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29
Euclidean distortion and the Sparsest Cut
 In Proceedings of the 37th Annual ACM Symposium on Theory of Computing
, 2005
"... BiLipschitz embeddings of finite metric spaces, a topic originally studied in geometric analysis and Banach space theory, became an integral part of theoretical computer science following work of Linial, London, and Rabinovich [29]. They presented an algorithmic version of a result of Bourgain [8] ..."
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Cited by 95 (20 self)
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BiLipschitz embeddings of finite metric spaces, a topic originally studied in geometric analysis and Banach space theory, became an integral part of theoretical computer science following work of Linial, London, and Rabinovich [29]. They presented an algorithmic version of a result of Bourgain [8] which shows that every
Nonembeddability theorems via Fourier analysis
"... Various new nonembeddability results (mainly into L1) are proved via Fourier analysis. In particular, it is shown that the Edit Distance on {0, 1}d has L1 distortion (log d) 12o(1). We also give new lower bounds on the L1 distortion of flat tori, quotients of the discrete hypercube under group ac ..."
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Cited by 42 (9 self)
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Various new nonembeddability results (mainly into L1) are proved via Fourier analysis. In particular, it is shown that the Edit Distance on {0, 1}d has L1 distortion (log d) 12o(1). We also give new lower bounds on the L1 distortion of flat tori, quotients of the discrete hypercube under group actions, and the transportation cost (Earthmover) metric.
Metric cotype
, 2005
"... We introduce the notion of metric cotype, a property of metric spaces related to a property of normed spaces, called Rademacher cotype. Apart from settling a long standing open problem in metric geometry, this property is used to prove the following dichotomy: A family of metric spaces F is either a ..."
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Cited by 28 (16 self)
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We introduce the notion of metric cotype, a property of metric spaces related to a property of normed spaces, called Rademacher cotype. Apart from settling a long standing open problem in metric geometry, this property is used to prove the following dichotomy: A family of metric spaces F is either almost universal (i.e., contains any finite metric space with any distortion> 1), or there exists α> 0, and arbitrarily large npoint metrics whose distortion when embedded in any member of F is at least Ω ((log n) α). The same property is also used to prove strong nonembeddability theorems of Lq into Lp, when q> max{2, p}. Finally we use metric cotype to obtain a new type of isoperimetric inequality on the discrete torus. 1
Trees and Markov convexity
 In Proceedings of the 17th annual ACMSIAM Symposium on Discrete Algorithms
, 2006
"... We show that an infinite weighted tree admits a biLipschitz embedding into Hilbert space if and only if it does not contain arbitrarily large complete binary trees with uniformly bounded distortion. We also introduce a new metric invariant called Markov convexity, and show how it can be used to com ..."
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Cited by 14 (5 self)
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We show that an infinite weighted tree admits a biLipschitz embedding into Hilbert space if and only if it does not contain arbitrarily large complete binary trees with uniformly bounded distortion. We also introduce a new metric invariant called Markov convexity, and show how it can be used to compute the Euclidean distortion of any metric tree up to universal factors. 1
Scaled Enflo Type is Equivalent to Rademacher Type
 Bull. London Math. Soc
"... We introduce the notion of scaled Enflo type of a metric space, and show that for Banach spaces, scaled Enflo type p is equivalent to Rademacher type p. 1 ..."
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Cited by 11 (6 self)
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We introduce the notion of scaled Enflo type of a metric space, and show that for Banach spaces, scaled Enflo type p is equivalent to Rademacher type p. 1
Metric Extension Operators, Vertex Sparsifiers and Lipschitz Extendability
"... We study vertex cut and flow sparsifiers that were recently introduced by Moitra (2009), and Leighton and Moitra (2010). We improve and generalize their results. We give a new polynomialtime algorithm for constructing O(log k / log log k) cut and flow sparsifiers, matching the best existential upper ..."
