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.

We show that recent results in [3] on risk bounds for regularized least-squares on reproducing kernel Hilbert spaces can be straightforwardly extended to the vector-valued regression setting. We first briefly introduce central concepts on operator-valued kernels. Then we show how risk bounds can be expressed in terms of a generalization of effective dimension.

Um espaço de Hilbert de reprodução (EHR) é um espaço de Hilbert de funções construído de maneira específica e única a partir de um núcleo positivo definido. As funções do EHR tem a seguinte peculiaridade: seus valores podem ser reproduzidos através de uma operação elementar envolvendo a própria função, o núcleo gerador e o produto interno do espaço. Neste trabalho, consideramos EHR gerados por núcleos positivos definidos sobre a esfera unitária m-dimensional usual. Analisamos quais propriedades são herdadas pelos elementos do espaço, quando o núcleo gerador possui alguma hipótese de diferenciabilidade. A análise é elaborada em duas frentes: com a noção de diferenciabilidade usual sobre a esfera e com uma noção de diferenciabilidade definida por uma operação multiplicativa genérica. Esta última inclui como caso particular as derivadas fracionárias e a derivada forte de Laplace-Beltrami. Em cada um dos casos consideramos ainda propriedades específicas do mergulho do EHR em espaços de funções suaves definidos pela diferenciabilidade utilizada / A reproducing kernel Hilbert space (EHR) is a Hilbert space of functions constructed in a unique manner from a fixed positive definite generating kernel. The values of a function in a reproducing kernel Hilbert space can be reproduced through an elementary operation involving the function itself, the generating kernel and the inner product of the space. In this work, we consider reproducing kernel Hilbert spaces generated by a positive definite kernel on the usual m-dimensional sphere. The main goal is to analyze differentiability properties inherited by the functions in the space when the generating kernel carries a differentiability assumption. That is done in two different cases: using the usual notion of differentiability on the sphere and using another one defined through multiplicative operators. The second case includes the Laplace-Beltrami derivative and fractional derivatives as well. In both cases we consider specific properties of the embeddings of the reproducing kernel Hilbert space into spaces of smooth functions induced by notion of differentiability used

Um espaço de Hilbert de reprodução (EHR) é um espaço de Hilbert de funções construído de maneira específica e única a partir de um núcleo positivo definido. As funções do EHR tem a seguinte peculiaridade: seus valores podem ser reproduzidos através de uma operação elementar envolvendo a própria função, o núcleo gerador e o produto interno do espaço. Neste trabalho, consideramos EHR gerados por núcleos positivos definidos sobre a esfera unitária m-dimensional usual. Analisamos quais propriedades são herdadas pelos elementos do espaço, quando o núcleo gerador possui alguma hipótese de diferenciabilidade. A análise é elaborada em duas frentes: com a noção de diferenciabilidade usual sobre a esfera e com uma noção de diferenciabilidade definida por uma operação multiplicativa genérica. Esta última inclui como caso particular as derivadas fracionárias e a derivada forte de Laplace-Beltrami. Em cada um dos casos consideramos ainda propriedades específicas do mergulho do EHR em espaços de funções suaves definidos pela diferenciabilidade utilizada / A reproducing kernel Hilbert space (EHR) is a Hilbert space of functions constructed in a unique manner from a fixed positive definite generating kernel. The values of a function in a reproducing kernel Hilbert space can be reproduced through an elementary operation involving the function itself, the generating kernel and the inner product of the space. In this work, we consider reproducing kernel Hilbert spaces generated by a positive definite kernel on the usual m-dimensional sphere. The main goal is to analyze differentiability properties inherited by the functions in the space when the generating kernel carries a differentiability assumption. That is done in two different cases: using the usual notion of differentiability on the sphere and using another one defined through multiplicative operators. The second case includes the Laplace-Beltrami derivative and fractional derivatives as well. In both cases we consider specific properties of the embeddings of the reproducing kernel Hilbert space into spaces of smooth functions induced by notion of differentiability used

This thesis compares and contrasts currents- and varifolds-based diffeomorphic image registration approaches for registering tree-like structures in the lung and surface of the lung. In these approaches, curve-like structures in the lung—for example, the skeletons of vessels and airways segmentation—and surface of the lung are represented by currents or varifolds in the dual space of a Reproducing Kernel Hilbert Space (RKHS). Currents and varifolds representations are discretized and are parameterized via of a collection of momenta. A momenta corresponds to a line segment via the coordinates of the center of the line segment and the tangent direction of the line segment at the center. A momentum corresponds to a mesh via the coordinates of the center of the mesh and the normal direction of the mesh at the center. The magnitude of the tangent vector for the line segment and the normal vector for the mesh are the length of the line segment and the area of the mesh respectively. A varifolds-based registration approach is similar to currents except that two varifolds representations are aligned independent of the tangent (normal) vector orientation. An advantage of varifolds over currents is that the orientation of the tangent vectors can be difficult to determine especially when the vessel and airway trees are not connected. In this thesis, we examine the image registration sensitivity and accuracy of currents- and varifolds-based registration as a function of the number and location of momenta used to represent tree like-structures in the lung and the surface of the lung. The registrations presented in this thesis were generated using the Deformetrica software package, which is publicly available at www.deformetrica.org.

