Approximate Polytope Membership Queries and Applications
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Iterated Nearest Neighbors and Finding Minimal Polytopes* 1
Iterated Nearest Neighb ors and Finding Minimal Polytop es y z David Eppstein Je Erickson Abstract Weintro duce a new metho d for nding several typ es of optimal k -p oint sets, minimizing p erimeter, diameter, circumradius, and related measures, by testing sets of the O (k ) nearest neighb ors to each p oint. We argue that this is b etter in a number of ways than previous algorithms, whichwere based on high order Voronoi diagrams. Our technique allows us for the rst time to eciently maintain minimal sets as new p oints are inserted, to generalize our algorithms to higher dimensions, to nd minimal convex k -vertex p olygons and p olytop es, and to improvemany previous results. Weachievemany of our results via a new algorithm for nding rectilinear nearest neighb ors in the plane in time O (n log n + kn). We also demonstrate a related technique for nding minimum area k -p oint sets in the plane, based on testing sets of nearest vertical neighbors to each line segment determined by a pair of p oints. A generalization of this technique also allows us to nd minimum volume and b oundary measure sets in arbitrary dimensions. 1 Intro duction Anumb er of recent pap ers have discussed problems of selecting, from a set of n p oints, the k points optimizing some particular criterion [2, 14, 20]. Criteria that have b een studied include diameter [2], variance [2], area of the convex hull [20], convex hull p erimeter [14,20], and rectilinear diameter and p erimeter [2]. -
Scenechecker: Boosting Scenario Verification Using Symmetry
SceneChecker: Boosting Scenario Verification using Symmetry Abstractions? Hussein Sibai , Yangge Li , and Sayan Mitra University of Illinois at Urbana-Champaign Coordinated Science Laboratory {sibai2,li213,mitras}@illinois.edu Abstract. We present SceneChecker, a tool for verifying scenarios involving vehicles executing complex plans in large cluttered workspaces. SceneChecker converts the scenario verification problem to a standard hybrid system verification problem, and solves it effectively by exploiting structural properties in the plan and the vehicle dynamics. SceneChecker uses symmetry abstractions, a novel refine- ment algorithm, and importantly, is built to boost the performance of any existing reachability analysis tool as a plug-in subroutine. We evaluated SceneChecker on several scenarios involving ground and aerial vehicles with nonlinear dynamics and neural network controllers, employing different kinds of symmetries, using different reachability subroutines, and following plans with hundreds of way- points in complex workspaces. Compared to two leading tools, DryVR and Flow*, SceneChecker shows 14× average speedup in verification time, even while using those very tools as reachability subroutines1. Keywords: Hybrid systems · Safety verification · Symmetry. 1 Introduction Remarkable progress has been made in safety verification of hybrid and cyber-physical systems in the last decade [2,3,4,5,6,7,8,9]. The methods and tools developed have been applied to check safety of aerospace, medical, and autonomous vehicle control systems [4,5,10,11]. The next barrier in making these techniques usable for more com- plex applications is to deal with what is colloquially called the scenario verification problem. A key part of the scenario verification problem is to check that a vehicle or an agent can execute a plan through a complex environment. -
Topics in Algorithmic, Enumerative and Geometric Combinatorics
Thesis for the Degree of Doctor of Philosophy Topics in algorithmic, enumerative and geometric combinatorics Ragnar Freij Division of Mathematics Department of Mathematical Sciences Chalmers University of Technology and University of Gothenburg G¨oteborg, Sweden 2012 Topics in algorithmic, enumerative and geometric combinatorics Ragnar Freij ISBN 978-91-7385-668-3 c Ragnar Freij, 2012. Doktorsavhandlingar vid Chalmers Tekniska H¨ogskola Ny serie Nr 3349 ISSN 0346-718X Department of Mathematical Sciences Chalmers University of Technology and University of Gothenburg SE-412 96 GOTEBORG,¨ Sweden Phone: +46 (0)31-772 10 00 [email protected] Printed in