Pointwise estimates for the Bergman kernel of the weighted Fock space

We prove upper pointwise estimates for the Bergman kernel of the weighted Fock space of entire functions in $L^{2}(e^{-2\phi}) $ where $\phi$ is a subharmonic function with $\Delta\phi$ a doubling measure. We derive estimates for the canonical solution operator to the inhomogeneous Cauchy-Riemann equation and we characterize the compactness of this operator in terms of $\Delta\phi$.

Kernel-PCA data integration with enhanced interpretability

Background Nowadays, combining the different sources of information to improve the biological knowledge available is a challenge in bioinformatics. One of the most powerful methods for integrating heterogeneous data types are kernel-based methods. Kernel-based data integration approaches consist of two basic steps: firstly the right kernel is chosen for each data set; secondly the kernels from the different data sources are combined to give a complete representation of the available data for a given statistical task. Results We analyze the integration of data from several sources of information using kernel PCA, from the point of view of reducing dimensionality. Moreover, we improve the interpretability of kernel PCA by adding to the plot the representation of the input variables that belong to any dataset. In particular, for each input variable or linear combination of input variables, we can represent the direction of maximum growth locally, which allows us to identify those samples with higher/lower values of the variables analyzed. Conclusions The integration of different datasets and the simultaneous representation of samples and variables together give us a better understanding of biological knowledge.

The Bonenblust-Hille inequality for homogeneous polynomials is hypercontractive

The Bohnenblust-Hille inequality says that the $\ell^{\frac{2m}{m+1}}$ -norm of the coefficients of an $m$-homogeneous polynomial $P$ on $\Bbb{C}^n$ is bounded by $\| P \|_\infty$ times a constant independent of $n$, where $\|\cdot \|_\infty$ denotes the supremum norm on the polydisc $\mathbb{D}^n$. The main result of this paper is that this inequality is hypercontractive, i.e., the constant can be taken to be $C^m$ for some $C>1$. Combining this improved version of the Bohnenblust-Hille inequality with other results, we obtain the following: The Bohr radius for the polydisc $\mathbb{D}^n$ behaves asymptotically as $\sqrt{(\log n)/n}$ modulo a factor bounded away from 0 and infinity, and the Sidon constant for the set of frequencies $\bigl\{ \log n: n \text{a positive integer} \le N\bigr\}$ is $\sqrt{N}\exp\{(-1/\sqrt{2}+o(1))\sqrt{\log N\log\log N}\}$.

Beurling-Landau densities of weighted Fekete sets and correlation kernel estimates

Let $Q$ be a suitable real function on $C$. An $n$-Fekete set corresponding to $Q$ is a subset ${Z_{n1}},\dotsb, Z_{nn}}$ of $C$ which maximizes the expression $\Pi^n_i_{

New kernel functions and learning methods for text and data mining

Recent advances in machine learning methods enable increasingly the automatic construction of various types of computer assisted methods that have been difficult or laborious to program by human experts. The tasks for which this kind of tools are needed arise in many areas, here especially in the fields of bioinformatics and natural language processing. The machine learning methods may not work satisfactorily if they are not appropriately tailored to the task in question. However, their learning performance can often be improved by taking advantage of deeper insight of the application domain or the learning problem at hand. This thesis considers developing kernel-based learning algorithms incorporating this kind of prior knowledge of the task in question in an advantageous way. Moreover, computationally efficient algorithms for training the learning machines for specific tasks are presented. In the context of kernel-based learning methods, the incorporation of prior knowledge is often done by designing appropriate kernel functions. Another well-known way is to develop cost functions that fit to the task under consideration. For disambiguation tasks in natural language, we develop kernel functions that take account of the positional information and the mutual similarities of words. It is shown that the use of this information significantly improves the disambiguation performance of the learning machine. Further, we design a new cost function that is better suitable for the task of information retrieval and for more general ranking problems than the cost functions designed for regression and classification. We also consider other applications of the kernel-based learning algorithms such as text categorization, and pattern recognition in differential display. We develop computationally efficient algorithms for training the considered learning machines with the proposed kernel functions. We also design a fast cross-validation algorithm for regularized least-squares type of learning algorithm. Further, an efficient version of the regularized least-squares algorithm that can be used together with the new cost function for preference learning and ranking tasks is proposed. In summary, we demonstrate that the incorporation of prior knowledge is possible and beneficial, and novel advanced kernels and cost functions can be used in algorithms efficiently.

