Discrete Geometry

Matoušek, Jiřı́; Rote, Günter; Bárány, Imre

doi:10.4171/owr/2011/44

Cited by 91 publications

(126 citation statements)

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“…This fact should be compared to the Johnson-Lindenstrauss lemma for finite subsets K ⊂ R n ( [12], see [17,Section 15.2]) which states that if m ≥ Cδ −2 log |K| then K can be Lipschitz embedded into R m as follows:…”

Section: 3mentioning

confidence: 99%

Dimension Reduction by Random Hyperplane Tessellations

Plan

Vershynin

2013

Discrete Comput Geom

141

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Abstract. Given a subset K of the unit Euclidean sphere, we estimate the minimal number m = m(K) of hyperplanes that generate a uniform tessellation of K, in the sense that the fraction of the hyperplanes separating any pair x, y ∈ K is nearly proportional to the Euclidean distance between x and y. Random hyperplanes prove to be almost ideal for this problem; they achieve the almost optimal bound m = O(w(K) 2 ) where w(K) is the Gaussian mean width of K. Using the map that sends x ∈ K to the sign vector with respect to the hyperplanes, we conclude that every bounded subset K of R n embeds into the Hamming cube {−1, 1} m with a small distortion in the Gromov-Haussdorff metric. Since for many sets K one has m = m(K) n, this yields a new discrete mechanism of dimension reduction for sets in Euclidean spaces.

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Section: 3mentioning

confidence: 99%

Dimension Reduction by Random Hyperplane Tessellations

Plan

Vershynin

2013

Discrete Comput Geom

141

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“…Otherwise, h is O((n/r) log n)-shallow w.h.p. (n) by a Chernoff bound (or the ε-net property of random samples [32,40,43]). Then w.h.p.…”

Section: Shallow Applicationsmentioning

confidence: 99%

“…For a given set H of m hyperplanes, a (1/r)-cutting [20,40,43] is a collection of disjoint cells such that each cell ∆ is crossed by at most m/r hyperplanes. Let A(H) denote the arrangement of H. We let H ∆ denote the subset of all hyperplanes of H that cross ∆.…”

Section: Introductionmentioning

confidence: 99%

Optimal Partition Trees

Chan

2012

Discrete Comput Geom

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We revisit one of the most fundamental classes of data structure problems in computational geometry: range searching. Back in SoCG'92, Matoušek gave a partition tree method for ddimensional simplex range searching achieving O(n) space and O(n 1−1/d ) query time. Although this method is generally believed to be optimal, it is complicated and requires O(n 1+ε ) preprocessing time for any fixed ε > 0. An earlier method by Matoušek (SoCG'91) requires O(n log n) preprocessing time but O(n 1−1/d log O(1) n) query time. We give a new method that achieves simultaneously O(n log n) preprocessing time, O(n) space, and O(n 1−1/d ) query time with high probability. Our method has several advantages:• It is conceptually simpler than Matoušek's SoCG'92 method. Our partition trees satisfy many ideal properties (e.g., constant degree, optimal crossing number at almost all layers, and disjointness of the children's cells at each node).• It leads to more efficient multilevel partition trees, which are important in many data structural applications (each level adds at most one logarithmic factor to the space and query bounds, better than in all previous methods).• A similar improvement applies to a shallow version of partition trees, yielding O(n log n) time, O(n) space, and O(n 1−1/ d/2 ) query time for halfspace range emptiness in even dimensions d ≥ 4.Numerous consequences follow (e.g., improved results for computing spanning trees with low crossing number, ray shooting among line segments, intersection searching, exact nearest neighbor search, linear programming queries, finding extreme points, . . . ).

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“…A celebrated result in this area is the Johnson-Lindenstrauss Lemma [7], showing that metrics induced by n points in an Euclidean space can be embedded with (1 + ε) distortion into O(ε −2 log n)-dimensional space. The reader may consult the book by Matoušek [13] for an excellent survey and exposition to many elegant results on metric embeddings.…”

Section: Introductionmentioning

confidence: 99%

Metric embedding, hyperbolic space, and social networks

Verbeek

Suri

2016

Computational Geometry

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We consider the problem of embedding an undirected graph into hyperbolic space with minimum distortion. A fundamental problem in its own right, it has also drawn a great deal of interest from applied communities interested in empirical analysis of large-scale graphs. In this paper, we establish a connection between distortion and quasi-cyclicity of graphs, and use it to derive lower and upper bounds on metric distortion. Two particularly simple and natural graphs with large quasi-cyclicity are n-node cycles and n × n square lattices, and our lower bound shows that any hyperbolic-space embedding of these graphs incurs a multiplicative distortion of at least Ω(n/ log n). This is in sharp contrast to Euclidean space, where both of these graphs can be embedded with only constant multiplicative distortion. We also establish a relation between quasi-cyclicity and δ-hyperbolicity of a graph as a way to prove upper bounds on the distortion. Using this relation, we show that graphs with small quasi-cyclicity can be embedded into hyperbolic space with only constant additive distortion. Finally, we also present an efficient (linear-time) randomized algorithm for embedding a graph with small quasi-cyclicity into hyperbolic space, so that with high probability at least a (1 − ε) fraction of the node-pairs has only constant additive distortion. Our results also give a plausible theoretical explanation for why social networks have been observed to embed well into hyperbolic space: they tend to have small quasi-cyclicity.

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Discrete Geometry

Cited by 91 publications

References 0 publications

Dimension Reduction by Random Hyperplane Tessellations

Dimension Reduction by Random Hyperplane Tessellations

Optimal Partition Trees

Metric embedding, hyperbolic space, and social networks

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