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

Stefan Geschke
2009 Annals of Pure and Applied Logic  
Finally we study decompositions 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.  ... 
doi:10.1016/j.apal.2008.09.014 fatcat:qyz25dradfdjlakkzb2ojcin6i

Book Review: Metric embeddings: bilipschitz and coarse embedddings into Banach spaces

Florent P. Baudier, William B. Johnson
2016 Bulletin of the American Mathematical Society  
This result spurred an explosion 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.  ... 
doi:10.1090/bull/1523 fatcat:efhiga3vunafnepoppyrf3ulqi

Learning Embeddings into Entropic Wasserstein Spaces [article]

Charlie Frogner, Farzaneh Mirzazadeh, Justin Solomon
2019 arXiv   pre-print
We examine empirically 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 ).  ... 
arXiv:1905.03329v1 fatcat:yzkljzpulvgq7hva5wiogmfyaa

Constant-Distortion Embeddings of Hausdorff Metrics into Constant-Dimensional l_p Spaces

Arturs Backurs, Anastasios Sidiropoulos, Marc Herbstritt
2016 International Workshop on Approximation Algorithms for Combinatorial Optimization  
For 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.  ... 
doi:10.4230/lipics.approx-random.2016.1 dblp:conf/approx/BackursS16 fatcat:oxrtvf6xpvctfhwflwsqw4w5ay

Linear properties of Banach spaces and low distortion embeddings of metric graphs [article]

Antonin Prochazka
2016 arXiv   pre-print
We characterize non-reflexive Banach 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.  ... 
arXiv:1603.00741v2 fatcat:mj5fcfntjzgoxnqqyapkoswo5a

Finite metric spaces and partitions [chapter]

Sariel Har-Peled
2011 Mathematical Surveys and Monographs  
Thus, given a finite 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.  ... 
doi:10.1090/surv/173/26 fatcat:2qhikpk56fhe3cn2nblu5uvu5y

On Low Distortion Embeddings of Statistical Distance Measures into Low Dimensional Spaces [chapter]

Arnab Bhattacharya, Purushottam Kar, Manjish Pal
2009 Lecture Notes in Computer Science  
In this paper, we investigate various statistical distance measures from 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.  ... 
doi:10.1007/978-3-642-03573-9_13 fatcat:l2kbthlcavb4vexfy2tdwnzl5i

Low distortion euclidean embeddings of trees

Nathan Linial, Avner Magen, Michael E. Saks
1998 Israel Journal of Mathematics  
We introduce 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.  ... 
doi:10.1007/bf02773475 fatcat:lq5madrmvjh6rfmqwfhay5zypq

Low distortion maps between point sets

Claire Kenyon, Yuval Rabani, Alistair Sinclair
2004 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing - STOC '04  
We initiate 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  ... 
doi:10.1145/1007352.1007398 dblp:conf/stoc/KenyonRS04 fatcat:a4eocusiqbbrndmgeixs67l4am

Low Distortion Maps Between Point Sets

Claire Kenyon, Yuval Rabani, Alistair Sinclair
2010 SIAM journal on computing (Print)  
We initiate 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  ... 
doi:10.1137/080712921 fatcat:jq5s3ig5srhpdfdbmokgrh264u

Approximating the Distortion [chapter]

Alexander Hall, Christos Papadimitriou
2005 Lecture Notes in Computer Science  
(STOC 04) compute 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.  ... 
doi:10.1007/11538462_10 fatcat:iber3qhstffwjc6lx7sf4mkrza

On the Impossibility of Dimension Reduction for Doubling Subsets of ℓ_p, p>2 [article]

Yair Bartal and Lee-Ad Gottlieb and Ofer Neiman
2013 arXiv   pre-print
A major open problem in 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  ... 
arXiv:1308.4996v1 fatcat:v3naovm6l5hdbjielgkya6p6hu

On the Impossibility of Dimension Reduction for Doubling Subsets of $\ell_{p}$

Yair Bartal, Lee-Ad Gottlieb, Ofer Neiman
2015 SIAM Journal on Discrete Mathematics  
A major open problem in 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  ... 
doi:10.1137/140977655 fatcat:znba4oy2xrambmkenbzjlqt2ey

Advances in metric embedding theory

Ittai Abraham, Yair Bartal, Ofer Neiman
2011 Advances in Mathematics  
One 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.  ... 
doi:10.1016/j.aim.2011.08.003 fatcat:uclcimbuo5astnzlh7ko4bmsw4

The geometry of graphs and some of its algorithmic applications

Nathan Linial, Eran London, Yuri Rabinovich
1995 Combinatorica  
Given a graph G we map its vertices to a normed 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.  ... 
doi:10.1007/bf01200757 fatcat:lhkhob7abvhc7bhcfzyci7uqoy
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