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Distortion lower bounds for line embeddings

Claire Mathieu, Charalampos Papamanthou
2008 Information Processing Letters  
In this paper, we show how we can derive lower bounds and also compute the exact distortion for the line embeddings of some special metrics, especially trees and graphs with certain structure.  ...  Using linear programming to formulate a simpler version of the problem gives an interesting intuition and direction concerning the computation of general lower bounds for distortion into the line.  ...  The authors would like to thank Aris Anagnostopoulos and Eli Upfal for their useful feedback at the early stages of this work.  ... 
doi:10.1016/j.ipl.2008.05.003 fatcat:tmds2cn4qrffxnqubigtil4cpq

Lower Bounds for Embedding into Distributions over Excluded Minor Graph Families [chapter]

Douglas E. Carroll, Ashish Goel
2004 Lecture Notes in Computer Science  
We show that this same lower bound holds for embeddings into distributions over any minor excluded family.  ...  It is also known that this bound is tight since there are expander graphs which cannot be embedded into distributions over trees with better than Ω(log n) distortion.  ...  We would also like to thank Elias Koutsoupias for valuable discussions, and Adam Meyerson and Shailesh Vaya for helpful comments on previous drafts.  ... 
doi:10.1007/978-3-540-30140-0_15 fatcat:5ihzwtzh6jbq3jarj3ea6dtrna

On the impossibility of dimension reduction in l1

Bo Brinkman, Moses Charikar
2005 Journal of the ACM  
We show strong lower bounds for general dimension reduction in 1 . We give an explicit family of n points in 1 such that any embedding with distortion δ requires n Ω(1/δ 2 ) dimensions.  ...  Further, embedding the points into 1 with 1 + ε distortion requires n 1 2 −O(ε log( 1 ε )) dimensions. Our proof establishes this lower bound for shortest path metrics of series-parallel graphs.  ...  How can one prove lower bounds on the stretch for arbitrary (i.e. nonlinear) line embeddings?  ... 
doi:10.1145/1089023.1089026 fatcat:dnkyc2bpdfhevk4javjr7uerfi

Near Linear Lower Bound for Dimension Reduction in L1

Alexandr Andoni, Moses S. Charikar, Ofer Neiman, Huy L. Nguyen
2011 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science  
In this work, we show the first near linear lower bounds for dimension reduction in 1. In particular, we show that 1 + distortion requires at least n 1−O(1/ log(1/ )) dimensions.  ...  In sharp contrast, there is a significant gap between upper and lower bounds for dimension reduction in 1. A recent result shows that distortion 1 + can be achieved with n/ 2 dimensions.  ...  This gives our main result showing a near linear lower bound for 1 + distortion embeddings of 1 .  ... 
doi:10.1109/focs.2011.87 dblp:conf/focs/AndoniCNN11 fatcat:n3ur4ibi5rfwjmci3zusyfd4xi

Lower Bounds for Embedding into Distributions over Excluded Minor Graph Families [article]

Douglas E. Carroll, Ashish Goel
2008 arXiv   pre-print
We show that this same lower bound holds for embeddings into distributions over any minor excluded family.  ...  It is also known that this bound is tight since there are expander graphs which cannot be embedded into distributions over trees with better than Omega(log n) distortion.  ...  We would also like to thank Elias Koutsoupias for valuable discussions, and Adam Meyerson and Shailesh Vaya for helpful comments on previous drafts.  ... 
arXiv:0807.4582v1 fatcat:4viasdxsv5fmnlummxn4jl5iqm

Combinatorial theorems about embedding trees on the real line

Amit Chakrabarti, Subhash Khot
2011 Journal of Graph Theory  
., distortion-minimizing) line embeddings for these metrics and prove their optimality via suitable lower bound arguments.  ...  Our results about these metrics show that the local density of a graph -an a priori reasonable lower bound on the optimum distortion -might in fact be arbitrarily smaller than the true optimum, even for  ...  Acknowledgments We would like to thank Anupam Gupta for getting us interested in the problems handled here and Bernard Chazelle and an anonymous referee for helpful comments on earlier drafts of the paper  ... 
doi:10.1002/jgt.20608 fatcat:vtkcxehxerggrnp2jf4lgajotq

Online Embedding of Metrics

Ilan Newman, Yuri Rabinovich, Susanne Albers
2020 Scandinavian Workshop on Algorithm Theory  
We establish some upper and lower bounds on the distortion of such embedding, and pose some challenging open questions.  ...  We study deterministic online embeddings of metric spaces into normed spaces of various dimensions and into trees.  ...  Unlike [6], we seek bounds independent of the aspect ratio. We also establish a lower bound of Ω(2 n/2 ) for online embedding metrics on n points into trees.  ... 
doi:10.4230/lipics.swat.2020.32 dblp:conf/swat/NewmanR20 fatcat:tff3vqozanf7heqmjjbdlzx2ru

Approximation Algorithms for Minimizing Average Distortion

Kedar Dhamdhere, Anupam Gupta, R. Ravi
2005 Theory of Computing Systems  
We give a constant-factor approximation for the problem of embedding general metrics into the line metric.  ...  embedding for the given input metric.  ...  on bounding the maximum distortion of embeddings.  ... 
doi:10.1007/s00224-005-1259-6 fatcat:477ackl7nrbbdesju6yhoz72pm

On Euclidean Embeddings and Bandwidth Minimization [chapter]

