Metric Dimension Reduction: A Snapshot of the Ribe Program

NAOR, ASSAF

doi:10.1142/9789813272880_0029

Cited by 27 publications

(40 citation statements)

References 209 publications

(526 reference statements)

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“…See [121] for an explanation of this terminology, as well as its motivation within an extensive long-term research program called the Ribe program. Explaining this larger context is beyond the scope of the present article, but accessible introductory surveys are available [89,128,12,143,63,132]. It suffices to say here that our work is yet another example of an application of the Ribe program in which insights that are inspired by Banach space theory are used to answer a geometric question about objects which a priori have nothing to do with Banach spaces.…”

Section: Theorem 2 There Exists a Metric Space Z That Does Not Embedmentioning

confidence: 99%

Nonpositive curvature is not coarsely universal

Eskenazis

Mendel

2019

Invent. math.

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We prove that not every metric space embeds coarsely into an Alexandrov space of nonpositive curvature. This answers a question of Gromov (1993) and is in contrast to the fact that any metric space embeds coarsely into an Alexandrov space of nonnegative curvature, as shown by Andoni, Naor and Neiman (2015). We establish this statement by proving that a metric space which is q-barycentric for some q ∈ [1,∞) has metric cotype q with sharp scaling parameter. Our proof utilizes nonlinear (metric space-valued) martingale inequalities and yields sharp bounds even for some classical Banach spaces. This allows us to evaluate the bi-Lipschitz distortion of the ℓ ∞ grid [m] n ∞ = ({1,... ,m} n , · ∞ ) into ℓ q for all q ∈ (2,∞), from which we deduce the following discrete converse to the fact that ℓ n ∞ embeds with distortion O(1) into ℓ q for q = O(logn). A rigidity theorem of Ribe (1976) implies that for every n ∈ N there exists m ∈ N such that if [m] n ∞ embeds into ℓ q with distortion O(1), then q is necessarily at least a universal constant multiple of log n. Ribe's theorem does not give an explicit upper bound on this m, but by the work of Bourgain (1987) it suffices to take m = n, and this was the previously best-known estimate for m. We show that the above discretization statement actually holds when m is a universal constant.

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Section: Theorem 2 There Exists a Metric Space Z That Does Not Embedmentioning

confidence: 99%

Nonpositive curvature is not coarsely universal

Eskenazis

Mendel

2019

Invent. math.

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“…This line of work has been initiated by Boutsidis, Zouzias, and Drineas [BZD10] and prior to this work, the best bounds were due to Cohen, Elder, Musco, Musco and Persu [CEM + 15]. Before stating our results, let us briefly recall the notion of Euclidean dimension reduction (see [Nao18] for a broad overview of the area).…”

Section: Introductionmentioning

confidence: 99%

Performance of Johnson-Lindenstrauss transform for k -means and k -medians clustering

Makarychev¹,

Makarychev

Razenshteyn

2019

Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

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Consider an instance of Euclidean k-means or k-medians clustering. We show that the cost of the optimal solution is preserved up to a factor of (1 + ε) under a projection onto a random O(log(k/ε)/ε 2 )-dimensional subspace. Further, the cost of every clustering is preserved within (1 + ε). More generally, our result applies to any dimension reduction map satisfying a mild sub-Gaussian-tail condition. Our bound on the dimension is nearly optimal. Additionally, our result applies to Euclidean k-clustering with the distances raised to the p-th power for any constant p.For k-means, our result resolves an open problem posed by Cohen, Elder, Musco, Musco, and Persu (STOC 2015); for k-medians, it answers a question raised by Kannan. * Supported by NSF CCF-1718820 and NSF Career CCF-1150062.Dimension reduction. The cornerstone dimension reduction statement for the Euclidean distance is the Johnson-Lindenstrauss Lemma [JL84]. For positive reals p, q, ε, we write p ≈ 1+ε q if 1 1+ε · p ≤ q ≤ (1 + ε) · p.Theorem 1.1 ([JL84]). There exists a family of random linear maps π m,d : R m → R d with the following properties. For every m ≥ 1, ε, δ ∈ (0; 1/2) and all x ∈ R m , we haveA straightforward corollary is that one is able to embed any n-point subset of a Euclidean space into an O log n ε 2 -dimensional space, while preserving all of the pairwise distances up to (1 + ε). This bound is known to be tight [Alo03, LN17]. The attractive feature of the dimension reduction procedure given by Theorem 1.1 is that it is data-oblivious i.e., the distribution over linear maps is independent of the set of points we apply it to.There are several constructions of families of random maps π m,d that satisfy Theorem 1.1: projections on a random subspace [JL84, DG03] and maps given by matrices with i.i.d. Gaussian and sub-Gaussian entries [IM98, Ach03, KM + 05]. All of these constructions satisfy a certain additional condition, which we will need later. Definition 1.2. A family of random linear maps π m,d : R m → R d is called sub-Gaussian-tailed if for every unit vector x ∈ R m and every t > 0, one has:Pr π∼π m,d

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“…See [Mat02], [Nao18], and [Ost13] for background on this notion. Let X be a finitedimensional Banach space and Y be an infinite-dimensional Banach space.…”

Section: Introductionmentioning

confidence: 99%

Bourgain discretization using Lebesgue-Bochner spaces

Ostrovskii

Randrianantoanina

2019

Quaestiones Mathematicae

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This paper is dedicated to the memory of our friend Joe Diestel . Lebesgue-Bochner spaces were one of the main passions of Joe. He started to work in this direction in his Ph.D. thesis [Die68], and devoted to Lebesgue-Bochner spaces a large part of his most popular, classical, Dunford-Schwartz-style monograph [DU77], joint with Jerry Uhl. AbstractWe study the Lebesgue-Bochner discretization property of Banach spaces Y , which ensures that the Bourgain's discretization modulus for Y has a good lower estimate. We prove that there exist spaces that do not have the Lebesgue-Bochner discretization property, and we give a class of examples of spaces that enjoy this property.

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Metric Dimension Reduction: A Snapshot of the Ribe Program

Cited by 27 publications

References 209 publications

Nonpositive curvature is not coarsely universal

Nonpositive curvature is not coarsely universal

Performance of Johnson-Lindenstrauss transform for k -means and k -medians clustering

Bourgain discretization using Lebesgue-Bochner spaces

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