9 Search Results for "Naor, Assaf"

Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2024.47
Upper Bounds on the 2-Colorability Threshold of Random d-Regular k-Uniform Hypergraphs for k ≥ 3

Authors: Evan Chang, Neel Kolhe, and Youngtak Sohn

Published in: LIPIcs, Volume 317, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)

Abstract
For a large class of random constraint satisfaction problems (csp), deep but non-rigorous theory from statistical physics predict the location of the sharp satisfiability transition. The works of Ding, Sly, Sun (2014, 2016) and Coja-Oghlan, Panagiotou (2014) established the satisfiability threshold for random regular k-nae-sat, random k-sat, and random regular k-sat for large enough k ≥ k₀ where k₀ is a large non-explicit constant. Establishing the same for small values of k ≥ 3 remains an important open problem in the study of random csps. In this work, we study two closely related models of random csps, namely the 2-coloring on random d-regular k-uniform hypergraphs and the random d-regular k-nae-sat model. For every k ≥ 3, we prove that there is an explicit d_⋆(k) which gives a satisfiability upper bound for both of the models. Our upper bound d_⋆(k) for k ≥ 3 matches the prediction from statistical physics for the hypergraph 2-coloring by Dall’Asta, Ramezanpour, Zecchina (2008), thus conjectured to be sharp. Moreover, d_⋆(k) coincides with the satisfiability threshold of random regular k-nae-sat for large enough k ≥ k₀ by Ding, Sly, Sun (2014).
Cite as

Evan Chang, Neel Kolhe, and Youngtak Sohn. Upper Bounds on the 2-Colorability Threshold of Random d-Regular k-Uniform Hypergraphs for k ≥ 3. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 47:1-47:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{chang_et_al:LIPIcs.APPROX/RANDOM.2024.47,
  author =	{Chang, Evan and Kolhe, Neel and Sohn, Youngtak},
  title =	{{Upper Bounds on the 2-Colorability Threshold of Random d-Regular k-Uniform Hypergraphs for k ≥ 3}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
  pages =	{47:1--47:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-348-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{317},
  editor =	{Kumar, Amit and Ron-Zewi, Noga},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.47},
  URN =		{urn:nbn:de:0030-drops-210402},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2024.47},
  annote =	{Keywords: Random constraint satisfaction problem, replica symmetry breaking, interpolation bound}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2024.68
Sparse High Dimensional Expanders via Local Lifts

Authors: Inbar Ben Yaacov, Yotam Dikstein, and Gal Maor

Published in: LIPIcs, Volume 317, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)

Abstract
High dimensional expanders (HDXs) are a hypergraph generalization of expander graphs. They are extensively studied in the math and TCS communities due to their many applications. Like expander graphs, HDXs are especially interesting for applications when they are bounded degree, namely, if the number of edges adjacent to every vertex is bounded. However, only a handful of constructions are known to have this property, all of which rely on algebraic techniques. In particular, no random or combinatorial construction of bounded degree high dimensional expanders is known. As a result, our understanding of these objects is limited. The degree of an i-face in an HDX is the number of (i+1)-faces that contain it. In this work we construct complexes whose higher dimensional faces have bounded degree. This is done by giving an elementary and deterministic algorithm that takes as input a regular k-dimensional HDX X and outputs another regular k-dimensional HDX X̂ with twice as many vertices. While the degree of vertices in X̂ grows, the degree of the (k-1)-faces in X̂ stays the same. As a result, we obtain a new "algebra-free" construction of HDXs whose (k-1)-face degree is bounded. Our construction algorithm is based on a simple and natural generalization of the expander graph construction by Bilu and Linial [Yehonatan Bilu and Nathan Linial, 2006], which build expander graphs using lifts coming from edge signings. Our construction is based on local lifts of high dimensional expanders, where a local lift is a new complex whose top-level links are lifts of the links of the original complex. We demonstrate that a local lift of an HDX is also an HDX in many cases. In addition, combining local lifts with existing bounded degree constructions creates new families of bounded degree HDXs with significantly different links than before. For every large enough D, we use this technique to construct families of bounded degree HDXs with links that have diameter ≥ D.
Cite as