Estimating extreme value cumulative distribution functions using bias-corrected kernel approaches

We propose a new kernel estimation of the cumulative distribution function based on transformation and on bias reducing techniques. We derive the optimal bandwidth that minimises the asymptotic integrated mean squared error. The simulation results show that our proposed kernel estimation improves alternative approaches when the variable has an extreme value distribution with heavy tail and the sample size is small.

Learning Preferences with Kernel-Based Methods

Learning of preference relations has recently received significant attention in machine learning community. It is closely related to the classification and regression analysis and can be reduced to these tasks. However, preference learning involves prediction of ordering of the data points rather than prediction of a single numerical value as in case of regression or a class label as in case of classification. Therefore, studying preference relations within a separate framework facilitates not only better theoretical understanding of the problem, but also motivates development of the efficient algorithms for the task. Preference learning has many applications in domains such as information retrieval, bioinformatics, natural language processing, etc. For example, algorithms that learn to rank are frequently used in search engines for ordering documents retrieved by the query. Preference learning methods have been also applied to collaborative filtering problems for predicting individual customer choices from the vast amount of user generated feedback. In this thesis we propose several algorithms for learning preference relations. These algorithms stem from well founded and robust class of regularized least-squares methods and have many attractive computational properties. In order to improve the performance of our methods, we introduce several non-linear kernel functions. Thus, contribution of this thesis is twofold: kernel functions for structured data that are used to take advantage of various non-vectorial data representations and the preference learning algorithms that are suitable for different tasks, namely efficient learning of preference relations, learning with large amount of training data, and semi-supervised preference learning. Proposed kernel-based algorithms and kernels are applied to the parse ranking task in natural language processing, document ranking in information retrieval, and remote homology detection in bioinformatics domain. Training of kernel-based ranking algorithms can be infeasible when the size of the training set is large. This problem is addressed by proposing a preference learning algorithm whose computation complexity scales linearly with the number of training data points. We also introduce sparse approximation of the algorithm that can be efficiently trained with large amount of data. For situations when small amount of labeled data but a large amount of unlabeled data is available, we propose a co-regularized preference learning algorithm. To conclude, the methods presented in this thesis address not only the problem of the efficient training of the algorithms but also fast regularization parameter selection, multiple output prediction, and cross-validation. Furthermore, proposed algorithms lead to notably better performance in many preference learning tasks considered.