G¨oteborg, Sweden, 2012 Topics in algorithmic, enumerative and geometric com- binatorics Ragnar Freij ABSTRACT This thesis presents five papers, studying enumerative and extremal problems on combinatorial structures. The first paper studies Forman’s discrete Morse theory in the case where a group acts on the underlying complex. We generalize the notion of a Morse matching, and obtain a theory that can be used to simplify the description of the G-homotopy type of a simplicial complex. As an application, we determine the S2 × Sn−2-homotopy type of the complex of non-connected graphs on n nodes. In the introduction, connections are drawn between the first paper and the evasiveness conjecture for monotone graph properties. In the second paper, we investigate Hansen polytopes of split graphs. By applying a partitioning technique, the number of nonempty faces is counted, and in particular we confirm Kalai’s 3d-conjecture for such polytopes. Further- more, a characterization of exactly which Hansen polytopes are also Hanner polytopes is given. -
Optimal Area-Sensitive Bounds for Polytope Approximation
Optimal Area-Sensitive Bounds for Polytope Approximation Sunil Arya∗ Guilherme D. da Fonseca† David M. Mount‡ Department of Computer Dept. de Informática Aplicada Department of Computer Science and Engineering Universidade Federal do Science and Institute for The Hong Kong University of Estado do Rio de Janeiro Advanced Computer Studies Science and Technology (UniRio) University of Maryland Clear Water Bay, Kowloon Rio de Janeiro, Brazil College Park, Maryland 20742 Hong Kong [email protected] [email protected] [email protected] ABSTRACT Mahler volume, it is possible to achieve the desired width- Approximating convex bodies is a fundamental question in based sampling. geometry and has applications to a wide variety of optimiza- tion problems. Given a convex body K in Rd for fixed d, Categories and Subject Descriptors the objective is to minimize the number of vertices or facets F.2.2 [Analysis of Algorithms and Problem Complex- of an approximating polytope for a given Hausdorff error ity]: Nonnumerical Algorithms and Problems—Geometrical ε. The best known uniform bound, due to Dudley (1974), problems and computations shows that O((diam(K)/ε)(d−1)/2) facets suffice. While this bound is optimal in the case of a Euclidean ball, it is far General Terms from optimal for skinny convex bodies. We show that, under the assumption that the width of the Algorithms, Theory body in any direction is at least ε, it is possible to approx- imate a convex body using O( area(K)/ε(d−1)/2) facets, Keywords where area(K) is the surface area of the body. -
Abstract Adaptive Sampling for Geometric
ABSTRACT Title of dissertation: ADAPTIVE SAMPLING FOR GEOMETRIC APPROXIMATION Ahmed Abdelkader Abdelrazek, Doctor of Philosophy, 2020 Dissertation directed by: Professor David M. Mount Computer Science Geometric approximation of multi-dimensional data sets is an essential algorith- mic component for applications in machine learning, computer graphics, and scientific computing. This dissertation promotes an algorithmic sampling methodology for a number of fundamental approximation problems in computational geometry. For each problem, the proposed sampling technique is carefully adapted to the geometry of the input data and the functions to be approximated. In particular, we study proximity queries in spaces of constant dimension and mesh generation in 3D. We start with polytope membership queries, where query points are tested for inclusion in a convex polytope. Trading-off accuracy for efficiency, we tolerate one-sided errors for points within an "-expansion of the polytope. We propose a sampling strategy for the placement of covering ellipsoids sensitive to the local shape of the polytope. The key insight is to realize the samples as Delone sets in the intrinsic Hilbert metric. Using this intrinsic formulation, we considerably simplify state-of-the-art techniques yielding an intuitive and optimal data structure. Next, we study nearest-neighbor queries which retrieve the most similar data point to a given query point. To accommodate more general measures of similarity, we consider non-Euclidean distances including convex distance functions and Bregman divergences. Again, we tolerate multiplicative errors retrieving any point no farther than (1 + ") times the distance to the nearest neighbor. We propose a sampling strategy sensitive to the local distribution of points and the gradient of the distance functions. -