Inbar Ben Yaacov, Yotam Dikstein, and Gal Maor. Sparse High Dimensional Expanders via Local Lifts. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 68:1-68:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{benyaacov_et_al:LIPIcs.APPROX/RANDOM.2024.68,
  author =	{Ben Yaacov, Inbar and Dikstein, Yotam and Maor, Gal},
  title =	{{Sparse High Dimensional Expanders via Local Lifts}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
  pages =	{68:1--68:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-348-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{317},
  editor =	{Kumar, Amit and Ron-Zewi, Noga},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.68},
  URN =		{urn:nbn:de:0030-drops-210612},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2024.68},
  annote =	{Keywords: High Dimensional Expanders, HDX, Spectral Expansion, Lifts, Covers, Explicit Constructions, Randomized Constructions, Deterministic Constructions}
}
Document
DOI: 10.4230/LIPIcs.CCC.2024.3
Solving Unique Games over Globally Hypercontractive Graphs

Authors: Mitali Bafna and Dor Minzer

Published in: LIPIcs, Volume 300, 39th Computational Complexity Conference (CCC 2024)

Abstract
We study the complexity of affine Unique-Games (UG) over globally hypercontractive graphs, which are graphs that are not small set expanders but admit a useful and succinct characterization of all small sets that violate the small-set expansion property. This class of graphs includes the Johnson and Grassmann graphs, which have played a pivotal role in recent PCP constructions for UG, and their generalizations via high-dimensional expanders. We show new rounding techniques for higher degree sum-of-squares (SoS) relaxations for worst-case optimization. In particular, our algorithm shows how to round "low-entropy" pseudodistributions, broadly extending the algorithmic framework of [Mitali Bafna et al., 2021]. At a high level, [Mitali Bafna et al., 2021] showed how to round pseudodistributions for problems where there is a "unique" good solution. We extend their framework by exhibiting a rounding for problems where there might be "few good solutions". Our result suggests that UG is easy on globally hypercontractive graphs, and therefore highlights the importance of graphs that lack such a characterization in the context of PCP reductions for UG.
Cite as

Mitali Bafna and Dor Minzer. Solving Unique Games over Globally Hypercontractive Graphs. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 3:1-3:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bafna_et_al:LIPIcs.CCC.2024.3,
  author =	{Bafna, Mitali and Minzer, Dor},
  title =	{{Solving Unique Games over Globally Hypercontractive Graphs}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{3:1--3:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-331-7},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{300},
  editor =	{Santhanam, Rahul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.3},
  URN =		{urn:nbn:de:0030-drops-203996},
  doi =		{10.4230/LIPIcs.CCC.2024.3},
  annote =	{Keywords: unique games, approximation algorithms}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.42
Path-Reporting Distance Oracles with Logarithmic Stretch and Linear Size

Authors: Shiri Chechik and Tianyi Zhang

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
Given an undirected graph G = (V, E, 𝐰) on n vertices with positive edge weights, a distance oracle is a space-efficient data structure that answers pairwise distance queries in fast runtime. The quality of a distance oracle is measured by three parameters: space, query time, and stretch. In a landmark paper by [Thorup and Zwick, 2001], they showed that for any integer parameter k ≥ 1, there exists a distance oracle with size O(kn^{1+1/k}), O(k) query time, and (2k-1)-stretch error on the approximate distances. After that, there has been a line of subsequent improvements which culminated in the optimal trade-off of O(n^{1+1/k}) space, O(1) query time, and (2k-1)-stretch [Chechik, 2015]. However, these line of constructions did not require that the distance oracle is capable of printing an actual path besides an approximate distance estimate, and there has been a performance gap between path-reporting distance oracles and ones that are not path-reporting. It is known that the earliest construction by [Thorup and Zwick, 2001] is path-reporting, but the parameters are worse by a factor of k. In a later construction by [Wulff-Nilsen, 2013], the query time was improved from O(k) to O(log k). Better trade-offs were discovered in [Elkin and Pettie, 2015] where the authors broke the O(kn^{1+1/k}) space barrier and achieved O(n^{1+1/k}log k) space with O(log k) query time, but their stretch was blown up to a polynomial O(k^{log_{4/3}7}); they also gave an alternative choice of O(n^{1+1/k}) space which is optimal, and O(k)-stretch which is also optimal up to a constant factor, but their query time rose exponentially to O(n^ε). In a recent work [Elkin and Shabat, 2023], the authors obtained significant improvements of O(n^{1+1/k}log k) space, O(k)-stretch, and O(log log k) query time, or O(n^{1+1/k}) space, O(klog k)-stretch, and O(log log k) query time. All the above constructions of path-reporting distance oracles share a common barrier; that is, they could not achieve optimal space O(n^{1+1/k}) and stretch O(k) simultaneously within logarithmic query time; for example, in the natural regime where k = ⌈log n⌉, previous distance oracles had to pay an extra factor of log log n either in the space or stretch. As our result, we bypass this barrier by a new construction of path-reporting distance oracles with O(n^{1+1/k}) space and O(k)-stretch and O(log log k) query time.
Cite as