Kernel-Based Ranking. Methods for Learning and Performance Estimation

Machine learning provides tools for automated construction of predictive models in data intensive areas of engineering and science. The family of regularized kernel methods have in the recent years become one of the mainstream approaches to machine learning, due to a number of advantages the methods share. The approach provides theoretically well-founded solutions to the problems of under- and overfitting, allows learning from structured data, and has been empirically demonstrated to yield high predictive performance on a wide range of application domains. Historically, the problems of classification and regression have gained the majority of attention in the field. In this thesis we focus on another type of learning problem, that of learning to rank. In learning to rank, the aim is from a set of past observations to learn a ranking function that can order new objects according to how well they match some underlying criterion of goodness. As an important special case of the setting, we can recover the bipartite ranking problem, corresponding to maximizing the area under the ROC curve (AUC) in binary classification. Ranking applications appear in a large variety of settings, examples encountered in this thesis include document retrieval in web search, recommender systems, information extraction and automated parsing of natural language. We consider the pairwise approach to learning to rank, where ranking models are learned by minimizing the expected probability of ranking any two randomly drawn test examples incorrectly. The development of computationally efficient kernel methods, based on this approach, has in the past proven to be challenging. Moreover, it is not clear what techniques for estimating the predictive performance of learned models are the most reliable in the ranking setting, and how the techniques can be implemented efficiently. The contributions of this thesis are as follows. First, we develop RankRLS, a computationally efficient kernel method for learning to rank, that is based on minimizing a regularized pairwise least-squares loss. In addition to training methods, we introduce a variety of algorithms for tasks such as model selection, multi-output learning, and cross-validation, based on computational shortcuts from matrix algebra. Second, we improve the fastest known training method for the linear version of the RankSVM algorithm, which is one of the most well established methods for learning to rank. Third, we study the combination of the empirical kernel map and reduced set approximation, which allows the large-scale training of kernel machines using linear solvers, and propose computationally efficient solutions to cross-validation when using the approach. Next, we explore the problem of reliable cross-validation when using AUC as a performance criterion, through an extensive simulation study. We demonstrate that the proposed leave-pair-out cross-validation approach leads to more reliable performance estimation than commonly used alternative approaches. Finally, we present a case study on applying machine learning to information extraction from biomedical literature, which combines several of the approaches considered in the thesis. The thesis is divided into two parts. Part I provides the background for the research work and summarizes the most central results, Part II consists of the five original research articles that are the main contribution of this thesis.

Excessive functions, Appell polynomials and optimal stopping

The main topic of the thesis is optimal stopping. This is treated in two research articles. In the first article we introduce a new approach to optimal stopping of general strong Markov processes. The approach is based on the representation of excessive functions as expected suprema. We present a variety of examples, in particular, the Novikov-Shiryaev problem for Lévy processes. In the second article on optimal stopping we focus on differentiability of excessive functions of diffusions and apply these results to study the validity of the principle of smooth fit. As an example we discuss optimal stopping of sticky Brownian motion. The third research article offers a survey like discussion on Appell polynomials. The crucial role of Appell polynomials in optimal stopping of Lévy processes was noticed by Novikov and Shiryaev. They described the optimal rule in a large class of problems via these polynomials. We exploit the probabilistic approach to Appell polynomials and show that many classical results are obtained with ease in this framework. In the fourth article we derive a new relationship between the generalized Bernoulli polynomials and the generalized Euler polynomials.

Functional characterization of acetylated Brazil nut (Bertholletia excelsa HBK) kernel globulin

Defatted Brazil nut kernel flour, a rich source of high quality proteins, is presently being utilized in the formulation of animal feeds. One of the possible ways to improve its utilization for human consumption is through improvement in its functional properties. In the present study, changes in some of the functional properties of Brazil nut kernel globulin were evaluated after acetylation at 58.6, 66.2 and 75.3% levels. The solubility of acetylated globulin was improved above pH 6.0 but was reduced in the pH range of 3.0-4.0. Water and oil absorption capacity, as well as the viscosity increased with increase in the level of acetylation. Level of modification also influenced the emulsifying capacity: decreased at pH 3.0, but increased at pH 7.0 and 9.0. Highest emulsion activity (approximately 62.2%) was observed at pH 3.0 followed by pH 9.0 and pH 7.0 and least (about 11.8%) at pH 5.0. Emulsion stability also followed similar behavior as that of emulsion activity.

Effects of the mechanical damage on the water absorption process by corn kernel

The purpose of this study was to investigate and model the water absorption process by corn kernels with different levels of mechanical damage Corn kernels of AG 1510 variety with moisture content of 14.2 (% d.b.) were used. Different mechanical damage levels were indirectly evaluated by electrical conductivity measurements. The absorption process was based on the industrial corn wet milling process, in which the product was soaked with a 0.2% sulfur dioxide (SO2) solution and 0.55% lactic acid (C3H6O3) in distilled water, under controlled temperatures of 40, 50, 60, and 70 ºC and different mechanical damage levels. The Peleg model was used for the analysis and modeling of water absorption process. The conclusion is that the structural changes caused by the mechanical damage to the corn kernels influenced the initial rates of water absorption, which were higher for the most damaged kernels, and they also changed the equilibrium moisture contents of the kernels. The Peleg model was well adjusted to the experimental data presenting satisfactory values for the analyzed statistic parameters for all temperatures regardless of the damage level of the corn kernels.

