Formal Desingularization of Surfaces - The Jung Method Revisited -

In this paper we propose the concept of formal desingularizations as a substitute for the resolution of algebraic varieties. Though a usual resolution of algebraic varieties provides more information on the structure of singularities there is evidence that the weaker concept is enough for many computational purposes. We give a detailed study of the Jung method and show how it facilitates an efficient computation of formal desingularizations for projective surfaces over a field of characteristic zero, not necessarily algebraically closed. The paper includes a generalization of Duval's Theorem on rational Puiseux parametrizations to the multivariate case and a detailed description of a system for multivariate algebraic power series computations.Comment: 33 pages, 2 figure

Quasiconvex Programming

We define quasiconvex programming, a form of generalized linear programming in which one seeks the point minimizing the pointwise maximum of a collection of quasiconvex functions. We survey algorithms for solving quasiconvex programs either numerically or via generalizations of the dual simplex method from linear programming, and describe varied applications of this geometric optimization technique in meshing, scientific computation, information visualization, automated algorithm analysis, and robust statistics.Comment: 33 pages, 14 figure

Multivariate Splines and Algebraic Geometry

Multivariate splines are effective tools in numerical analysis and approximation theory. Despite an extensive literature on the subject, there remain open questions in finding their dimension, constructing local bases, and determining their approximation power. Much of what is currently known was developed by numerical analysts, using classical methods, in particular the so-called Bernstein-B´ezier techniques. Due to their many interesting structural properties, splines have become of keen interest to researchers in commutative and homological algebra and algebraic geometry. Unfortunately, these communities have not collaborated much. The purpose of the half-size workshop is to intensify the interaction between the different groups by bringing them together. This could lead to essential breakthroughs on several of the above problems

Doctor of Philosophy

dissertationA broad range of applications capture dynamic data at an unprecedented scale. Independent of the application area, finding intuitive ways to understand the dynamic aspects of these increasingly large data sets remains an interesting and, to some extent, unsolved research problem. Generically, dynamic data sets can be described by some, often hierarchical, notion of feature of interest that exists at each moment in time, and those features evolve across time. Consequently, exploring the evolution of these features is considered to be one natural way of studying these data sets. Usually, this process entails the ability to: 1) define and extract features from each time step in the data set; 2) find their correspondences over time; and 3) analyze their evolution across time. However, due to the large data sizes, visualizing the evolution of features in a comprehensible manner and performing interactive changes are challenging. Furthermore, feature evolution details are often unmanageably large and complex, making it difficult to identify the temporal trends in the underlying data. Additionally, many existing approaches develop these components in a specialized and standalone manner, thus failing to address the general task of understanding feature evolution across time. This dissertation demonstrates that interactive exploration of feature evolution can be achieved in a non-domain-specific manner so that it can be applied across a wide variety of application domains. In particular, a novel generic visualization and analysis environment that couples a multiresolution unified spatiotemporal representation of features with progressive layout and visualization strategies for studying the feature evolution across time is introduced. This flexible framework enables on-the-fly changes to feature definitions, their correspondences, and other arbitrary attributes while providing an interactive view of the resulting feature evolution details. Furthermore, to reduce the visual complexity within the feature evolution details, several subselection-based and localized, per-feature parameter value-based strategies are also enabled. The utility and generality of this framework is demonstrated by using several large-scale dynamic data sets