Shiri Chechik and Tianyi Zhang. Path-Reporting Distance Oracles with Logarithmic Stretch and Linear Size. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 42:1-42:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{chechik_et_al:LIPIcs.ICALP.2024.42,
  author =	{Chechik, Shiri and Zhang, Tianyi},
  title =	{{Path-Reporting Distance Oracles with Logarithmic Stretch and Linear Size}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{42:1--42:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.42},
  URN =		{urn:nbn:de:0030-drops-201859},
  doi =		{10.4230/LIPIcs.ICALP.2024.42},
  annote =	{Keywords: graph algorithms, shortest paths, distance oracles}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.101
On the Space Usage of Approximate Distance Oracles with Sub-2 Stretch

Authors: Tsvi Kopelowitz, Ariel Korin, and Liam Roditty

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
For an undirected unweighted graph G = (V,E) with n vertices and m edges, let d(u,v) denote the distance from u ∈ V to v ∈ V in G. An (α,β)-stretch approximate distance oracle (ADO) for G is a data structure that given u,v ∈ V returns in constant (or near constant) time a value dˆ(u,v) such that d(u,v) ≤ dˆ(u,v) ≤ α⋅ d(u,v) + β, for some reals α > 1, β. Thorup and Zwick [Mikkel Thorup and Uri Zwick, 2005] showed that one cannot beat stretch 3 with subquadratic space (in terms of n) for general graphs. Pǎtraşcu and Roditty [Mihai Pǎtraşcu and Liam Roditty, 2010] showed that one can obtain stretch 2 using O(m^{1/3}n^{4/3}) space, and so if m is subquadratic in n then the space usage is also subquadratic. Moreover, Pǎtraşcu and Roditty [Mihai Pǎtraşcu and Liam Roditty, 2010] showed that one cannot beat stretch 2 with subquadratic space even for graphs where m = Õ(n), based on the set-intersection hypothesis. In this paper we explore the conditions for which an ADO can beat stretch 2 while using subquadratic space. In particular, we show that if the maximum degree in G is Δ_G ≤ O(n^{1/k-ε}) for some 0 < ε ≤ 1/k, then there exists an ADO for G that uses Õ(n^{2-(kε)/3) space and has a (2,1-k)-stretch. For k = 2 this result implies a subquadratic sub-2 stretch ADO for graphs with Δ_G ≤ O(n^{1/2-ε}). Moreover, we prove a conditional lower bound, based on the set intersection hypothesis, which states that for any positive integer k ≤ log n, obtaining a sub-(k+2)/k stretch for graphs with Δ_G = Θ(n^{1/k}) requires Ω̃(n²) space. Thus, for graphs with maximum degree Θ(n^{1/2}), obtaining a sub-2 stretch requires Ω̃(n²) space.
Cite as

