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Low-distortion embeddings of infinite metric spaces into the real line
2009
Annals of Pure and Applied Logic
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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
2016
Bulletin of the American Mathematical Society
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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]
2019
arXiv
pre-print
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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
2016
International Workshop on Approximation Algorithms for Combinatorial Optimization
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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]
2016
arXiv
pre-print
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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]
2011
Mathematical Surveys and Monographs
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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]
2009
Lecture Notes in Computer Science
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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
1998
Israel Journal of Mathematics
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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
2004
Proceedings of the thirty-sixth annual ACM symposium on Theory of computing - STOC '04
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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
2010
SIAM journal on computing (Print)
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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]
2005
Lecture Notes in Computer Science
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(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]
2013
arXiv
pre-print
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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 $\ell_{p}$
2015
SIAM Journal on Discrete Mathematics
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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
2011
Advances in Mathematics
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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
1995
Combinatorica
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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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