224 research outputs found

    Classroom Examples of Robustness Problems in Geometric Computations

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    International audienceThe algorithms of computational geometry are designed for a machine model with exact real arithmetic. Substituting floating point arithmetic for the assumed real arithmetic may cause implementations to fail. Although this is well known, there is no comprehensive documentation of what can go wrong and why. In this extended abstract, we study a simple incremental algorithm for planar convex hulls and give examples which make the algorithm fail in all possible ways. We also show how to construct failure-examples semi-systematically and discuss the geometry of the floating point implementation of the orientation predicate. We hope that our work will be useful for teaching computational geometry. The full paper is available at http://hal.inria.fr/inria-00344310/. It contains further examples, more theory, and color pictures. We strongly recommend to read the full paper instead of this extended abstract

    Adaptive Precision Floating-Point Arithmetic and Fast Robust Geometric Predicates

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    Exact computer arithmetic has a variety of uses including, but not limited to, the robust implementation of geometric algorithms. This report has three purposes. The first is to offer fast software-level algorithms for exact addition and multiplication of arbitrary precision floating-point values. The second is to propose a technique for adaptive-precision arithmetic that can often speed these algorithms when one wishes to perform multiprecision calculations that do not always require exact arithmetic, but must satisfy some error bound. The third is to provide a practical demonstration of these techniques, in the form of implementations of several common geometric calculations whose required degree of accuracy depends on their inputs. These robust geometric predicates are adaptive; their running time depends on the degree of uncertainty of the result, and is usually small. These algorithms work on computers whose floating-point arithmetic uses radix two and exact rounding, including machines complying with the IEEE 754 standard. The inputs to the predicates may be arbitrary single or double precision floating-point numbers. C code is publicly available for the 2D and 3D orientation and incircle tests, an

    Computations of Delaunay and Higher Order Triangulations, with Applications to Splines

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    Digital data that consist of discrete points are frequently captured and processed by scientific and engineering applications. Due to the rapid advance of new data gathering technologies, data set sizes are increasing, and the data distributions are becoming more irregular. These trends call for new computational tools that are both efficient enough to handle large data sets and flexible enough to accommodate irregularity. A mathematical foundation that is well-suited for developing such tools is triangulation, which can be defined for discrete point sets with little assumption about their distribution. The potential benefits from using triangulation are not fully exploited. The challenges fundamentally stem from the complexity of the triangulation structure, which generally takes more space to represent than the input points. This complexity makes developing a triangulation program a delicate task, particularly when it is important that the program runs fast and robustly over large data. This thesis addresses these challenges in two parts. The first part concentrates on techniques designed for efficiently and robustly computing Delaunay triangulations of three kinds of practical data: the terrain data from LIDAR sensors commonly found in GIS, the atom coordinate data used for biological applications, and the time varying volume data generated from from scientific simulations. The second part addresses the problem of defining spline spaces over triangulations in two dimensions. It does so by generalizing Delaunay configurations, defined as follows. For a given point set P in two dimensions, a Delaunay configuration is a pair of subsets (T, I) from P, where T, called the boundary set, is a triplet and I, called the interior set, is the set of points that fall in the circumcircle through T. The size of the interior set is the degree of the configuration. As recently discovered by Neamtu (2004), for a chosen point set, the set of all degree k Delaunay configurations can be associated with a set of degree k plus 1 splines that form the basis of a spline space. In particular, for the trivial case of k equals 0, the spline space coincides with the PL interpolation functions over the Delaunay triangulation. Neamtu’s definition of the spline space relies only on a few structural properties of the Delaunay configurations. This raises the question whether there exist other sets of configurations with identical structural properties. If there are, then these sets of configurations—let us call them generalized configurations from hereon—can be substituted for Delaunay configurations in Neamtu’s definition of spline space thereby yielding a family of splines over the same point set

    Non-acyclicity of coset lattices and generation of finite groups

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    Convex hulls in concept induction

