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41
Improved lower bounds for embeddings into L1
 SIAM J. COMPUT.
, 2009
"... We improve upon recent lower bounds on the minimum distortion of embedding certain finite metric spaces into L1. In particular, we show that for every n ≥ 1, there is an npoint metric space of negative type that requires a distortion of Ω(log log n) for such an embedding, implying the same lower bo ..."
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Cited by 40 (5 self)
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We improve upon recent lower bounds on the minimum distortion of embedding certain finite metric spaces into L1. In particular, we show that for every n ≥ 1, there is an npoint metric space of negative type that requires a distortion of Ω(log log n) for such an embedding, implying the same lower bound on the integrality gap of a wellknown semidefinite programming relaxation for sparsest cut. This result builds upon and improves the recent lower bound of (log log n) 1/6−o(1) due to Khot and Vishnoi [The unique games conjecture, integrality gap for cut problems and the embeddability of negative type metrics into l1, in Proceedings of the 46th Annual IEEE Symposium
Differentiating maps into L1 and the geometry of BV functions
 Ann. Math
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Compression bounds for Lipschitz maps from the Heisenberg group to L1
 Acta mathematica
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On the unique games conjecture
 In FOCS
, 2005
"... This article surveys recently discovered connections between the Unique Games Conjecture and computational complexity, algorithms, discrete Fourier analysis, and geometry. 1 ..."
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Cited by 15 (1 self)
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This article surveys recently discovered connections between the Unique Games Conjecture and computational complexity, algorithms, discrete Fourier analysis, and geometry. 1
Minmax graph partitioning and small set expansion
, 2011
"... We study graph partitioning problems from a minmax perspective, in which an input graph on n vertices should be partitioned into k parts, and the objective is to minimize the maximum number of edges leaving a single part. The two main versions we consider are: (i) the k parts need to be of equal s ..."
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Cited by 14 (2 self)
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We study graph partitioning problems from a minmax perspective, in which an input graph on n vertices should be partitioned into k parts, and the objective is to minimize the maximum number of edges leaving a single part. The two main versions we consider are: (i) the k parts need to be of equal size, and (ii) the parts must separate a set of k given terminals. We consider a common generalization of these two problems, and design for it an O ( √ log n log k)approximation algorithm. This improves over an O(log 2 n) approximation for the second version due to Svitkina and Tardos [22], and roughly O(k log n) approximation for the first version that follows from other previous work. We also give an improved O(1)approximation algorithm for graphs that exclude any fixed minor. Our algorithm uses a new procedure for solving the SmallSet Expansion problem. In this problem, we are given a graph G and the goal is to find a nonempty set S ⊆ V of size S  ≤ ρn with minimum edgeexpansion. We give an O ( √ log n log (1/ρ)) bicriteria approximation algorithm for the general case of SmallSet Expansion, and O(1) approximation algorithm for graphs that exclude any fixed minor.
ASSOUAD’S THEOREM WITH DIMENSION INDEPENDENT OF THE SNOWFLAKING
"... It is shown that for every K> 0 and ε ∈ (0, 1/2) there exist N = N(K) ∈ N and D = D(K, ε) ∈ (1, ∞) with the following properties. For every metric space (X, d) with doubling constant at most K, the metric space (X, d 1−ε) admits a biLipschitz embedding into R N with distortion at most D. The ..."
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Cited by 10 (4 self)
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It is shown that for every K> 0 and ε ∈ (0, 1/2) there exist N = N(K) ∈ N and D = D(K, ε) ∈ (1, ∞) with the following properties. For every metric space (X, d) with doubling constant at most K, the metric space (X, d 1−ε) admits a biLipschitz embedding into R N with distortion at most D. The classical Assouad embedding theorem makes the same assertion, but with N → ∞ as ε → 0.
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 10 (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.
Sharp quantitative nonembeddability of the Heisenberg group into superreflexive Banach spaces
, 2010
"... Let H denote the discrete Heisenberg group, equipped with a word metric dW associated to some finite symmetric generating set. We show that if (X, ‖ · ‖) is a pconvex Banach space then for any Lipschitz function f: H → X there exist x, y ∈ H with dW (x, y) arbitrarily large and ‖f(x) − f(y)‖ dW ..."
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Cited by 8 (4 self)
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Let H denote the discrete Heisenberg group, equipped with a word metric dW associated to some finite symmetric generating set. We show that if (X, ‖ · ‖) is a pconvex Banach space then for any Lipschitz function f: H → X there exist x, y ∈ H with dW (x, y) arbitrarily large and ‖f(x) − f(y)‖ dW (x, y) log log dW (x, y) log dW (x, y)