John Dunagan, Santosh Vempala
2001 Lecture Notes in Computer Science  
, (2) an O(log 3 n) approximation in O(n log n ) time using a new constraint set, (3) a lower bound of Θ( √ log n) on the least possible volume distortion for Euclidean metrics, (4) a new embedding with  ...  We study Euclidean embeddings of Euclidean metrics and present the following four results: (1) an O(log 3 n √ log log n) approximation for minimum bandwidth in conjunction with a semi-definite relaxation  ...  A Lower Bound on Volume Distortion Our bandwidth algorithm and its analysis motivate the question of whether there are embeddings with better volume distortion.  ... 
doi:10.1007/3-540-44666-4_26 fatcat:i6nzbbdb5fcidmldunnllqdxj4

Plane embeddings of planar graph metrics

MohammadHossein Bateni, MohammadTaghi Hajiaghayi, Erik D. Demaine, Mohammad Moharrami
2006 Proceedings of the twenty-second annual symposium on Computational geometry - SCG '06  
the best known approximation algorithm for minimizing distortion.  ...  We also prove that some planar graph metrics require Ω(n) distortion in any crossingfree straight-line embedding into the plane, suggesting a separation between low-distortion plane embedding and the well-studied  ...  Acknowledgments We thank Anastasios Sidiropoulos for helpful discussions.  ... 
doi:10.1145/1137856.1137887 dblp:conf/compgeom/BateniHDM06 fatcat:jmjkjef6cndqpcyy65wpbyfd5i

Plane Embeddings of Planar Graph Metrics

MohammadHossein Bateni, Erik D. Demaine, MohammadTaghi Hajiaghayi, Mohammad Moharrami
2007 Discrete & Computational Geometry  
provides the best known approximation algorithm for minimizing distortion.  ...  Finally, on the upper-bound side, we prove that all outerplanar graph metrics can be embedded into the plane with O( √ n) distortion, generalizing the previous results on trees (both the worst-case bound  ...  Acknowledgments We thank Anastasios Sidiropoulos for helpful discussions. We also thank the anonymous referees for their helpful comments on the paper.  ... 
doi:10.1007/s00454-007-1353-4 fatcat:ktdaurszkrbazmsik5bk5yoh2m

Dimensionality reduction: theoretical perspective on practical measures

Yair Bartal, Nova Fandina, Ofer Neiman
2019 Neural Information Processing Systems  
The main consequences of our work are nearly tight bounds on the absolute values of all distortion criteria, as well as first approximation algorithms with provable guarantees.  ...  This paper can be viewed as providing a comprehensive theoretical framework for analyzing different distortion measurement criteria, with the lens of practical applicability, and in particular for Machine  ...  Moreover, in (40) a lower bound of n Ω(1/k) on the worst case distortion of any embedding in k dimensions was proven.  ... 
dblp:conf/nips/BartalFN19 fatcat:5yuaklr3c5e6nfgzcfldecwzfe

On the Power and Limits of Distance-Based Learning

Periklis A. Papakonstantinou, Jia Xu, Guang Yang
2016 International Conference on Machine Learning  
We develop new geometric techniques and prove strong learning lower bounds. These provable limits hold even when we allow learners and classifiers to get advice by one or more experts.  ...  metric structure, and (ii) the concepts we wish to learn are low-distortion embeddings.  ...  the Sino-Danish Center for the Theory of Interactive Computation and from the Center for Research in Foundations of Electronic Markets (CFEM), supported by the Danish Strategic Research Council.  ... 
dblp:conf/icml/Papakonstantinou16 fatcat:by5ixdeztrh5joebbemesqxbj4

Probabilistic embeddings of bounded genus graphs into planar graphs

Piotr Indyk, Anastasios Sidiropoulos
2007 Proceedings of the twenty-third annual symposium on Computational geometry - SCG '07  
1) O(1) Planar O(1)-Genus [Carroll,Goel'04] Ω(logn) O(logn) Treewidth-(k-3) Treewidth-k [Karp89] Ω(1) O(1) Line Embedding into L 1 : If all planar graphs embed into L 1 with distortion γ, then all boundedgenus  ...  exists an a-approximation for A on planar graphs, then there exists an O(a)-approximation for A on bounded-genus graphs. ), such that ∀ u,v ∈ X, -∀ M i ∈ F, D i (u,v) ≥ D(u,v) -E N ∈ F [Indyk,S'06] Ω(  ... 
doi:10.1145/1247069.1247107 dblp:conf/compgeom/IndykS07 fatcat:kwk664vkhvbqzh5s2jvoxah3jm

Reconstructing approximate tree metrics

Ittai Abraham, Mahesh Balakrishnan, Fabian Kuhn, Dahlia Malkhi, Venugopalan Ramasubramanian, Kunal Talwar
2007 Proceedings of the twenty-sixth annual ACM symposium on Principles of distributed computing - PODC '07  
In addition, we prove a lower bound on approximate distance labelings of ε-4PC metrics, and give tight bounds for tree embeddings with additive error guarantees.  ...  We study embeddings of ε-4PC metric spaces into trees and prove tight upper and lower bounds.  ...  Nevertheless, all these results apply for arbitrary metrics and hence are bounded by the worst case linear distortion lower bounds.  ... 
doi:10.1145/1281100.1281110 dblp:conf/podc/AbrahamBKMRT07 fatcat:leop4y25p5egzc4l6he6qocf24
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