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    Classification learning is dominated by systems which induce large numbers of small axis-orthogonal decision surfaces. This strongly biases such systems towards particular hypothesis types but there is reason believe that many domains have underlying concepts which do not involve axis orthogonal surfaces. Further, the multiplicity of small decision regions mitigates against any holistic appreciation of the theories produced by these systems, notwithstanding the fact that many of the small regions are individually comprehensible. This thesis investigates modeling concepts as large geometric structures in n-dimensional space. Convex hulls are a superset of the set of axis orthogonal hyperrectangles into which axis orthogonal systems partition the instance space. In consequence, there is reason to believe that convex hulls might provide a more flexible and general learning bias than axis orthogonal regions. The formation of convex hulls around a group of points of the same class is shown to be a usable generalisation and is more general than generalisations produced by axis-orthogonal based classifiers, without constructive induction, like decision trees, decision lists and rules. The use of a small number of large hulls as a concept representation is shown to provide classification performance which can be better than that of classifiers which use a large number of small fragmentary regions for each concept. A convex hull based classifier, CH1, has been implemented and tested. CH1 can handle categorical and continuous data. Algorithms for two basic generalisation operations on hulls, inflation and facet deletion, are presented. The two operations are shown to improve the accuracy of the classifier and provide moderate classification accuracy over a representative selection of typical, largely or wholly continuous valued machine learning tasks. The classifier exhibits superior performance to well-known axis-orthogonal-based classifiers when presented with domains where the underlying decision surfaces are not axis parallel. The strengths and weaknesses of the system are identified. One particular advantage is the ability of the system to model domains with approximately the same number of structures as there are underlying concepts. This leads to the possibility of extraction of higher level mathematical descriptions of the induced concepts, using the techniques of computational geometry, which is not possible from a multiplicity of small regions

    Feasible Form Parameter Design of Complex Ship Hull Form Geometry

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    This thesis introduces a new methodology for robust form parameter design of complex hull form geometry via constraint programming, automatic differentiation, interval arithmetic, and truncated hierarchical B- splines. To date, there has been no clearly stated methodology for assuring consistency of general (equality and inequality) constraints across an entire geometric form parameter ship hull design space. In contrast, the method to be given here can be used to produce guaranteed narrowing of the design space, such that infeasible portions are eliminated. Furthermore, we can guarantee that any set of form parameters generated by our method will be self consistent. It is for this reason that we use the title feasible form parameter design. In form parameter design, a design space is represented by a tuple of design parameters which are extended in each design space dimension. In this representation, a single feasible design is a consistent set of real valued parameters, one for every component of the design space tuple. Using the methodology to be given here, we pick out designs which consist of consistent parameters, narrowed to any desired precision up to that of the machine, even for equality constraints. Furthermore, the method is developed to enable the generation of complex hull forms using an extension of the basic rules idea to allow for automated generation of rules networks, plus the use of the truncated hierarchical B-splines, a wavelet-adaptive extension of standard B-splines and hierarchical B-splines. The adaptive resolution methods are employed in order to allow an automated program the freedom to generate complex B-spline representations of the geometry in a robust manner across multiple levels of detail. Thus two complementary objectives are pursued: ensuring feasible starting sets of form parameters, and enabling the generation of complex hull form geometry

    Interpretable Network Representations

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    Networks (or interchangeably graphs) have been ubiquitous across the globe and within science and engineering: social networks, collaboration networks, protein-protein interaction networks, infrastructure networks, among many others. Machine learning on graphs, especially network representation learning, has shown remarkable performance in network-based applications, such as node/graph classification, graph clustering, and link prediction. Like performance, it is equally crucial for individuals to understand the behavior of machine learning models and be able to explain how these models arrive at a certain decision. Such needs have motivated many studies on interpretability in machine learning. For example, for social network analysis, we may need to know the reasons why certain users (or groups) are classified or clustered together by the machine learning models, or why a friend recommendation system considers some users similar so that they are recommended to connect with each other. Therefore, an interpretable network representation is necessary and it should carry the graph information to a level understandable by humans. Here, we first introduce our method on interpretable network representations: the network shape. It provides a framework to represent a network with a 3-dimensional shape, and one can customize network shapes for their need, by choosing various graph sampling methods, 3D network embedding methods and shape-fitting methods. In this thesis, we introduce the two types of network shape: a Kronecker hull which represents a network as a 3D convex polyhedron using stochastic Kronecker graphs as the network embedding method, and a Spectral Path which represents a network as a 3D path connecting the spectral moments of the network and its subgraphs. We demonstrate that network shapes can capture various properties of not only the network, but also its subgraphs. For instance, they can provide the distribution of subgraphs within a network, e.g., what proportion of subgraphs are structurally similar to the whole network? Network shapes are interpretable on different levels, so one can quickly understand the structural properties of a network and its subgraphs by its network shape. Using experiments on real-world networks, we demonstrate that network shapes can be used in various applications, including (1) network visualization, the most intuitive way for users to understand a graph; (2) network categorization (e.g., is this a social or a biological network?); (3) computing similarity between two graphs. Moreover, we utilize network shapes to extend biometrics studies to network data, by solving two problems: network identification (Given an anonymized graph, can we identify the network from which it is collected? i.e., answering questions such as ``where is this anonymized graph sampled from, Twitter or Facebook? ) and network authentication (If one claims the graph is sampled from a certain network, can we verify this claim?). The overall objective of the thesis is to provide a compact, interpretable, visualizable, comparable and efficient representation of networks
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