Optimal Payments in Dominant-Strategy Mechanisms for Single-Parameter Domains
Optimal Payments in Dominant-Strategy Mechanisms for Single-Parameter Domains Victor Naroditskiy Maria Polukarov Nicholas R. Jennings School of Electronics and Computer Science, University of Southampton, United Kingdom fvn,mp3,[email protected] Abstract We study dominant-strategy mechanisms in allocation domains where agents have one-dimensional types and quasi-linear utilities. Taking an allocation function as an input, we present an algorithmic technique for finding opti- mal payments in a class of mechanism design problems, including utilitarian and egalitarian allocation of homogeneous items with nondecreasing marginal costs. Our results link optimality of payment functions to a geometric condi- tion involving triangulations of polytopes. When this condition is satisfied, we constructively show the existence of an optimal payment function that is piecewise linear in agent types. 1. Introduction A celebrated positive result in dominant-strategy implementation is the class of Groves mechanism [10] for quasi-linear environments. For unrestricted type spaces, the Groves class describes all deterministic dominant-strategy mechanisms [21]. When types are given by single numbers|in single-parameter domains|it characterizes all efficient mechanisms. Mechanisms within the Groves class differ from one another in the amounts of payments they col- lect from the agents. Until recently, most of the attention in the literature was given to a single Groves mechanism called VCG.1 Our work develops a 1Also known as the Vickrey-Clarke-Groves, the pivotal, or the Clarke mechanism. technique for finding the best mechanism from the Groves class (and a more general class that includes non-efficient mechanisms) for a given objective. To see the need to optimize over mechanisms in the Groves class, consider allocating free items among a group of participants each having a private value for receiving an item (or, more generally, scenarios with no residual claimant absorbing the surplus or covering the deficit). -
Approximate Polytope Membership Queries | SIAM Journal on Computing
SIAM J. COMPUT. c 2018 Society for Industrial and Applied Mathematics Vol. 47, No. 1, pp. 1{51 ∗ APPROXIMATE POLYTOPE MEMBERSHIP QUERIES SUNIL ARYAy , GUILHERME D. DA FONSECAz , AND DAVID M. MOUNTx Abstract. In the polytope membership problem, a convex polytope K in d is given, and R d the objective is to preprocess K into a data structure so that, given any query point q 2 R , it is possible to determine efficiently whether q 2 K. We consider this problem in an approximate setting. Given an approximation parameter ", the query can be answered either way if the distance from q to K's boundary is at most " times K's diameter. We assume that the dimension d is fixed, and K is presented as the intersection of n halfspaces. Previous solutions to approximate polytope membership were based on straightforward applications of classic polytope approximation techniques by Dudley [Approx. Theory, 10 (1974), pp. 227{236] and Bentley, Faust, and Preparata [Commun. ACM, 25 (1982), pp. 64{68]. The former is optimal in the worst case with respect to space, and the latter is optimal with respect to query time. We present four main results. First, we show how to combine the two above techniques to obtain a simple space-time trade-off. Second, we present an algorithm that dramatically improves this trade-off. In particular, for any constant α ≥ 4, this data structure achieves query time roughly O(1="(d−1)/α) and space roughly O (1="(d−1)(1−Ω(log α)/α)). We do not know whether this space bound is tight, but our third result showsp that there is a convex body such (d−1)(1−O( α)/α that our algorithm achieves a space of at least Ω(1=" ). -
Ten Problems in Geometry