Characterization, physicochemical stability, and evaluation of in vitro digestibility of solid lipid microparticles produced with palm kernel oil and tristearin

Solid lipid particles have been investigated by food researchers due to their ability to enhance the incorporation and bioavailability of lipophilic bioactives in aqueous formulations. The objectives of this study were to evaluate the physicochemical stability and digestibility of lipid microparticles produced with tristearin and palm kernel oil. The motivation for conducting this study was the fact that mixing lipids can prevent the expulsion of the bioactive from the lipid core and enhance the digestibility of lipid structures. The lipid microparticles containing different palm kernel oil contents were stable after 60 days of storage according to the particle size and zeta potential data. Their calorimetric behavior indicated that they were composed of a very heterogeneous lipid matrix. Lipid microparticles were stable under various conditions of ionic strength, sugar concentration, temperature, and pH. Digestibility assays indicated no differences in the release of free fatty acids, which was approximately 30% in all analises. The in vitro digestibility tests showed that the amount of palm kernel in the particles did not affect the percentage of lipolysis, probably due to the high amount of surfactants used and/or the solid state of the microparticles.

Prime Rational Functions and Integral Polynomials

Let f(x) be a complex rational function. In this work, we study conditions under which f(x) cannot be written as the composition of two rational functions which are not units under the operation of function composition. In this case, we say that f(x) is prime. We give sufficient conditions for complex rational functions to be prime in terms of their degrees and their critical values, and we derive some conditions for the case of complex polynomials. We consider also the divisibility of integral polynomials, and we present a generalization of a theorem of Nieto. We show that if f(x) and g(x) are integral polynomials such that the content of g divides the content of f and g(n) divides f(n) for an integer n whose absolute value is larger than a certain bound, then g(x) divides f(x) in Z[x]. In addition, given an integral polynomial f(x), we provide a method to determine if f is irreducible over Z, and if not, find one of its divisors in Z[x].

Certain problems concerning polynomials and transcendental entire functions of exponential type

Soit $\displaystyle P(z):=\sum_{\nu=0}^na_\nu z^{\nu}$ un polynôme de degré $n$ et $\displaystyle M:=\sup_{|z|=1}|P(z)|.$ Sans aucne restriction suplémentaire, on sait que $|P'(z)|\leq Mn$ pour $|z|\leq 1$ (inégalité de Bernstein). Si nous supposons maintenant que les zéros du polynôme $P$ sont à l'extérieur du cercle $|z|=k,$ quelle amélioration peut-on apporter à l'inégalité de Bernstein? Il est déjà connu [{\bf \ref{Mal1}}] que dans le cas où $k\geq 1$ on a $$(*) \qquad |P'(z)|\leq \frac{n}{1+k}M \qquad (|z|\leq 1),$$ qu'en est-il pour le cas où $k < 1$? Quelle est l'inégalité analogue à $(*)$ pour une fonction entière de type exponentiel $\tau ?$ D'autre part, si on suppose que $P$ a tous ses zéros dans $|z|\geq k \, \, (k\geq 1),$ quelle est l'estimation de $|P'(z)|$ sur le cercle unité, en terme des quatre premiers termes de son développement en série entière autour de l'origine. Cette thèse constitue une contribution à la théorie analytique des polynômes à la lumière de ces questions.

Development of Shape Descriptors Based on Legendre Polynomials and Content-Based Retrieval of Scoliosis Images

The wealth of information available freely on the web and medical image databases poses a major problem for the end users: how to find the information needed? Content –Based Image Retrieval is the obvious solution.A standard called MPEG-7 was evolved to address the interoperability issues of content-based search.The work presented in this thesis mainly concentrates on developing new shape descriptors and a framework for content – based retrieval of scoliosis images.New region-based and contour based shape descriptor is developed based on orthogonal Legendre polymomials.A novel system for indexing and retrieval of digital spine radiographs with scoliosis is presented.