Tsvi Kopelowitz, Ariel Korin, and Liam Roditty. On the Space Usage of Approximate Distance Oracles with Sub-2 Stretch. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 101:1-101:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{kopelowitz_et_al:LIPIcs.ICALP.2024.101,
  author =	{Kopelowitz, Tsvi and Korin, Ariel and Roditty, Liam},
  title =	{{On the Space Usage of Approximate Distance Oracles with Sub-2 Stretch}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{101:1--101:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.101},
  URN =		{urn:nbn:de:0030-drops-202443},
  doi =		{10.4230/LIPIcs.ICALP.2024.101},
  annote =	{Keywords: Graph algorithms, Approximate distance oracle, data structures, shortest path}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2024.55
Dimensionality of Hamming Metrics and Rademacher Type

Authors: Alexandros Eskenazis

Published in: LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024)

Abstract
Let X be a finite-dimensional normed space. We prove that if the Hamming cube {-1,1}ⁿ embeds into X with bi-Lipschitz distortion at most D ≥ 1, then dim(X) ≳ sup_{p ∈ [1,2]} n^p/(D^p 𝖳_p(X)^p), where 𝖳_p(X) is the Rademacher type p constant of X. This estimate yields a mutual refinement of distortion lower bounds which follow from works of Oleszkiewicz (1996) and Ivanisvili, van Handel and Volberg (2020). The proof relies on a combination of semigroup techniques on the biased hypercube with the Borsuk-Ulam theorem from algebraic topology.
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Alexandros Eskenazis. Dimensionality of Hamming Metrics and Rademacher Type. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 55:1-55:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{eskenazis:LIPIcs.SoCG.2024.55,
  author =	{Eskenazis, Alexandros},
  title =	{{Dimensionality of Hamming Metrics and Rademacher Type}},
  booktitle =	{40th International Symposium on Computational Geometry (SoCG 2024)},
  pages =	{55:1--55:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-316-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{293},
  editor =	{Mulzer, Wolfgang and Phillips, Jeff M.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.55},
  URN =		{urn:nbn:de:0030-drops-200004},
  doi =		{10.4230/LIPIcs.SoCG.2024.55},
  annote =	{Keywords: Hamming cube, Rademacher type, metric embeddings, Borsuk-Ulam theorem}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2022.40
ε-Isometric Dimension Reduction for Incompressible Subsets of 𝓁_p

Authors: Alexandros Eskenazis

Published in: LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022)

Abstract
Fix p ∈ [1,∞), K ∈ (0,∞) and a probability measure μ. We prove that for every n ∈ ℕ, ε ∈ (0,1) and x₁,…,x_n ∈ L_p(μ) with ‖max_{i ∈ {1,…,n}}|x_i|‖_{L_p(μ)} ≤ K, there exists d ≤ (32e² (2K)^{2p}log n)/ε² and vectors y₁,…, y_n ∈ 𝓁_p^d such that ∀i,j∈{1,…,n}, ‖x_i-x_j‖^p_{L_p(μ)}-ε ≤ ‖y_i-y_j‖_{𝓁_p^d}^p ≤ ‖x_i-x_j‖^p_{L_p(μ)}+ε. Moreover, the argument implies the existence of a greedy algorithm which outputs {y_i}_{i = 1}ⁿ after receiving {x_i}_{i = 1}ⁿ as input. The proof relies on a derandomized version of Maurey’s empirical method (1981) combined with a combinatorial idea of Ball (1990) and a suitable change of measure. Motivated by the above embedding, we introduce the notion of ε-isometric dimension reduction of the unit ball B_E of a normed space (E,‖⋅‖_E) and we prove that B_{𝓁_p} does not admit ε-isometric dimension reduction by linear operators for any value of p≠2.
Cite as

Alexandros Eskenazis. ε-Isometric Dimension Reduction for Incompressible Subsets of 𝓁_p. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 40:1-40:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{eskenazis:LIPIcs.SoCG.2022.40,
  author =	{Eskenazis, Alexandros},
  title =	{{\epsilon-Isometric Dimension Reduction for Incompressible Subsets of 𝓁\underlinep}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{40:1--40:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-227-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{224},
  editor =	{Goaoc, Xavier and Kerber, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.40},
  URN =		{urn:nbn:de:0030-drops-160486},
  doi =		{10.4230/LIPIcs.SoCG.2022.40},
  annote =	{Keywords: Dimension reduction, \epsilon-isometric embedding, Maurey’s empirical method, change of measure}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2017.50
A Spectral Gap Precludes Low-Dimensional Embeddings

