DNA microarrays are capable of measuring expression levels of thousands of genes, even the whole genome, in a single experiment. Based on this, they have been widely used to extend the studies of cancerous tissues to a genomic level. One of the main goals in DNA microarray experiments is to identify a set of relevant genes such that the desired outputs of the experiment mostly depend on this set, to the exclusion of the rest of the genes. This is motivated by the fact that the biological process in cell typically involves only a subset of genes, and not the whole genome. The task of selecting a subset of relevant genes is called feature (gene) selection. Herein, we propose a feature selection algorithm for gene expression data. It is based on the Hilbert-Schmidt independence criterion, and partly motivated by Rank-One Downdate (R1D) and the Singular Value Decomposition (SVD). The algorithm is computationally very fast and scalable to large data sets, and can be applied to response variables of arbitrary type (categorical and continuous). Experimental results of the proposed technique are presented on some synthetic and well-known microarray data sets. Later, we discuss the capability of HSIC in providing a general framework which encapsulates many widely used techniques for dimensionality reduction, clustering and metric learning. We will use this framework to explain two metric learning algorithms, namely the Fisher discriminant analysis (FDA) and closed form metric learning (CFML). As a result of this framework, we are able to propose a new metric learning method. The proposed technique uses the concepts from normalized cut spectral clustering and is associated with an underlying convex optimization problem.

The basic philosophy of Functional Data Analysis (FDA) is to think of the observed data functions as elements of a possibly infinite-dimensional function space. Most of the current research topics on FDA focus on advancing theoretical tools and extending existing multivariate techniques to accommodate the infinite-dimensional nature of data. This dissertation reports contributions on both fronts, where a unifying inverse regression theory for both the multivariate setting (Li 1991) and functional data from a Reproducing Kernel Hilbert Space (RKHS) prospective is developed. We proposed a functional multiple-index model which models a real response variable as a function of a few predictor variables called indices. These indices are random elements of the Hilbert space spanned by a second order stochastic process and they constitute the so-called Effective Dimensional Reduction Space (EDRS). To conduct inference on the EDRS, we discovered a fundamental result which reveals the geometrical association between the EDRS and the RKHS of the process. Two inverse regression procedures, a “slicing” approach and a kernel approach, were introduced to estimate the counterpart of the EDRS in the RKHS. Further the estimate of the EDRS was achieved via the transformation from the RKHS to the original Hilbert space. To construct an asymptotic theory, we introduced an isometric mapping from the empirical RKHS to the theoretical RKHS, which can be used to measure the distance between the estimator and the target. Some general computational issues of FDA were discussed, which led to the smoothed versions of the functional inverse regression methods. Simulation studies were performed to evaluate the performance of the inference procedures and applications to biological and chemometrical data analysis were illustrated.

Riemann-Hilbert-Probleme sind Randwertaufgaben für im Einheitskreis $\mathbb D$ holomorphe Funktionen $w$, deren Randwerte $w(t)$ auf gewissen Kurven $M_t$ liegen sollen. Ein Teil der Untersuchungen ist dem Fall explizit gegebener Kurven gewidmet. Dabei werden bekannte Resultate über glatte Kurven auf stetige Restriktionskurven erweitert, und die Existenz von Lösungen in gewissen Hardy-Räumen gezeigt. Die Eindeutigkeitsfrage führt auf ein Gegenbeispiel, das zugleich eine Vermutung aus einer Dissertation von Belch widerlegt. Der andere Teil der Untersuchungen ist dem klassischen Fall geschlossener Restriktionskurven gewidmet. Hier steht statt der Abschwächung von Glattheitsvoraussetzungen die Formulierung geeigneter Nebenbedingungen im Mittelpunkt. Die Abhängigkeit der Lösung von Zusatzbedingungen erweist sich als Verallgemeinerung des Verhaltens von Blaschkeprodukten. Für drei Interpolationpunkte kann charakterisiert werden, wann durch sie eine Lösung mit Windungszahl 1 verläuft, durch $k$ Interpolationspunkte wird die Existenz einer Lösung mit Windungszahl $k-1$ gezeigt.

