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Journal ArticleDOI

ź-nets and simplex range queries

01 Dec 1987-Discrete and Computational Geometry (Springer New York)-Vol. 2, Iss: 1, pp 127-151
TL;DR: The concept of an ɛ-net of a set of points for an abstract set of ranges is introduced and sufficient conditions that a random sample is an Â-net with any desired probability are given.
Abstract: We demonstrate the existence of data structures for half-space and simplex range queries on finite point sets ind-dimensional space,dÂ?2, with linear storage andO(nÂ?) query time, $$\alpha = \frac{{d(d - 1)}}{{d(d - 1) + 1}} + \gamma for all \gamma > 0$$ . These bounds are better than those previously published for alldÂ?2. Based on ideas due to Vapnik and Chervonenkis, we introduce the concept of an Â?-net of a set of points for an abstract set of ranges and give sufficient conditions that a random sample is an Â?-net with any desired probability. Using these results, we demonstrate how random samples can be used to build a partition-tree structure that achieves the above query time.

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Citations
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TL;DR: This work formally study the problem of classification under adversarial perturbations, both from the learner's perspective, and from the viewpoint of a third-party who aims at certifying the robustness of a given black-box classifier.
Abstract: We formally study the problem of classification under adversarial perturbations, both from the learner's perspective, and from the viewpoint of a third-party who aims at certifying the robustness of a given black-box classifier. We analyze a PAC-type framework of semi-supervised learning and identify possibility and impossibility results for proper learning of VC-classes in this setting. We further introduce and study a new setting of black-box certification under limited query budget. We analyze this for various classes of predictors and types of perturbation. We also consider the viewpoint of a black-box adversary that aims at finding adversarial examples, showing that the existence of an adversary with polynomial query complexity implies the existence of a robust learner with small sample complexity.

13 citations

Book ChapterDOI
15 Dec 2010
TL;DR: An algorithm is presented that, for any e> 0, finds an overlap at least the optimum minus e and reports a translation realizing it and computes the maximum overlap of the perturbed polytopes.
Abstract: We study the problem of maximizing the overlap of two convex polytopes under translation in \({\mathbb R}^d\) for some constant d ≥ 3. Let n be the number of bounding hyperplanes of the polytopes. We present an algorithm that, for any e> 0, finds an overlap at least the optimum minus e and reports a translation realizing it. The running time is \(O(n^{{\lfloor d/2 \rfloor}+1} \log^d n)\) with probability at least 1 − n − O(1), which can be improved to O(nlog3.5 n) in \({\mathbb R}^3\). The time complexity analysis depends on a bounded incidence condition that we enforce with probability one by randomly perturbing the input polytopes. This causes an additive error e, which can be made arbitrarily small by decreasing the perturbation magnitude. Our algorithm in fact computes the maximum overlap of the perturbed polytopes. All bounds and their big-O constants are independent of e.

13 citations


Cites background from "ź-nets and simplex range queries"

  • ...Lemmas 1 and 2 below state the results on the e-net theory [ 10 ] and cuttings [6] that we use heavily....

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Journal ArticleDOI
TL;DR: This survey describes recent progress in the area of mass partition problems in addition to its connections to topology, discrete geometry, and computer science.
Abstract: Mass partition problems describe the partitions we can induce on a family of measures or finite sets of points in Euclidean spaces by dividing the ambient space into pieces. In this survey we describe recent progress in the area in addition to its connections to topology, discrete geometry, and computer science.

13 citations

Book ChapterDOI
17 Jun 2015
TL;DR: In this article, it was shown that for any positive integer m, every m-fold covering of a region with unit disks splits into two coverings, provided that every point is covered by at most c 2 m/2 sets.
Abstract: A system of sets forms an m-fold covering of a set X if every point of X belongs to at least m of its members. A 1-fold covering is called a covering. The problem of splitting multiple coverings into several coverings was motivated by classical density estimates for sphere packings as well as by the planar sensor cover problem. It has been the prevailing conjecture for 35i¾?years settled in many special cases that for every plane convex body C, there exists a constant $$m=mC$$ such that every m-fold covering of the plane with translates of C splits into 2 coverings. In the present paper, it is proved that this conjecture is false for the unit disk. The proof can be generalized to construct, for every m, an unsplittable m-fold covering of the plane with translates of any open convex body C which has a smooth boundary with everywhere positive curvature. Somewhat surprisingly, unbounded open convex sets C do not misbehave, they satisfy the conjecture: every 3-fold covering of any region of the plane by translates of such a set C splits into two coverings. To establish this result, we prove a general coloring theorem for hypergraphs of a special type: shift-chains. We also show that there is a constant $$c>0$$ such that, for any positive integer m, every m-fold covering of a region with unit disks splits into two coverings, provided that every point is covered by at most$$c2^{m/2}$$ sets.

13 citations

References
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Book ChapterDOI
TL;DR: This chapter reproduces the English translation by B. Seckler of the paper by Vapnik and Chervonenkis in which they gave proofs for the innovative results they had obtained in a draft form in July 1966 and announced in 1968 in their note in Soviet Mathematics Doklady.
Abstract: This chapter reproduces the English translation by B. Seckler of the paper by Vapnik and Chervonenkis in which they gave proofs for the innovative results they had obtained in a draft form in July 1966 and announced in 1968 in their note in Soviet Mathematics Doklady. The paper was first published in Russian as Вапник В. Н. and Червоненкис А. Я. О равномерноЙ сходимости частот появления событиЙ к их вероятностям. Теория вероятностеЙ и ее применения 16(2), 264–279 (1971).

