The Euclidean distortion of flat tori

Ishay Haviv, Oded Regev

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

We show that for every n-dimensional lattice ℒ the torus ℝn/ℒ can be embedded with distortion O(n·√log n) into a Hilbert space. This improves the exponential upper bound of O(n 3n/2) due to Khot and Naor (FOCS 2005, Math. Annal. 2006) and gets close to their lower bound of Ω(√n). We also obtain tight bounds for certain families of lattices. Our main new ingredient is an embedding that maps any point u ∈ ℝn/ℒ to a Gaussian function centered at u in the Hilbert space L2n/ℒ. The proofs involve Gaussian measures on lattices, the smoothing parameter of lattices and Korkine-Zolotarev bases.

Original languageEnglish (US)
Title of host publicationApproximation, Randomization, and Combinatorial Optimization
Subtitle of host publicationAlgorithms and Techniques - 13th International Workshop, APPROX 2010 and 14th International Workshop, RANDOM 2010, Proceedings
Pages232-245
Number of pages14
DOIs
StatePublished - 2010
Event13th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2010 and 14th International Workshop on Randomization and Computation, RANDOM 2010 - Barcelona, Spain
Duration: Sep 1 2010Sep 3 2010

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume6302 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Other

Other13th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2010 and 14th International Workshop on Randomization and Computation, RANDOM 2010
CountrySpain
CityBarcelona
Period9/1/109/3/10

Keywords

  • Embedding
  • Lattice
  • Torus

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

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