Ten Problems in Geometry Moritz W. Schmitt G¨unter M. Ziegler∗ Institut f¨urMathematik Institut f¨urMathematik Freie Universit¨atBerlin Freie Universit¨atBerlin Arnimallee 2 Arnimallee 2 14195 Berlin, Germany 14195 Berlin, Germany [email protected] [email protected] May 1, 2011 Abstract We describe ten problems that wait to be solved. Introduction Geometry is a field of knowledge, but it is at the same time an active field of research | our understanding of space, about shapes, about geometric structures develops in a lively dialog, where problems arise, new questions are asked every day. Some of the problems are settled nearly immediately, some of them need years of careful study by many authors, still others remain as challenges for decades. Contents 1 Unfolding polytopes 2 2 Almost disjoint triangles 4 3 Representing polytopes with small coordinates 6 4 Polyhedra that tile space 8 5 Fatness 10 6 The Hirsch conjecture 12 7 Unimodality for cyclic polytopes 14 8 Decompositions of the cube 16 9 A problem concerning the ball and the cube 19 10 The 3d conjecture 21 ∗Partially supported by DFG, Research Training Group \Methods for Discrete Structures" 1 1 Unfolding polytopes In 1525 Albrecht D¨urer'sfamous geometry masterpiece \Underweysung der Messung mit dem Zirckel und Richtscheyt"1 was published in Nuremberg. Its fourth part contains many drawings of nets of 3-dimensional polytopes. Implicitly it contains the following conjecture: Every 3-dimensional convex polytope can be cut open along a spanning tree of its graph and then unfolded into the plane without creating overlaps. -
Focm 2017 Foundations of Computational Mathematics Barcelona, July 10Th-19Th, 2017 Organized in Partnership With
FoCM 2017 Foundations of Computational Mathematics Barcelona, July 10th-19th, 2017 http://www.ub.edu/focm2017 Organized in partnership with Workshops Approximation Theory Computational Algebraic Geometry Computational Dynamics Computational Harmonic Analysis and Compressive Sensing Computational Mathematical Biology with emphasis on the Genome Computational Number Theory Computational Geometry and Topology Continuous Optimization Foundations of Numerical PDEs Geometric Integration and Computational Mechanics Graph Theory and Combinatorics Information-Based Complexity Learning Theory Plenary Speakers Mathematical Foundations of Data Assimilation and Inverse Problems Multiresolution and Adaptivity in Numerical PDEs Numerical Linear Algebra Karim Adiprasito Random Matrices Jean-David Benamou Real-Number Complexity Alexei Borodin Special Functions and Orthogonal Polynomials Mireille Bousquet-Mélou Stochastic Computation Symbolic Analysis Mark Braverman Claudio Canuto Martin Hairer Pierre Lairez Monique Laurent Melvin Leok Lek-Heng Lim Gábor Lugosi Bruno Salvy Sylvia Serfaty Steve Smale Andrew Stuart Joel Tropp Sponsors Shmuel Weinberger 2 FoCM 2017 Foundations of Computational Mathematics Barcelona, July 10th{19th, 2017 Books of abstracts 4 FoCM 2017 Contents Presentation . .7 Governance of FoCM . .9 Local Organizing Committee . .9 Administrative and logistic support . .9 Technical support . 10 Volunteers . 10 Workshops Committee . 10 Plenary Speakers Committee . 10 Smale Prize Committee . 11 Funding Committee . 11 Plenary talks . 13 Workshops . 21 A1 { Approximation Theory Organizers: Albert Cohen { Ron Devore { Peter Binev . 21 A2 { Computational Algebraic Geometry Organizers: Marta Casanellas { Agnes Szanto { Thorsten Theobald . 36 A3 { Computational Number Theory Organizers: Christophe Ritzenhaler { Enric Nart { Tanja Lange . 50 A4 { Computational Geometry and Topology Organizers: Joel Hass { Herbert Edelsbrunner { Gunnar Carlsson . 56 A5 { Geometric Integration and Computational Mechanics Organizers: Fernando Casas { Elena Celledoni { David Martin de Diego . -
Arxiv:Math/0610904V3 [Math.MG] 2 Jul 2008
From the Mahler conjecture to Gauss linking integrals Greg Kuperberg∗ Department of Mathematics, University of California, Davis, CA 95616 Dedicated to my father, on no particular occasion We establish a version of the bottleneck conjecture, which in turn implies a partial solution to the Mahler conjecture on the product v(K) = (Vol K)(Vol K ) of the volume of a symmetric convex body K Rn and its ◦ ∈ polar body K◦. The Mahler conjecture asserts that the Mahler volume v(K) is minimized (non-uniquely) when K is an n-cube. The bottleneck conjecture (in its least general form) asserts that the volume of a certain domain π nγ K♦ K K◦ is minimized when K is an ellipsoid. It implies the Mahler