Light transport simulation aims at the numerical computation of the propagation of visible electromagnetic energy in macroscopic environments. In this thesis, we develop the foundations for a modern theory of light transport simulation, unveiling the geometric structure of the continuous theory and providing a formulation of computational techniques that furnishes remarkably efficacy with only local information. Utilizing recent results from various communities, we develop the physical and mathematical structure of light transport from Maxwell's equations by studying a lifted representation of electromagnetic theory on the cotangent bundle. At the short wavelength limit, this yields a Hamiltonian description on six-dimensional phase space, with the classical formulation over the space of "positions and directions" resulting from a reduction to the five-dimensional cosphere bundle. We establish the connection between light transport and geometrical optics by a non-canonical Legendre transform, and we derive classical concepts from radiometry, such as radiance and irradiance, by considering measurements of the light energy density. We also show that in idealized environments light transport is a Lie-Poisson system for the group of symplectic diffeomorphisms, unveiling a tantalizing similarity between light transport and fluid dynamics. Using Stone's theorem, we also derive a functional analytic description of light transport. This bridges the gap to existing formulations in the literature and naturally leads to computational questions. We then address one of the central challenges for light transport simulation in everyday environments with scattering surfaces: how are efficient computations possible when the light energy density can only be evaluated pointwise? Using biorthogonal and possibly overcomplete bases formed by reproducing kernel functions, we develop a comprehensive theory for computational techniques that are restricted to pointwise information, subsuming for example sampling theorems, interpolation formulas, quadrature rules, density estimation schemes, and Monte Carlo integration. The use of overcomplete representations makes us thereby robust to imperfect information, as is often unavoidable in practical applications, and numerical optimization of the sampling locations leads to close to optimal techniques, providing performance which considerably improves over the state of the art in the literature.

Light transport simulation aims at the numerical computation of the propagation of visible electromagnetic energy in macroscopic environments. In this thesis, we develop the foundations for a modern theory of light transport simulation, unveiling the geometric structure of the continuous theory and providing a formulation of computational techniques that furnishes remarkably efficacy with only local information. Utilizing recent results from various communities, we develop the physical and mathematical structure of light transport from Maxwell's equations by studying a lifted representation of electromagnetic theory on the cotangent bundle. At the short wavelength limit, this yields a Hamiltonian description on six-dimensional phase space, with the classical formulation over the space of "positions and directions" resulting from a reduction to the five-dimensional cosphere bundle. We establish the connection between light transport and geometrical optics by a non-canonical Legendre transform, and we derive classical concepts from radiometry, such as radiance and irradiance, by considering measurements of the light energy density. We also show that in idealized environments light transport is a Lie-Poisson system for the group of symplectic diffeomorphisms, unveiling a tantalizing similarity between light transport and fluid dynamics. Using Stone's theorem, we also derive a functional analytic description of light transport. This bridges the gap to existing formulations in the literature and naturally leads to computational questions. We then address one of the central challenges for light transport simulation in everyday environments with scattering surfaces: how are efficient computations possible when the light energy density can only be evaluated pointwise? Using biorthogonal and possibly overcomplete bases formed by reproducing kernel functions, we develop a comprehensive theory for computational techniques that are restricted to pointwise information, subsuming for example sampling theorems, interpolation formulas, quadrature rules, density estimation schemes, and Monte Carlo integration. The use of overcomplete representations makes us thereby robust to imperfect information, as is often unavoidable in practical applications, and numerical optimization of the sampling locations leads to close to optimal techniques, providing performance which considerably improves over the state of the art in the literature.

Modern data collection methods are now frequently returning observations that should be viewed as the result of digitized recording or sampling from stochastic processes rather than vectors of finite length. In spite of great demands, only a few classification methodologies for such data have been suggested and supporting theory is quite limited. The focus of this dissertation is on discrimination and classification in this infinite dimensional setting. The methodology and theory we develop are based on the abstract canonical correlation concept of Eubank and Hsing (2005), and motivated by the fact that Fisher's discriminant analysis method is intimately tied to canonical correlation analysis. Specifically, we have developed a theoretical framework for discrimination and classification of sample paths from stochastic processes through use of the Loeve-Parzen isomorphism that connects a second order process to the reproducing kernel Hilbert space generated by its covariance kernel. This approach provides a seamless transition between the finite and infinite dimensional settings and lends itself well to computation via smoothing and regularization. In addition, we have developed a new computational procedure and illustrated it with simulated data and Canadian weather data.

We develop a theoretical analysis of generalization performances of regularized least-squares on reproducing kernel Hilbert spaces for supervised learning. We show that the concept of effective dimension of an integral operator plays a central role in the definition of a criterion for the choice of the regularization parameter as a function of the number of samples. In fact, a minimax analysis is performed which shows asymptotic optimality of the above-mentioned criterion.

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