Contributions to Vine-Copula Modeling

144 p.Regular vine-copula models (R-vines) are a powerful statistical tool for modeling thedependence structure of multivariate distribution functions. In particular, they allow modelingdierent types of dependencies among random variables independently of their marginaldistributions, which is deemed the most valued characteristic of these models. In this thesis, weinvestigate the theoretical properties of R-vines for representing dependencies and extend theiruse to solve supervised classication problems. We focus on three research directions.!In the rst line of research, the relationship between the graphical representations of R-vines!ÁREA LÍNEA1 2 0 3 0 4ÁREA LÍNEA1 2 0 3 1 7ÁREA LÍNEAÁREA LÍNEA!and Bayesian polytree networks is analyzed in terms of how conditional pairwise independence!relationships are represented by both models. In order to do that, we use an extended graphical!representation of R-vines in which the R-vine graph is endowed with further expressiveness,being possible to distinguish between edges representing independence and dependencerelationships. Using this representation, a separation criterion in the R-vine graph, called Rseparation,is dened. The proposed criterion is used in designing methods for building thegraphical structure of polytrees from that of R-vines, and vice versa. Moreover, possiblecorrespondences between the R-vine graph and the associated R-vine copula as well as dierentproperties of R-separation are analyzed. In the second research line, we design methods forlearning the graphical structure of R-vines from dependence lists. The main challenge of thistask lies in the extremely large size of the search space of all possible R-vine structures. Weprovide two strategies to solve the problem of learning R-vines that represent the largestnumber of dependencies in a list. The rst approach is a 0 -1 linear programming formulation forbuilding truncated R-vines with only two trees. The second approach is an evolutionaryalgorithm, which is able to learn complete and truncated R-vines. Experimental results show thesuccess of this strategy in solving the optimization problem posed. In the third research line, weintroduce a supervised classication approach where the dependence structure of the problemfeatures is modeled through R-vines. The ecacy of these classiers is validated in a mentaldecoding problem and in an image recognition task. While Rvines have been extensivelyapplied in elds such as economics, nance and statistics, only recently have they found theirplace in classication tasks. This contribution represents a step forward in understanding R-vinesand the prospect of extending their use to other machine learning tasks

More on the O(n) model on random maps via nested loops: loops with bending energy

We continue our investigation of the nested loop approach to the O(n) model on random maps, by extending it to the case where loops may visit faces of arbitrary degree. This allows to express the partition function of the O(n) loop model as a specialization of the multivariate generating function of maps with controlled face degrees, where the face weights are determined by a fixed point condition. We deduce a functional equation for the resolvent of the model, involving some ring generating function describing the immediate vicinity of the loops. When the ring generating function has a single pole, the model is amenable to a full solution. Physically, such situation is realized upon considering loops visiting triangles only and further weighting these loops by some local bending energy. Our model interpolates between the two previously solved cases of triangulations without bending energy and quadrangulations with rigid loops. We analyze the phase diagram of our model in details and derive in particular the location of its non-generic critical points, which are in the universality classes of the dense and dilute O(n) model coupled to 2D quantum gravity. Similar techniques are also used to solve a twisting loop model on quadrangulations where loops are forced to make turns within each visited square. Along the way, we revisit the problem of maps with controlled, possibly unbounded, face degrees and give combinatorial derivations of the one-cut lemma and of the functional equation for the resolvent.Comment: 40 pages, 9 figures, final accepted versio

DEPLOYING, IMPROVING AND EVALUATING EDGE BUNDLING METHODS FOR VISUALIZING LARGE GRAPHS

A tremendous increase in the scale of graphs has been witnessed in a wide range of fields, which demands efficient and effective visualization techniques to assist users in better understandings of large graphs. Conventional node-link diagrams are often used to visualize graphs, whereas excessive edge crossings can easily incur severe visual clutter in the node-link diagram of a large graph. Edge bundling can effectively remedy visual clutter and reveal high-level graph structures. Although significant efforts have been devoted to developing edge bundling, three challenging problems remain. First, edge bundling techniques are often computationally expensive and are not easy to deploy for web-based applications. The state-of-the-art edge bundling methods often require special system supports and techniques such as high-end GPU acceleration for large graphs, which makes these methods less portable, especially for ubiquitous mobile devices. Second, the quantitative quality of edge bundling results is barely assessed in the literature. Currently, the comparison of edge bundling mainly focuses on computational performance and perceptual results. Third, although the family of edge bundling techniques has a rich set of bundling layout, there is a lack of a generic method to generate different styles of edge bundling. In this research, I aim to address these problems and have made the following contributions. First, I provide an efficient framework to deploy edge bundling for web-based platforms by exploiting standard graphics hardware functions and libraries. My framework can generate high-quality edge bundling results on web-based platforms, and achieve a speedup of 50X compared to the previous state-of-the-art edge bundling method on a graph with half of a million edges. Second, I propose a new moving least squares based approach to lower the algorithm complexity of edge bundling. In addition, my approach can generate better bundling results compared to other methods based on a quality metric. Third, I provide an information-theoretic metric to evaluate the edge bundling methods. I leverage information theory in this metric. With my information-theoretic metric, domain users can choose appropriate edge bundling methods with proper parameters for their applications. Last but not least, I present a deep learning framework for edge bundling visualizations. Through a training process that learns the results of a specific edge bundling method, my deep learning framework can infer the final layout of the edge bundling method. My deep learning framework is a generic framework that can generate the corresponding results of different edge bundling methods. Adviser: Hongfeng Y