ź-nets and simplex range queries
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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TL;DR: In this paper, it was shown that the VC-dimension of convex sets in the plane is bounded by 3, and this is tight for intersecting segments in the convex set.
Abstract: A family S of convex sets in the plane defines a hypergraph H = (S, E) as follows. Every subfamily S' of S defines a hyperedge of H if and only if there exists a halfspace h that fully contains S' , and no other set of S is fully contained in h. In this case, we say that h realizes S'. We say a set S is shattered, if all its subsets are realized. The VC-dimension of a hypergraph H is the size of the largest shattered set.
We show that the VC-dimension for pairwise disjoint convex sets in the plane is bounded by 3, and this is tight. In contrast, we show the VC-dimension of convex sets in the plane (not necessarily disjoint) is unbounded. We provide a quadratic lower bound in the number of pairs of intersecting sets in a shattered family of convex sets in the plane. We also show that the VC-dimension is unbounded for pairwise disjoint convex sets in R^d , for d > 2. We focus on, possibly intersecting, segments in the plane and determine that the VC-dimension is always at most 5. And this is tight, as we construct a set of five segments that can be shattered. We give two exemplary applications. One for a geometric set cover problem and one for a range-query data structure problem, to motivate our findings.
1 citations
Posted Content•
TL;DR: It is proved a lower bound that any data structure that answers queries with error O(log n) must use Ω(n log n) bits, and the upper bound cannot be improved in general by more than an O( log log 1/e) factor.
Abstract: We study the problem of $2$-dimensional orthogonal range counting with additive error. Given a set $P$ of $n$ points drawn from an $n\times n$ grid and an error parameter $\eps$, the goal is to build a data structure, such that for any orthogonal range $R$, it can return the number of points in $P\cap R$ with additive error $\eps n$. A well-known solution for this problem is the {\em $\eps$-approximation}, which is a subset $A\subseteq P$ that can estimate the number of points in $P\cap R$ with the number of points in $A\cap R$. It is known that an $\eps$-approximation of size $O(\frac{1}{\eps} \log^{2.5} \frac{1}{\eps})$ exists for any $P$ with respect to orthogonal ranges, and the best lower bound is $\Omega(\frac{1}{\eps} \log \frac{1}{\eps})$. The $\eps$-approximation is a rather restricted data structure, as we are not allowed to store any information other than the coordinates of the points in $P$. In this paper, we explore what can be achieved without any restriction on the data structure. We first describe a simple data structure that uses $O(\frac{1}{\eps}(\log^2\frac{1} {\eps} + \log n) )$ bits and answers queries with error $\eps n$. We then prove a lower bound that any data structure that answers queries with error $\eps n$ must use $\Omega(\frac{1}{\eps}(\log^2\frac{1} {\eps} + \log n) )$ bits. Our lower bound is information-theoretic: We show that there is a collection of $2^{\Omega(n\log n)}$ point sets with large {\em union combinatorial discrepancy}, and thus are hard to distinguish unless we use $\Omega(n\log n)$ bits.
1 citations
01 Jan 2009
TL;DR: A new and arguably simpler proof of the well known centerpoint theorem in any dimension is given and the same idea is used to prove an optimal generalization of the centerpoint to two points in the plane.
Abstract: This thesis deals with strong and weak ǫ-nets in geometry and related problems. In the first half of the thesis we look at strong ǫ-nets and the closely related problem of finding minimum hitting sets. We give a new technique for proving the existence of small ǫ-nets for several geometric range spaces. Our technique also gives efficient algorithms to compute small ǫ-nets. By a well known reduction due to Bronimann and Goodrich [10], our results imply constant factor approximation algorithms for the corresponding minimum hitting set problems. We show how the approximation factor given by this standard technique can be improved by giving the first polynomial time approximation scheme for some of the minimum hitting set problems. The algorithm is a very simple and is based on local search. In the second half of the thesis, we turn to weak ǫ-nets, a very important generalization of the idea of strong ǫ-nets for convex ranges. We first consider the simplest example of a weak ǫ-net, namely the centerpoint. We give a new and arguably simpler proof of the well known centerpoint theorem (and also Helly’s theorem) in any dimension and use the same idea to prove an optimal generalization of the centerpoint to two points in the plane. Our technique also gives several improved results for small weak ǫ-nets in the plane. We finally look at the general weak ǫ-net problem is d-dimensions. A long standing conjecture states that weak ǫ-nets of size O(ǫ−1polylogǫ−1) exist for convex sets in any dimension. It turns out that if the conjecture is true then it should be possible to construct a weak ǫ-net from a small number of input points. We show that this is indeed true and it is possible to construct a weak ǫ-net from O(ǫ−1polylogǫ−1) input points. We also show an interesting connection between weak and strong ǫ-nets which shows how random sampling can be used to construct weak ǫ-nets.
1 citations
Cites background from "ź-nets and simplex range queries"
...The ǫ-net theorem (Welzl and Haussler [26]) states that there exists an ǫ-net of sizeO(dǫ−1 log(ǫ−1)) for any range space with VC-dimension d....
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...The idea of strongǫ-nets was extended to weakǫ-nets for convex sets by Haussler and Welzl in their seminal paper [26]....
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...The concept of weakǫ-nets with respect to convex ranges was introduced by Haussler and Welzl [26] and the notion has found several applicationsn discrete and combinatorial geometry (see Matousek’s book for several examples [35])....
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...This range space has constant VC-dimension (dep en ing onk), and from the result of Haussler and Welzl [26], it follows that a random sa mple of sizeO(ǫ−1 log(ǫ−1)) is anǫ-net forRk with some constant probability....
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...Haussler and Welzl [26], who introduced the notion ofǫ-nets, showed that range spaces with a small VC dimension admit a smallǫ-net....
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Posted Content•
TL;DR: In this paper, a new, efficient version of the hypergraph container theorems that is suited for hypergraphs with large uniformities has been proposed, which is a refined approach to constructing containers that employs simple ideas from high-dimensional convex geometry.
Abstract: We prove a new, efficient version of the hypergraph container theorems that is suited for hypergraphs with large uniformities. The main novelty is a refined approach to constructing containers that employs simple ideas from high-dimensional convex geometry. The existence of smaller families of containers for independent sets in such hypergraphs, which is guaranteed by the new theorem, allows us to improve upon the best currently known bounds for several problems in extremal graph theory, discrete geometry, and Ramsey theory.
1 citations
Posted Content•
TL;DR: This paper proposes a framework for reasoning about learning under arbitrary quantizations, and proves the convergence of quantization-aware versions of the Perceptron and Frank-Wolfe algorithms.
Abstract: There is a mismatch between the standard theoretical analyses of statistical machine learning and how learning is used in practice. The foundational assumption supporting the theory is that we can represent features and models using real-valued parameters. In practice, however, we do not use real numbers at any point during training or deployment. Instead, we rely on discrete and finite quantizations of the reals, typically floating points. In this paper, we propose a framework for reasoning about learning under arbitrary quantizations. Using this formalization, we prove the convergence of quantization-aware versions of the Perceptron and Frank-Wolfe algorithms. Finally, we report the results of an extensive empirical study of the impact of quantization using a broad spectrum of datasets.
1 citations
Cites background from "ź-nets and simplex range queries"
...Then, the standard sample complexity bounds [Haussler and Welzl, 1987, Vapnik and Chervonenkis, 1971, Li et al., 2001] apply directly to the quantized versions as claimed....
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References
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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 1987TL;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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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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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
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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