Le quotient géométrique d'une variété lisse par l'action d'un groupe fini préservant le volume est une variété singulière. La correspondance de McKay relie la géométrie des résolutions crépantes du quotient et la géométrie de l'action sur la variété lisse. Sous certaines hypothèses, le schéma de Hilbert équivariant de la variété lisse est une résolution crépante. Nous interprétons ce schéma en terme de grassmannienne d'algèbres équivariante, afin d'en déduire une description explicite. D'après la conjecture de Ruan, modulo une déformation quantique, l'anneau de cohomologie d'une résolution crépante est isomorphe à l'anneau de cohomologie orbifold du quotient. Pour le quotient d'une variété de dimension trois locale (espace vectoriel avec action linéaire) ou compacte, nous calculons l'anneau de cohomologie des résolutions crépantes. Dans le cas local, un exemple montre la nécessité de la déformation quantique dans la conjecture. Dans le cas compact, l'analogie entre les deux anneaux conforte la conjecture.

Cette thèse porte sur l'étude de systèmes de type gaz particules, tant d'un point de vue mathématique que physique et numérique. Par ailleurs, quelques aspects de la turbulence en lien avec ces systèmes et le modèle k- sont étudiés.

Thesis (Ph. D.)--Ohio State University, 2009. / Title from first page of PDF file. Includes bibliographical references (p. 40-42).

Zugl.: München, Techn. Univ., Diss., 2009

This thesis proposes new analysis tools for simulation models in the presence of data. To achieve a representation close to reality, simulation models are typically endowed with a set of inputs, termed parameters, that represent several controllable, stochastic or unknown components of the system. Because these models often utilize computationally expensive procedures, even modern supercomputers require a nontrivial amount of time, money, and energy to run for complex systems. Existing statistical frameworks avoid repeated evaluations of deterministic models through an emulator, constructed by conducting an experiment on the code. In high dimensional scenarios, the traditional framework for emulator-based analysis can fail due to the computational burden of inference. This thesis proposes a new class of experiments where inference from half a million observations is possible in seconds versus the days required for the traditional technique. In a case study presented in this thesis, the parameter of interest is a function as opposed to a scalar or a set of scalars, meaning the problem exists in the high dimensional regime. This work develops a new modeling strategy to nonparametrically study the functional parameter using Bayesian inference. Stochastic simulations are also investigated in the thesis. I describe the development of emulators through a framework termed quantile kriging, which allows for non-parametric representations of the stochastic behavior of the output whereas previous work has focused on normally distributed outputs. Furthermore, this work studied asymptotic properties of this methodology that yielded practical insights. Under certain regulatory conditions, there is the following result: By using an experiment that has the appropriate ratio of replications to sets of different inputs, we can achieve an optimal rate of convergence. Additionally, this method provided the basic tool for the study of defect patterns and a case study is explored.

The Hilbert scheme of \(n\) points in the projective plane parameterizes degree \(n\) zero-dimensional subschemes of the projective plane. We examine the dual cones of effective divisors and moving curves on the Hilbert scheme. By studying interpolation, restriction, and stability properties of certain vector bundles on the plane we fully determine these cones for just over three fourths of all values of \(n\). A general Steiner bundle on \(\mathbb{P}^N\) is a vector bundle \(E\) admitting a resolution of the form \(0 \rightarrow \mathcal{O}_{\mathbb{P}^N} (−1)^s {M \atop \rightarrow} \mathcal{O}^{s+r}_{\mathbb{P}^N} \rightarrow E \rightarrow 0\), where the map \(M\) is general. We complete the classification of slopes of semistable Steiner bundles on \(\mathbb{P}^N\) by showing every admissible slope is realized by a bundle which restricts to a balanced bundle on a rational curve. The proof involves a basic question about multiplication of polynomials on \(\mathbb{P}^1\) which is interesting in its own right. / Mathematics

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