A Simple Proof of Optimal Epsilon Nets

Mustafa, Nabil H.; Dutta, Kunal; Ghosh, Arijit

doi:10.1007/s00493-017-3564-5

Cited by 14 publications

(17 citation statements)

References 32 publications

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“…The first part of the following Theorem is an improvement of a recent theorem from [30] (see also the related discussions there), and the technique of the proof is similar. We should note, however, that the technique of the proof is also closely related to the peeling technique which originates from the empirical processes theory and which is widely used in the Learning Theory [7,8,35,36].…”

Section: A New Bound For ǫ-Netsmentioning

confidence: 92%

“…The upper bound in Theorem 4 (i) is sharp for constant d, since it is a strengthening of an upper bound from [30] (see Section 5), which was recently shown to be tight in some specific cases in [22]. The upper bound in Theorem 4 (ii) may be stated in terms of τ i , but the formulation gets rather complicated, so we decided to omit it.…”

Section: A New Bound For ǫ-Netsmentioning

confidence: 96%

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When are epsilon-nets small?

Kupavskii

Zhivotovskiy

2020

Journal of Computer and System Sciences

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Given a range space (X , R), where X is a set equipped with probability measure P and R ⊂ 2 X is a family of measurable subsets, and ǫ > 0, an ǫ-net is a subset of X in the support of P, which intersects each R ∈ R with P(R) ≥ ǫ. In many interesting situations the size of ǫ-nets depends only on ǫ together with different complexity measures. The aim of this paper is to give a systematic treatment of such complexity measures arising in Discrete and Computational Geometry and Statistical Learning, and to bridge the gap between the results appearing in these two fields. As a byproduct, we obtain several new upper bounds on the sizes of ǫ-nets that generalize/improve the best known general guarantees. Some of our results deal with improvements in logarithmic factors (which is a subject of several classical problems in Learning Theory and Computational Geometry), while others work in regimes when ǫ-nets of size o( 1 ǫ ) exist. Inspired by results in Statistical Learning, we also give a short proof of the Haussler's upper bound on packing numbers [20]. ǫ-nets. Combinatorial and geometric point of viewConsider a set X equipped with probability measure P and a family of measurable subsets R ⊂ 2 X , where 2 X is the power set of X . For simplicity and to avoid potential measurability issues we assume that the set X is finite. The pair (X , R) is called a range space. For a fixed ǫ > 0, a subset S ⊂ X ∩ supp(P) is an ǫ-net, 1 if R ∩ S = ∅ for each R ∈ R with P(R) ≥ ǫ. ǫ-nets often arise in the context of Computational

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Section: A New Bound For ǫ-Netsmentioning

confidence: 92%

Section: A New Bound For ǫ-Netsmentioning

confidence: 96%

When are epsilon-nets small?

Kupavskii

Zhivotovskiy

2020

Journal of Computer and System Sciences

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“…Given a set P of points in R d , and a parameter ε > 0, a set Q ⊆ P is an ε-net for P with respect to half-spaces if any half-space containing at least ε • |P| points of P contains at least one point of Q. It is well known that there always exists such an ε-net of size independent of |P| (see also [13]). Given an integer k 1, a set P ⊆ P is called a k-set of P if |P | = k and if there exists a halfspace h in R d such that P = P ∩ h. A set P ⊆ P is called a ( k)-set if P is an l-set for some l k. The next well-known theorem gives an upper bound on the number of ( k)-sets in a point set (see also [18]).…”

Section: Proof Of Theorem 13mentioning

confidence: 99%

On a Problem of Danzer

Mustafa¹,

Ray²

2018

Combinator. Probab. Comp.

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Let C be a bounded convex object in ℝd, and let P be a set of n points lying outside C. Further, let cp, cq be two integers with 1 ⩽ cq ⩽ cp ⩽ n - ⌊d/2⌋, such that every cp + ⌊d/2⌋ points of P contain a subset of size cq + ⌊d/2⌋ whose convex hull is disjoint from C. Then our main theorem states the existence of a partition of P into a small number of subsets, each of whose convex hulls are disjoint from C. Our proof is constructive and implies that such a partition can be computed in polynomial time.In particular, our general theorem implies polynomial bounds for Hadwiger--Debrunner (p, q) numbers for balls in ℝd. For example, it follows from our theorem that when p > q = (1+β)⋅d/2 for β > 0, then any set of balls satisfying the (p, q)-property can be hit by O((1+β)2d2p1+1/β logp) points. This is the first improvement over a nearly 60 year-old exponential bound of roughly O(2d).Our results also complement the results obtained in a recent work of Keller, Smorodinsky and Tardos where, apart from improvements to the bound on HD(p, q) for convex sets in ℝd for various ranges of p and q, a polynomial bound is obtained for regions with low union complexity in the plane.

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“…Kaufmann [8], introduced and provided the theory of fuzzy hypergraphs as a generalisation of concept of hypergraphs, in such a way that fuzzy hypergraphs have important applications to decision making, mobile network and similar applications [9,10]. Further materials regarding graphs and hypergraphs are available in the literature too [11][12][13][14][15][16][17][18][19][20][21][22][23].…”

Section: Introductionmentioning

confidence: 99%

On KM-Fuzzy Metric Hypergraphs

Hamidi

Jahanpanah

Radfar

2020

Fuzzy Information and Engineering

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This paper, applies the concept of KM-fuzzy metric spaces and introduces a novel concept of KM-fuzzy metric hypergraphs based on KM-fuzzy metric spaces. In special cases, we add some conditions to axioms of KM-fuzzy metric hypergraphs(to obtain of elementary hypergraphs, C-accessible hypergraphs, Cor-able hypergraphs, fuzzy hypergraphs) and so obtain locally strong KM-fuzzy metric hypergraphs and strong KM-fuzzy metric hypergraphs. This study, investigates on the finite KM-fuzzy metric spaces with respect to metrics, KM-fuzzy metrics and constructs KM-fuzzy metric spaces on any given non-empty sets. It tries to extend the concept of KM-fuzzy metric spaces to union of KM-fuzzy metric spaces and product of KM-fuzzy metric spaces and in this regard investigates on union and product of KM-fuzzy metric hypergraphs.

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A Simple Proof of Optimal Epsilon Nets

Cited by 14 publications

References 32 publications

When are epsilon-nets small?

When are epsilon-nets small?

On a Problem of Danzer

On KM-Fuzzy Metric Hypergraphs

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