Authors: Assaf Naor

Published in: LIPIcs, Volume 77, 33rd International Symposium on Computational Geometry (SoCG 2017)

Abstract
We prove that if an n-vertex O(1)-expander embeds with average distortion D into a finite dimensional normed space X, then necessarily the dimension of X is at least n^{c/D} for some universal constant c>0. This is sharp up to the value of the constant c, and it improves over the previously best-known estimate dim(X)> c(log n)^2/D^2 of Linial, London and Rabinovich, strengthens a theorem of Matousek, and answers a question of Andoni, Nikolov, Razenshteyn and Waingarten.
Cite as

Assaf Naor. A Spectral Gap Precludes Low-Dimensional Embeddings. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 50:1-50:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{naor:LIPIcs.SoCG.2017.50,
  author =	{Naor, Assaf},
  title =	{{A Spectral Gap Precludes Low-Dimensional Embeddings}},
  booktitle =	{33rd International Symposium on Computational Geometry (SoCG 2017)},
  pages =	{50:1--50:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-038-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{77},
  editor =	{Aronov, Boris and Katz, Matthew J.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2017.50},
  URN =		{urn:nbn:de:0030-drops-71822},
  doi =		{10.4230/LIPIcs.SoCG.2017.50},
  annote =	{Keywords: Metric embeddings, dimensionality reduction, expander graphs, nonlinear spectral gaps, nearest neighbor search, complex interpolation, Markov type.}
}
Document
DOI: 10.4230/LIPIcs.ICALP.2016.83
Impossibility of Sketching of the 3D Transportation Metric with Quadratic Cost

Authors: Alexandr Andoni, Assaf Naor, and Ofer Neiman

Published in: LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

Abstract
Transportation cost metrics, also known as the Wasserstein distances W_p, are a natural choice for defining distances between two pointsets, or distributions, and have been applied in numerous fields. From the computational perspective, there has been an intensive research effort for understanding the W_p metrics over R^k, with work on the W_1 metric (a.k.a earth mover distance) being most successful in terms of theoretical guarantees. However, the W_2 metric, also known as the root-mean square (RMS) bipartite matching distance, is often a more suitable choice in many application areas, e.g. in graphics. Yet, the geometry of this metric space is currently poorly understood, and efficient algorithms have been elusive. For example, there are no known non-trivial algorithms for nearest-neighbor search or sketching for this metric. In this paper we take the first step towards explaining the lack of efficient algorithms for the W_2 metric, even over the three-dimensional Euclidean space R^3. We prove that there are no meaningful embeddings of W_2 over R^3 into a wide class of normed spaces, as well as that there are no efficient sketching algorithms for W_2 over R^3 achieving constant approximation. For example, our results imply that: 1) any embedding into L1 must incur a distortion of Omega(sqrt(log(n))) for pointsets of size n equipped with the W_2 metric; and 2) any sketching algorithm of size s must incur Omega(sqrt(log(n))/sqrt(s)) approximation. Our results follow from a more general statement, asserting that W_2 over R^3 contains the 1/2-snowflake of all finite metric spaces with a uniformly bounded distortion. These are the first non-embeddability/non-sketchability results for W_2.
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Alexandr Andoni, Assaf Naor, and Ofer Neiman. Impossibility of Sketching of the 3D Transportation Metric with Quadratic Cost. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 83:1-83:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{andoni_et_al:LIPIcs.ICALP.2016.83,
  author =	{Andoni, Alexandr and Naor, Assaf and Neiman, Ofer},
  title =	{{Impossibility of Sketching of the 3D Transportation Metric with Quadratic Cost}},
  booktitle =	{43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)},
  pages =	{83:1--83:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-013-2},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{55},
  editor =	{Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.83},
  URN =		{urn:nbn:de:0030-drops-62028},
  doi =		{10.4230/LIPIcs.ICALP.2016.83},
  annote =	{Keywords: Transportation metric, embedding, snowflake, sketching}
}
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