30 research outputs found

    COMPOSE: Compacted object sample extraction a framework for semi-supervised learning in nonstationary environments

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    An increasing number of real-world applications are associated with streaming data drawn from drifting and nonstationary distributions. These applications demand new algorithms that can learn and adapt to such changes, also known as concept drift. Proper characterization of such data with existing approaches typically requires substantial amount of labeled instances, which may be difficult, expensive, or even impractical to obtain. In this thesis, compacted object sample extraction (COMPOSE) is introduced - a computational geometry-based framework to learn from nonstationary streaming data - where labels are unavailable (or presented very sporadically) after initialization. The feasibility and performance of the algorithm are evaluated on several synthetic and real-world data sets, which present various different scenarios of initially labeled streaming environments. On carefully designed synthetic data sets, we also compare the performance of COMPOSE against the optimal Bayes classifier, as well as the arbitrary subpopulation tracker algorithm, which addresses a similar environment referred to as extreme verification latency. Furthermore, using the real-world National Oceanic and Atmospheric Administration weather data set, we demonstrate that COMPOSE is competitive even with a well-established and fully supervised nonstationary learning algorithm that receives labeled data in every batch

    IST Austria Thesis

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    In this thesis we study persistence of multi-covers of Euclidean balls and the geometric structures underlying their computation, in particular Delaunay mosaics and Voronoi tessellations. The k-fold cover for some discrete input point set consists of the space where at least k balls of radius r around the input points overlap. Persistence is a notion that captures, in some sense, the topology of the shape underlying the input. While persistence is usually computed for the union of balls, the k-fold cover is of interest as it captures local density, and thus might approximate the shape of the input better if the input data is noisy. To compute persistence of these k-fold covers, we need a discretization that is provided by higher-order Delaunay mosaics. We present and implement a simple and efficient algorithm for the computation of higher-order Delaunay mosaics, and use it to give experimental results for their combinatorial properties. The algorithm makes use of a new geometric structure, the rhomboid tiling. It contains the higher-order Delaunay mosaics as slices, and by introducing a filtration function on the tiling, we also obtain higher-order α-shapes as slices. These allow us to compute persistence of the multi-covers for varying radius r; the computation for varying k is less straight-foward and involves the rhomboid tiling directly. We apply our algorithms to experimental sphere packings to shed light on their structural properties. Finally, inspired by periodic structures in packings and materials, we propose and implement an algorithm for periodic Delaunay triangulations to be integrated into the Computational Geometry Algorithms Library (CGAL), and discuss the implications on persistence for periodic data sets

    Toward Controllable and Robust Surface Reconstruction from Spatial Curves

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    Reconstructing surface from a set of spatial curves is a fundamental problem in computer graphics and computational geometry. It often arises in many applications across various disciplines, such as industrial prototyping, artistic design and biomedical imaging. While the problem has been widely studied for years, challenges remain for handling different type of curve inputs while satisfying various constraints. We study studied three related computational tasks in this thesis. First, we propose an algorithm for reconstructing multi-labeled material interfaces from cross-sectional curves that allows for explicit topology control. Second, we addressed the consistency restoration, a critical but overlooked problem in applying algorithms of surface reconstruction to real-world cross-sections data. Lastly, we propose the Variational Implicit Point Set Surface which allows us to robustly handle noisy, sparse and non-uniform inputs, such as samples from spatial curves

    Courbure discrète : théorie et applications

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    International audienceThe present volume contains the proceedings of the 2013 Meeting on discrete curvature, held at CIRM, Luminy, France. The aim of this meeting was to bring together researchers from various backgrounds, ranging from mathematics to computer science, with a focus on both theory and applications. With 27 invited talks and 8 posters, the conference attracted 70 researchers from all over the world. The challenge of finding a common ground on the topic of discrete curvature was met with success, and these proceedings are a testimony of this wor

    Decomposition Methods for Nonlinear Optimization and Data Mining

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    We focus on two central themes in this dissertation. The first one is on decomposing polytopes and polynomials in ways that allow us to perform nonlinear optimization. We start off by explaining important results on decomposing a polytope into special polyhedra. We use these decompositions and develop methods for computing a special class of integrals exactly. Namely, we are interested in computing the exact value of integrals of polynomial functions over convex polyhedra. We present prior work and new extensions of the integration algorithms. Every integration method we present requires that the polynomial has a special form. We explore two special polynomial decomposition algorithms that are useful for integrating polynomial functions. Both polynomial decompositions have strengths and weaknesses, and we experiment with how to practically use them. After developing practical algorithms and efficient software tools for integrating a polynomial over a polytope, we focus on the problem of maximizing a polynomial function over the continuous domain of a polytope. This maximization problem is NP-hard, but we develop approximation methods that run in polynomial time when the dimension is fixed. Moreover, our algorithm for approximating the maximum of a polynomial over a polytope is related to integrating the polynomial over the polytope. We show how the integration methods can be used for optimization. The second central topic in this dissertation is on problems in data science. We first consider a heuristic for mixed-integer linear optimization. We show how many practical mixed-integer linear have a special substructure containing set partition constraints. We then describe a nice data structure for finding feasible zero-one integer solutions to systems of set partition constraints. Finally, we end with an applied project using data science methods in medical research.Comment: PHD Thesis of Brandon Dutr

    Collection of abstracts of the 24th European Workshop on Computational Geometry

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    International audienceThe 24th European Workshop on Computational Geomety (EuroCG'08) was held at INRIA Nancy - Grand Est & LORIA on March 18-20, 2008. The present collection of abstracts contains the 63 scientific contributions as well as three invited talks presented at the workshop

    Geometric and Topological Methods for Applications to Materials and Data Skeletonisation

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    Crystal Structure Prediction (CSP) aims to speed up functional materials discovery by using supercomputers to predict whether an input molecule can form stable crystal struc- tures with desirable properties. The process produces large datasets where each entry is a simulated arrangement of copies of the input molecule to form a crystal. However, these datasets have little structure themselves, and it is the aim of this thesis to contribute towards simplifying and analysing such datasets. Crystals are unbounded collections of atoms or molecules, extending infinitely in the space they lie within. As such, rigorously quantifying the geometric similarity of crystal structures, and even just identifying identical structures, is a challenging problem. To solve it, we seek a continuous, complete, isometry classification of crystals. Consequently, by modelling crystals as periodic point sets, we introduce the density fingerprint, which is invariant under isometries, Lipschitz continuous, and complete for an open and dense space of crystal structures. Such a classification will be able to identify and remove near- duplicates from these large CSP datasets, and potentially even guide future searches. We describe how this fingerprint can be computed using periodic higher Voronoi zones. This geometric concept of concentric regions around a fixed centre characterises relative positions of points from the centre in a periodic point set. We present an algorithm to compute these zones in addition to proving key structural properties. We later discuss research into skeletonisation algorithms, proving theoretical guarantees of the homological persistent skeleton (HoPeS), subsequently formulating and performing an experimental comparison of HoPeS with other relevant algorithms. Such algorithms, if effectively used, can be applied to large datasets including those produced by CSP to reveal the shape of the data, helping to highlight regions of interest and branches that merit further study
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