3,939 citations


"ź-nets and simplex range queries" refers background or methods or result in this paper

  • ...The drawback is that the constants, if deri~,ed from the results in [ 17 ], can be quite large....

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  • ...More generally, we characterize the classes of ranges for which there exists a function f(E) for e S0 such that any finite point set A has an e-net of size f(e), independently of the size of A. These are precisely the classes of ranges with finite Vapnik-Chervonenkis dimension, known as Vapnik-Chervonenkis classes [ 17 ], [9], [19], [1]....

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  • ...The key concepts and proof techniques of this section are based on the pioneering work of Vapnik and Chervonenkis [ 17 ]....

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  • ...Example 5. Let A be a set of n points in E 2. Since the dimension of (E 2, H~-) is 2, the results in [ 17, Theorem 2 ] show that there exists a 0.01-approximation V of A for positive half-planes (and thus for all half-planes) with I VI = 2,525,039....

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  • ...Using the related notion of an e-approxirnation (directly from [ 17 ]), we also point out trivial data structures of constant size that give approximate solutions to the counting problem for halfspaces in constant time (compare [13])....

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Book
01 Jan 1987
TL;DR: This book offers a modern approach to computational geo- metry, an area thatstudies the computational complexity of geometric problems with an important role in this study.
Abstract: This book offers a modern approach to computational geo- metry, an area thatstudies the computational complexity of geometric problems. Combinatorial investigations play an important role in this study.

2,284 citations


"ź-nets and simplex range queries" refers background in this paper

  • ...We conclude this section by examining the relationship between the notion of an e-net and the established notion of a centerpoint [21], [11] in combinatorial geometry....

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  • ..., [11] for a general treatment of arrangements....

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Journal ArticleDOI
TL;DR: This paper will answer the question in the affirmative by determining the exact upper bound of T if T is a family of subsets of some infinite set S then either there exists to each number n a set A ⊂ S with |A| = n such that |T ∩ A| = 2n or there exists some number N such that •A| c for each A⩾ N and some constant c.

1,029 citations


"ź-nets and simplex range queries" refers background in this paper

  • ...Now the assertion can be seen as the dual formulation of Caratheodry's theorem (see [ 15 ], Theorem 2.3.5), which states that if a point x is in the convex hull of a set A in E d, then there exists a subset A' of A such that JA'I -< d + 1 and x is in the convex hull of A'. []...

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Journal ArticleDOI
TL;DR: In this article, the convergence of a stochastic process indexed by a Gaussian process to a certain Gaussian processes indexed by the supremum norm was studied in a Donsker class.
Abstract: Let $(X, \mathscr{A}, P)$ be a probability space. Let $X_1, X_2,\cdots,$ be independent $X$-valued random variables with distribution $P$. Let $P_n := n^{-1}(\delta_{X_1} + \cdots + \delta_{X_n})$ be the empirical measure and let $ u_n := n^\frac{1}{2}(P_n - P)$. Given a class $\mathscr{C} \subset \mathscr{a}$, we study the convergence in law of $ u_n$, as a stochastic process indexed by $\mathscr{C}$, to a certain Gaussian process indexed by $\mathscr{C}$. If convergence holds with respect to the supremum norm $\sup_{C \in \mathscr{C}}|f(C)|$, in a suitable (usually nonseparable) function space, we call $\mathscr{C}$ a Donsker class. For measurability, $X$ may be a complete separable metric space, $\mathscr{a} =$ Borel sets, and $\mathscr{C}$ a suitable collection of closed sets or open sets. Then for the Donsker property it suffices that for some $m$, and every set $F \subset X$ with $m$ elements, $\mathscr{C}$ does not cut all subsets of $F$ (Vapnik-Cervonenkis classes). Another sufficient condition is based on metric entropy with inclusion. If $\mathscr{C}$ is a sequence $\{C_m\}$ independent for $P$, then $\mathscr{C}$ is a Donsker class if and only if for some $r, \sigma_m(P(C_m)(1 - P(C_m)))^r < \infty$.

555 citations

Journal ArticleDOI
TL;DR: A new formulation of the notion of duality that allows the unified treatment of a number of geometric problems is used, to solve two long-standing problems of computational geometry and to obtain a quadratic algorithm for computing the minimum-area triangle with vertices chosen amongn points in the plane.
Abstract: This paper uses a new formulation of the notion of duality that allows the unified treatment of a number of geometric problems. In particular, we are able to apply our approach to solve two long-standing problems of computational geometry: one is to obtain a quadratic algorithm for computing the minimum-area triangle with vertices chosen amongn points in the plane; the other is to produce an optimal algorithm for the half-plane range query problem. This problem is to preprocessn points in the plane, so that given a test half-plane, one can efficiently determine all points lying in the half-plane. We describe an optimalO(k + logn) time algorithm for answering such queries, wherek is the number of points to be reported. The algorithm requiresO(n) space andO(n logn) preprocessing time. Both of these results represent significant improvements over the best methods previously known. In addition, we give a number of new combinatorial results related to the computation of line arrangements.

286 citations


"ź-nets and simplex range queries" refers methods in this paper

  • ...It should be noted that better bounds are possible for reporting in two dimensions (specifically O(log n + t) time, where t is the number of points reported [3]), but these techniques only work for half-planes....

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