Metric Embeddings

Ostrovskii, Mikhail I.

doi:10.1515/9783110264012

Cited by 50 publications

(31 citation statements)

References 8 publications

(9 reference statements)

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“…We now list some open problems. The case p = 2 was originally raised by Ostrovskii in [2], who asked:…”

Section: Further Remarks and Open Problemsmentioning

confidence: 99%

“…The general question of Ostrovskii, given in [1], still remains open: Question 4. Let X be an infinite-dimensional Banach space containing p isomorphically.…”

Section: Further Remarks and Open Problemsmentioning

confidence: 99%

“…Our starting point is the following question due to Ostrovskii [1]: Question 1. Suppose 1 < p < ∞, and that X is a Banach space that contains an isomorphic copy of p .…”

Section: Introductionmentioning

confidence: 99%

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On embeddings of finite subsets of ℓ_{𝑝}

Kilbane

2018

Proc. Amer. Math. Soc.

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“…We now list some open problems. The case p = 2 was originally raised by Ostrovskii in [2], who asked:…”

Section: Further Remarks and Open Problemsmentioning

confidence: 99%

“…The general question of Ostrovskii, given in [1], still remains open: Question 4. Let X be an infinite-dimensional Banach space containing p isomorphically.…”

Section: Further Remarks and Open Problemsmentioning

confidence: 99%

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On embeddings of finite subsets of ℓ_{𝑝}

Kilbane

2018

Proc. Amer. Math. Soc.

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“…Metric dimension reduction. Using standard terminology from metric embeddings (see [Ost13]), we say that a mapping between metric spaces f : (M, d M ) → (N, d N ) is a bi-Lipschitz embedding with distortion at most α ∈ [1, ∞) if there exists a scaling factor σ ∈ (0, ∞) such that ∀ x, y ∈ M, σ d M (x, y) ≤ d N f (x), f (y) ≤ ασ d M (x, y).…”

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confidence: 99%

$\varepsilon$-isometric dimension reduction for incompressible subsets of $\ell_p$

Eskenazis¹

2021

Preprint

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Fix p ∈ [1, ∞), K ∈ (0, ∞) and a probability measure µ. We prove that for every n ∈ N, ε ∈ (0, 1) and x 1 , . . . , x n ∈ L p (µ) with max i∈{1,...,n} |x i | L p (µ) ≤ K, there exists d ≤ 32e 2 (2K) 2p log n ε 2 and vectors y 1 , . . . , y n ∈ d p such thatMoreover, the argument implies the existence of a greedy algorithm which outputs {y i } n i=1 after receiving {x i } n i=1 as input. The proof relies on a derandomized version of Maurey's empirical method (1981) combined with a combinatorial idea of Ball (1990) and classical factorization theory of L p (µ) spaces. Motivated by the above embedding, we introduce the notion of ε-isometric dimension reduction of the unit ball B E of a normed space (E, • E ) and we prove that B p does not admit ε-isometric dimension reduction by linear operators for any value of p 2.

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“…In view of the extensive literature on the Sparsest Cut Problem, it would be needlessly repetitive to recount here the rich and multifaceted impact of this optimization problem on computer science and mathematics; see instead the articles [1,59], the surveys [16,62,73,85], Chapter 10 of the monograph [32], Chapter 15 of the monograph [68], Chapter 1 of the monograph [80], and the references therein. It su ces to say that by tuning the choice of matrices C, D to the problem at hand, the minimization in (1) nds a partition of the "universe" {1, .…”

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confidence: 99%

The integrality gap of the Goemans-Linial SDP relaxation for sparsest cut is at least a constant multiple of √log n

Young

2017

Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing

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We prove that the integrality gap of the Goemans-Linial semide nite programming relaxation for the Sparsest Cut Problem is Ω( log n) on inputs with n vertices, thus matching the upper bound (log n) 1 2 +o(1) of [3] up to lower-order factors. This statement is a consequence of the following new isoperimetric-type inequality. Consider the 8-regular graph whose vertex set is the 5-dimensional integer grid Z 5 and where each vertex (a, b, c, d, e) ∈ Z 5 is connected to the 8 vertices (a ± 1, b, c, d, e), (a, b ± 1, c, d, e), (a, b, c ± 1, d, e ± a), (a, b, c, d ± 1, e ± b). This graph is known as the Cayley graph of the 5-dimensional discrete Heisenberg group. Given Ω ⊆ Z 5 , denote the size of its edge boundary in this graph (a.k.a.We show that every subset Ω ⊆ Z 5 satis es |∂ v Ω| = O(|∂ h Ω|). This vertical-versushorizontal isoperimetric inequality yields the above-stated integrality gap for Sparsest Cut and answers several geometric and analytic questions of independent interest.The theorem stated above is the culmination of a program that was pursued in the works [7, 19-22, 53, 56] whose aim is to understand the performance of the Goemans-Linial semide nite program through the embeddability properties of Heisenberg groups. These investigations have mathematical signi cance even beyond their established relevance to approximation algorithms and combinatorial optimization. In particular they contribute to a range of mathematical disciplines including functional analysis, geometric group theory, harmonic analysis, sub-Riemannian geometry, geometric measure theory, ergodic theory, group representations, * The research presented here was conducted under the auspices of the Simons Algorithms and Geometry (A&G) Think Tank. A full version of this extended abstract, titled "Vertical perimeter versus horizontal perimeter," that contains complete proofs and additional results is available at https://arxiv.org/abs/1701.00620. Nevertheless, this extended abstract contains material that is not included in the full version.

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Metric Embeddings

Cited by 50 publications

References 8 publications

On embeddings of finite subsets of ℓ_{𝑝}

On embeddings of finite subsets of ℓ_{𝑝}

$\varepsilon$-isometric dimension reduction for incompressible subsets of $\ell_p$

The integrality gap of the Goemans-Linial SDP relaxation for sparsest cut is at least a constant multiple of √log n

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