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 article, a weak e-net theorem for convexity spaces with bounded Radon number is given. But it is not a weak one, as shown in this paper.
Abstract: A basic measure of the combinatorial complexity of a convexity space is its Radon number. In this paper we answer a question of Kalai, by showing a fractional Helly theorem for convexity spaces with bounded Radon number. As a consequence we also get a weak e-net theorem for convexity spaces with bounded Radon number. This answers a question of Bukh and extends a recent result of Moran and Yehudayoff.
5 citations
09 Jan 1995
TL;DR: This paper extends the e-approximation for the 2-label space originally considered by Vapnik and Chervonenkis to that for the k- label space and applies it to the randomized algorithm for the assignment problem by Tokuyama and Nakano.
Abstract: In learning a geometric concept from examples, examples are mostly classified into two, positive examples and negative examples which are contained and not contained, respectively, in the concept. However, there exist cases where examples are classified into k classes. For example, clustering a concept space by the Voronoi diagram generated by k points is a very common tool used in image processing and many other areas. We call such a space a k-label space. The case of positive and negative examples corresponds to the 2-label space. In this paper, we first extend the e-approximation for the 2-label space originally considered by Vapnik and Chervonenkis [10] (see also [1, 5]) to that for the k-label space. The generalized e-approximation is then applied to the randomized algorithm for the assignment problem by Tokuyama and Nakano
5 citations
Cites background or methods from "ź-nets and simplex range queries"
...for every concept R E 7~, where Member(9, R) is the number of elements of R in Y. Haussler and Welzl [ 5 ] defined an e-approximation for only a finite set X and a non-probabilistic concept space....
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...Vapnik and Chervonenkis [10] and others [1, 5 ] play a central role in discussing the learning complexity of the space....
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...In this paper, we first extend the e-approximation for the 2-1abel space originally considered by Vapnik and Chervonenkis [10] (see also [1, 5 ]) to that for the k-label space....
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Posted Content•
TL;DR: In this paper, the density of subgraphs of hypercubes of Cartesian products was studied and shown to be polylogarithmic in the size of adjacency labeling schemes.
Abstract: In this paper, we extend two classical results about the density of subgraphs of hypercubes to subgraphs $G$ of Cartesian products $G_1\times\cdots\times G_m$ of arbitrary connected graphs. Namely, we show that $\frac{|E(G)|}{|V(G)|}\le \lceil 2\max\{ \text{dens}(G_1),\ldots,\text{dens}(G_m)\} \rceil\log|V(G)|$, where $\text{dens}(H)$ is the maximum ratio $\frac{|E(H')|}{|V(H')|}$ over all subgraphs $H'$ of $H$. We introduce the notions of VC-dimension $\text{VC-dim}(G)$ and VC-density $\text{VC-dens}(G)$ of a subgraph $G$ of a Cartesian product $G_1\times\cdots\times G_m$, generalizing the classical Vapnik-Chervonenkis dimension of set-families (viewed as subgraphs of hypercubes). We prove that if $G_1,\ldots,G_m$ belong to the class ${\mathcal G}(H)$ of all finite connected graphs not containing a given graph $H$ as a minor, then for any subgraph $G$ of $G_1\times\cdots\times G_m$ a sharper inequality $\frac{|E(G)|}{|V(G)|}\le \text{VC-dim}(G)\alpha(H)$ holds, where $\alpha(H)$ is the density of the graphs from ${\mathcal G}(H)$. We refine and sharpen those two results to several specific graph classes. We also derive upper bounds (some of them polylogarithmic) for the size of adjacency labeling schemes of subgraphs of Cartesian products.
5 citations
24 Aug 1993
TL;DR: This work considers the recursion theoretic properties of Ω with special emphasis on r.e. sets, the class of sets A such that for some n the n-fold characteristic function of A can be computed with less than n errors.
Abstract: The notion of frequency computation captures the class Ω of all sets A such that for some n the n-fold characteristic function of A can be computed with less than n errors. Alternatively, it can be computed by a total oracle machine with less than n queries to the oracle. We consider the recursion theoretic properties of Ω with special emphasis on r.e. sets.
5 citations
Dissertation•
01 Jan 1991
TL;DR: This thesis analyses some of the more mathematical aspects of the Probably Approximately Correct (PAC) model of computational learning theory and finds a sufficient sample-size involving the Vapnik-Chervonenkis dimension of the hypothesis space is derived.
Abstract: This thesis analyses some of the more mathematical aspects of the Probably Approximately Correct (PAC) model of computational learning theory. The main concern is with the sample size required for valid learning in the PAC model. A sufficient sample-size involving the Vapnik-Chervonenkis (VC) dimension of the hypothesis space is derived; this improves the best previously known bound of this nature. Learnability results and sufficient sample-sizes can in many cases be derived from results of Vapnik on the uniform convergence (in probability) of relative frequencies of events to their probabilities, when the collection of events has finite VC dimension. Two simple new combinatorial proofs of each of two of Vapnik's results are proved here and the results are then applied to the theory of learning stochastic concepts, where again improved sample-size bounds are obtained. The PAC model of learning is a distribution-free model; the resulting sample sizes are not permitted to depend on the usually fixed but unknown probability distribution on the input space. Results of Ben-David, Benedek and Mansour are described, presenting a theory for distribution-dependent learnability. The conditions under which a feasible upper bound on sample-size can be obtained are investigated, introducing the concept of polynomial Xo-finite dimension. The theory thus far is then applied to the learnability of formal concepts, defined by Wille. A learning algorithm is also presented for this problem. Extending the theory of learnability to the learnability of functions which have range in some arbitrary set, learnability results and sample-size bounds, depending on a generalization of the VC dimension, are obtained and these results are applied to the theory of artificial neural networks. Specifically, a sufficient sample-size for valid generalization in multiple-output feedforward linear threshold networks is found.
5 citations
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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