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Low-distortion embeddings of infinite metric spaces into the real line

2009
*
Annals of Pure and Applied Logic
*

Finally we study decompositions

doi:10.1016/j.apal.2008.09.014
fatcat:qyz25dradfdjlakkzb2ojcin6i
*of**infinite*separable*metric**spaces**into*subsets that, for some K > 1, K -bi-Lipschitz embed*into**the**real**line*. ... We present a proof*of*a Ramsey-type theorem for*infinite**metric**spaces*due to Matoušek. ... Acknowledgement*The*research for this paper was supported by G.I.F. Research Grant No. I-802-195.6/2003. ...##
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Book Review: Metric embeddings: bilipschitz and coarse embedddings into Banach spaces

2016
*
Bulletin of the American Mathematical Society
*

This result spurred an explosion

doi:10.1090/bull/1523
fatcat:efhiga3vunafnepoppyrf3ulqi
*of*interest in*the*theory*of**low*-*distortion*embeddability*into**low*-dimensional*spaces*. ... In Chapter 3*the*technique*of*using stochastic padded decompositions to produce bi-Lipschitz*embeddings**of*finite*metric**spaces*with*low**distortion*is explained. ...##
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Learning Embeddings into Entropic Wasserstein Spaces
[article]

2019
*
arXiv
*
pre-print

We examine empirically

arXiv:1905.03329v1
fatcat:yzkljzpulvgq7hva5wiogmfyaa
*the*representational capacity*of*our learned Wasserstein*embeddings*, showing that they can embed a wide variety*of**metric*structures with smaller*distortion*than an equivalent Euclidean ... on a*low*-dimensional*space*. ... R n with*the*Euclidean*metric*, for example, embeds*into**the*L 1*metric*with*low**distortion*, while*the*reverse is not true (Deza & Laurent, 2009 ). ...##
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Constant-Distortion Embeddings of Hausdorff Metrics into Constant-Dimensional l_p Spaces

2016
*
International Workshop on Approximation Algorithms for Combinatorial Optimization
*

For

doi:10.4230/lipics.approx-random.2016.1
dblp:conf/approx/BackursS16
fatcat:oxrtvf6xpvctfhwflwsqw4w5ay
*the*case*of*pointsets*of*size s in*the**real**line**of*bounded resolution, we obtain a probabilistic*embedding**into*O(s log s) 1 with*distortion*O(s). ... In contrast, we obtain an*embedding**of*a*space**of**infinite*size*into*constant-dimensional ∞ . ... We study*embeddings**of**the*Hausdorff*metric*over finite subsets*of*Euclidean*space*. This is an*infinite**space*since there are*infinitely*many possible subsets even in*the**real**line*. ...##
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Linear properties of Banach spaces and low distortion embeddings of metric graphs
[article]

2016
*
arXiv
*
pre-print

We characterize non-reflexive Banach

arXiv:1603.00741v2
fatcat:mj5fcfntjzgoxnqqyapkoswo5a
*spaces*by a*low*-*distortion*(resp. isometric) embeddability*of*a certain*metric*graph up to a renorming. ... Also we study non-linear sufficient conditions for ℓ_1^n being (1+ε)-isomorphic to a subspace*of*a Banach*space*X. ... A part*of*this paper was written during*the*conference "Banach*spaces*and their applications in analysis" at CIRM. ...##
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Finite metric spaces and partitions
[chapter]

2011
*
Mathematical Surveys and Monographs
*

Thus, given a finite

doi:10.1090/surv/173/26
fatcat:2qhikpk56fhe3cn2nblu5uvu5y
*metric**space*, and a computational task, it is natural to try to map*the*given*metric**space**into*a new*metric*where*the*task at hand becomes easy. ... Underling our discussion*of**metric**spaces*are algorithmic applications.*The*hardness*of*various computational problems depends heavily on*the*structure*of**the*finite*metric**space*. ... Acknowledgments*The*presentation in this write-up follows closely*the*insightful suggestions*of*Manor Mendel. ...##
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On Low Distortion Embeddings of Statistical Distance Measures into Low Dimensional Spaces
[chapter]

2009
*
Lecture Notes in Computer Science
*

In this paper, we investigate various statistical distance measures from

doi:10.1007/978-3-642-03573-9_13
fatcat:l2kbthlcavb4vexfy2tdwnzl5i
*the*point*of*view*of*discovering*low**distortion**embeddings**into**low*-dimensional*spaces*. ... We provide explicit constructions*of*point sets under*the*Bhattacharyya and*the*Kullback-Leibler divergences whose*embeddings**into*any*metric**space*incur arbitrarily large*distortions*. ...*The*lack*of*inherent "geometric" properties make them harder candidates for*low**distortion*and*low*-dimensionality*embeddings**into**metric**spaces*. ...##
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Low distortion euclidean embeddings of trees

1998
*
Israel Journal of Mathematics
*

We introduce

doi:10.1007/bf02773475
fatcat:lq5madrmvjh6rfmqwfhay5zypq
*the*notation c2 (X, d) for*the*least*distortion*with which*the**metric**space*(X, d) may be*embedded*in a Euclidean*space*. ... We consider*the*problem*of**embedding*a certain finite*metric**space*to*the*Euclidean*space*, trying to keep*the*bi-Lipschitz constant as small as possible. ... Note added in proof: After*the*completion*of*this work, we were informed that J. ...##
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Low distortion maps between point sets