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Cited by 10 (1 self)
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We study vertex cut and flow sparsifiers that were recently introduced by Moitra (2009), and Leighton and Moitra (2010). We improve and generalize their results. We give a new polynomialtime algorithm for constructing O(log k / log log k) cut and flow sparsifiers, matching the best existential upper bound on the quality of a sparsifier, and improving the previous algorithmic upper bound of O(log 2 k / log log k). We show that flow sparsifiers can be obtained from linear operators approximating minimum metric extensions. We introduce the notion of (linear) metric extension operators, prove that they exist, and give an exact polynomialtime algorithm for finding optimal operators. We then establish a direct connection between flow and cut sparsifiers and Lipschitz extendability of maps in Banach spaces, a notion studied in functional analysis since 1950s. Using this connection, we prove a lower bound of Ω ( √ log k / log log k) for flow sparsifiers and a lower bound of Ω ( 4 √ log k / log log k) for cut sparsifiers. We show that if a certain open question posed by Ball in 1992 has a positive answer, then there exist Õ( √ log k) cut sparsifiers. On the other hand, any lower bound on cut sparsifiers better than ˜ Ω ( √ log k) would imply a negative answer to this question. 1
Embeddings of discrete groups and the speed of random walks
, 2007
"... Let G be a group generated by a finite set S and equipped with the associated leftinvariant word metric dG. For a Banach space X let α ∗ X (G) (respectively α # X (G)) be the supremum over all α ≥ 0 such that there exists a Lipschitz mapping (respectively an equivariant mapping) f: G → X and c> 0 s ..."
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Cited by 8 (3 self)
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Let G be a group generated by a finite set S and equipped with the associated leftinvariant word metric dG. For a Banach space X let α ∗ X (G) (respectively α # X (G)) be the supremum over all α ≥ 0 such that there exists a Lipschitz mapping (respectively an equivariant mapping) f: G → X and c> 0 such that for all x, y ∈ G we have ‖ f (x) − f (y) ‖ ≥ c · dG(x, y) α. In particular, the Hilbert compression exponent (respectively the equivariant Hilbert compression exponent) of G is α ∗ (G) ≔ α ∗ (G) (respectively
The wreath product of Z with Z has Hilbert compression exponent 2 3
"... Let G be a finitely generated group, equipped with the word metric d associated with some finite set of generators. The Hilbert compression exponent of G is the supremum over all α ≥ 0 such that there exists a Lipschitz mapping f: G → L2 and a constant c> 0 such that for all x, y ∈ G we have ‖ f (x) ..."
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Cited by 8 (2 self)
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Let G be a finitely generated group, equipped with the word metric d associated with some finite set of generators. The Hilbert compression exponent of G is the supremum over all α ≥ 0 such that there exists a Lipschitz mapping f: G → L2 and a constant c> 0 such that for all x, y ∈ G we have ‖ f (x) − f (y)‖2 ≥ cd(x, y) α. In [2] it was shown that the Hilbert compression exponent of the wreath product Z ≀ Z is at most 3 2 2
The Euclidean distortion of the lamplighter group
, 2007
"... We show that the cyclic lamplighter group C2 ≀ Cn embeds into Hilbert space with distortion O ( √ log n). This matches the lower bound proved by Lee, Naor and Peres in [14], answering a question posed in that paper. Thus the Euclidean distortion of C2 ≀ Cn is Θ ( √ log n). Our embedding is constru ..."
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Cited by 7 (4 self)
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We show that the cyclic lamplighter group C2 ≀ Cn embeds into Hilbert space with distortion O ( √ log n). This matches the lower bound proved by Lee, Naor and Peres in [14], answering a question posed in that paper. Thus the Euclidean distortion of C2 ≀ Cn is Θ ( √ log n). Our embedding is constructed explicitly in terms of the irreducible representations of the group. Since the optimal Euclidean embedding of a finite group can always be chosen to be equivariant, as shown by Aharoni, Maurey and Mityagin [1] and by Gromov (see [9]), such representationtheoretic considerations suggest a general tool for obtaining upper and lower bounds on Euclidean embeddings of finite groups.
Markov type of Alexandrov spaces of nonnegative curvature
, 2007
"... We prove that Alexandrov spaces X of nonnegative curvature have Markov type 2 in the sense of Ball. As a corollary, any Lipschitz continuous map from a subset of X into a 2uniformly convex Banach space is extended as a Lipschitz continuous map on the entire space X. ..."
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Cited by 7 (1 self)
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We prove that Alexandrov spaces X of nonnegative curvature have Markov type 2 in the sense of Ball. As a corollary, any Lipschitz continuous map from a subset of X into a 2uniformly convex Banach space is extended as a Lipschitz continuous map on the entire space X.