conjecture up to a factor of 4 n, ⊆ × 4 where γn is a monotonic factor that begins at π and converges to √2. This strengthens a result of Bourgain and Milman, who showed that there is a constant c such that the Mahler conjecture is true up to a factor of cn. The proof uses a version of the Gauss linking integral to obtain a constant lower bound on Vol K♦, with equality when K is an ellipsoid. It applies to a more general conjecture concerning the join of any two necks of the pseudospheres of an indefinite inner product space. Because the calculations are similar, we will also n 1 n 1 analyze traditional Gauss linking integrals in the sphere S − and in hyperbolic space H − . 1. INTRODUCTION Theorem 1.3 (Bourgain, Milman). There is a constant c > 0 such that for any n and any centrally-symmetric convex body n K of dimension n, If K R is a centrally symmetric convex body, let K◦ de- note its⊂ dual or polar body. -
A Concentration Result of Estimating Phi-Divergence Using Data
A Concentration Result of Estimating Phi-Divergence using Data Dependent Partition Fengqiao Luo1 and Sanjay Mehrotra2 1,2Department of Industrial Engineering and Management Science, Northwestern University September 18, 2018 Abstract Estimation of the φ-divergence between two unknown probability distributions using empirical data is a fundamental problem in information theory and statistical learning. We consider a multi-variate generalization of the data dependent partitioning method for estimating divergence between the two unknown distributions. Under the assumption that the distribution satisfies a power law of decay, we provide a convergence rate result for this method on the number of samples and hyper-rectangles required to ensure the estimation error is bounded by a given level with a given probability. 1 Introduction d Let P and Q be two probability distributions on defined on (R , Rd ), where Rd is the Borel measure on B B Rd. The φ-divergence of Q from P is defined as: dP Dφ(P Q)= φ dQ. (1) || Rd dQ Z The φ-divergence family includes the Kullback-Leibler (KL) divergence (Kullback and Leibler, 1951), Hellinger distance (Nikulin, 2001), total variation distance, χ2-divergence, α-divergence among others. Many other information-theoretic quantities such as entropy and mutual information can be formulated as special cases of φ-divergence. When the distributions P and Q are unknown, the estimate of D (P Q) based on the i.i.d. φ || samples from P and Q is a fundamental problem in information theory and statistics. The estimation of arXiv:1801.00852v1 [math.PR] 2 Jan 2018 φ-divergence as well as entropy and mutual information has many important applications. -
Isotropic Constants and Mahler Volumes
Isotropic constants and Mahler volumes Bo’az Klartag Abstract This paper contains a number of results related to volumes of projective perturba- tions of convex bodies and the Laplace transform on convex cones. First, it is shown that a sharp version of Bourgain’s slicing conjecture implies the Mahler conjecture for convex bodies that are not necessarily centrally-symmetric. Second, we find that by slightly translating the polar of a centered convex body, we may obtain another body with a bounded isotropic constant. Third, we provide a counter-example to a conjecture by Kuperberg on the distribution of volume in a body and in its polar. 1 Introduction This paper describes interrelations between duality and distribution of volume in convex bodies. A convex body is a compact, convex subset K Rn whose interior Int(K) is non-empty. If 0 Int(K), then the polar body is defined by⊆ ∈ K◦ = y Rn ; x K, x, y 1 . { ∈ ∀ ∈ h i ≤ } The polar body K◦ is itself a convex body with the origin in its interior, and moreover (K◦)◦ = K. The Mahler volume of a convex body K Rn with the origin in its interior is defined as ⊆ ◦ s(K)= V oln(K) V oln(K ), arXiv:1710.08084v2 [math.MG] 1 Mar 2018 · where V oln is n-dimensional volume. In the class of convex bodies with barycenter at the origin, the Mahler volume is maximized for ellipsoids, as proven by Santal´o[29], see also Meyer and Pajor [20]. The Mahler conjecture suggests that for any convex body K Rn containing the origin in its interior, ⊆ (n + 1)n+1 s(K) s(∆n)= , (1) ≥ (n!)2 where ∆n Rn is any simplex whose vertices span Rn and add up to zero.