2004
*
Proceedings of the thirty-sixth annual ACM symposium on Theory of computing - STOC '04
*

We initiate

doi:10.1145/1007352.1007398
dblp:conf/stoc/KenyonRS04
fatcat:a4eocusiqbbrndmgeixs67l4am
*the*study*of**the*minimum*distortion*problem: given as input two n-point*metric**spaces*, find a bijection between them with minimum*distortion*. ... We present a polynomial time algorithm that finds an optimal bijection between two*line**metrics*, provided*the**distortion*is less than 3 + 2 √ 2. ... From a computer science perspective,*the*core question*of*this*line**of*work has been to bound*the**distortion**of*an injection from an input*metric**space**into*an implicit*infinite*host*space*(for example ...##
###
Low Distortion Maps Between Point Sets

2010
*
SIAM journal on computing (Print)
*

We initiate

doi:10.1137/080712921
fatcat:jq5s3ig5srhpdfdbmokgrh264u
*the*study*of**the*minimum*distortion*problem: given as input two n-point*metric**spaces*, find a bijection between them with minimum*distortion*. ... We present a polynomial time algorithm that finds an optimal bijection between two*line**metrics*, provided*the**distortion*is less than 3 + 2 √ 2. ... From a computer science perspective,*the*core question*of*this*line**of*work has been to bound*the**distortion**of*an injection from an input*metric**space**into*an implicit*infinite*host*space*(for example ...##
###
Approximating the Distortion
[chapter]

2005
*
Lecture Notes in Computer Science
*

(STOC 04) compute

doi:10.1007/11538462_10
fatcat:iber3qhstffwjc6lx7sf4mkrza
*the**distortion*between one-dimensional finite point sets when*the**distortion*is small; Papadimitriou and Safra (SODA 05) show that*the*problem is NP-hard to approximate within a factor ... We also introduce additive*distortion*, and show that it can be easily approximated within a factor*of*two. ... Most Computer Science related work in this area focuses on*the*setting where a given finite*metric**space*is to be*embedded**into*an*infinite*host*space*, usually a*low*dimensional Euclidean*space*. ...##
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On the Impossibility of Dimension Reduction for Doubling Subsets of ℓ_p, p>2
[article]

2013
*
arXiv
*
pre-print

A major open problem in

arXiv:1308.4996v1
fatcat:v3naovm6l5hdbjielgkya6p6hu
*the*field*of**metric**embedding*is*the*existence*of*dimension reduction for n-point subsets*of*Euclidean*space*, such that both*distortion*and dimension depend only on*the*doubling ... In particular, we introduce an n-point subset*of*ℓ_p with doubling constant O(1), and demonstrate that any*embedding**of**the*set*into*ℓ_p^d with*distortion*D must have D>Ω((c n/d)^1/2-1/p). ... There are several results on*embedding**metric**spaces*with*low*intrinsic dimension*into**low*dimensional normed*space*with*low**distortion*: Assouad [Ass83] showed that*the*snowflakes*of*doubling*metrics*...##
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On the Impossibility of Dimension Reduction for Doubling Subsets of $\ell_{p}$

2015
*
SIAM Journal on Discrete Mathematics
*

A major open problem in

doi:10.1137/140977655
fatcat:znba4oy2xrambmkenbzjlqt2ey
*the*field*of**metric**embedding*is*the*existence*of*dimension reduction for n-point subsets*of*Euclidean*space*, such that both*distortion*and dimension depend only on*the*doubling ... In particular, we introduce an n-point subset*of*p with doubling constant O(1), and demonstrate that any*embedding**of**the*set*into*d p with*distortion*D must have D ≥ Ω c log n d 1 2 − 1 p . * Hebrew University ... There are several results on*embedding**metric**spaces*with*low*intrinsic dimension*into**low*dimensional normed*space*with*low**distortion*: Assouad [Ass83] showed that*the*snowflakes*of*doubling*metrics*...##
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Advances in metric embedding theory

2011
*
Advances in Mathematics
*

One

doi:10.1016/j.aim.2011.08.003
fatcat:uclcimbuo5astnzlh7ko4bmsw4
*of**the*main cornerstones in finite*metric**embedding*theory is a celebrated theorem*of*Bourgain which states that every finite*metric**space*on n points embeds in Euclidean*space*with O(log n)*distortion*... Indeed, in most practical applications*of**metric**embedding**the*main criteria for*the*quality*of*an*embedding*is its average*distortion*over all pairs. ...*Embedding**infinite**spaces**into*l p As in*the*finite case, we first construct an*embedding**into**the**real**line*, that is good in expectation. Lemma 26. ...##
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The geometry of graphs and some of its algorithmic applications

1995
*
Combinatorica
*

Given a graph G we map its vertices to a normed

doi:10.1007/bf01200757
fatcat:lhkhob7abvhc7bhcfzyci7uqoy
*space*in an attempt to (i) Keep down*the*dimension*of**the*host*space*and (ii) Guarantee a small*distortion*, i.e., make sure that *Supported in part by grants ... There are several ways to model graphs geometrically and our main concern here is with geometric representations that respect*the**metric**of**the*(possibly weighted) graph. ... Acknowledgments*The*authors wish to thank Mike Saks, Micha A. Perles and Gil Kalai for stimulating discussions. Thanks are also due to Alex Samorodnitsky. ...
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