Sequencing (MPSS), Serial Analysis of Gene Expression (SAGE) or other next generation sequencing technologies. We adopt two Bayesian hierarchical models—one parametric and the other semiparametric with a Dirichlet process prior that has the ability to borrow strength across related signatures, where a signature is a specific arrangement of the nucleotides. We utilize the discreteness of the Dirichlet process prior to cluster signatures that exhibit similar differential expression profiles. Tests for differential expression are carried out using non-parametric approaches, while controlling the false discovery rate. Next, we consider ways to combine expression data from different studies, possibly produced by different technologies resulting in mixed type responses, such as Microarrays and MPSS. Depending on the technology, the expression data can be continuous or discrete and can have different technology dependent noise characteristics. Adding to the difficulty, genes can have an arbitrary correlation structure both within and across studies. Performing several hypothesis tests for differential expression could also lead to false discoveries. We propose to address all the above challenges using a Hierarchical Dirichlet process with a spike-and-slab base prior on the random effects, while smoothing splines model the unknown link functions that map different technology dependent manifestations to latent processes upon which inference is based. Finally, we propose an algorithm for controlling different error measures in a Bayesian multiple testing under generic loss functions, including the widely used uniform loss function. We do not make any specific assumptions about the underlying probability model but require that indicator variables for the individual hypotheses are available as a component of the inference. Given this information, we recast multiple hypothesis testing as a combinatorial optimization problem and in particular, the 0-1 knapsack problem which can be solved efficiently using a variety of algorithms, both approximate and exact in nature.

In this thesis a number of improvements have been established for specific methods which utilize sequential Monte Carlo (SMC), aka. Particle filtering (PF) techniques. The first problem is the Bayesian multi-target tracking (MTT) problem for which we propose the use of non-parametric Bayesian models that are based on time varying extension of Dirichlet process (DP) models. The second problem studied in this thesis is an important application area for the proposed DP based MTT method / the tracking of vocal tract resonance frequencies of the speech signals. Lastly, we investigate SMC based parameter estimation problem of nonlinear non-Gaussian state space models in which we provide a performance improvement for the path density based methods by utilizing regularization techniques.

The increased collection of high-dimensional data in various fields has raised a strong interest in clustering algorithms and variable selection procedures. In this disserta- tion, I propose a model-based method that addresses the two problems simultane- ously. I use Dirichlet process mixture models to define the cluster structure and to introduce in the model a latent binary vector to identify discriminating variables. I update the variable selection index using a Metropolis algorithm and obtain inference on the cluster structure via a split-merge Markov chain Monte Carlo technique. I evaluate the method on simulated data and illustrate an application with a DNA microarray study. I also show that the methodology can be adapted to the problem of clustering functional high-dimensional data. There I employ wavelet thresholding methods in order to reduce the dimension of the data and to remove noise from the observed curves. I then apply variable selection and sample clustering methods in the wavelet domain. Thus my methodology is wavelet-based and aims at clustering the curves while identifying wavelet coefficients describing discriminating local features. I exemplify the method on high-dimensional and high-frequency tidal volume traces measured under an induced panic attack model in normal humans.

We study the Ambarzumyan problem for Sturm-Liouville operator defined on graph. The classical Ambarzumyan Theorem states that if the Neumann eigenvalues of the Sturm-Liouville operator defined on the interval [0,£k] are exactly {n^2: n ∈ N ⋃ {0} }, then the potential q=0. In 2005, Pivovarchik proved two similar theorems with uniform lengths a for the Sturm-Liouville operator defined on a 3-star graphs. Then Wu considered the Ambarzumyan problem for graphs of nonuniform length in his thesis. In this thesis, we shall study the Ambarzumyan problem on more complicated trees, namely, 4-star graphs and caterpillar graphs with edges of different lengths. We manage to solve the Ambarzumyan problem for both Neumann eigenvalues and Dirichlet eigenvalues. In particular, the whole spectrum can be partitioned into several parts. Each part forms the solution to one Ambarzumyan problem. For example, for a 4-star graphs with edge lengths a, a, 2a, 2a form the solution to 3 different Ambarzumyan problems for the Neumann eigenvalues.

Thèse de doctorat : Physique : Metz : 1999. / 1999METZ047S. 53 ref.