Near linear lower bound for dimension reduction in ℓ1

Alexandr Andoni, Moses S. Charikar, Ofer Neiman, Huy L. Nguyen

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

17 Scopus citations

Abstract

Given a set of n points in ℓ 1, how many dimensions are needed to represent all pair wise distances within a specific distortion? This dimension-distortion tradeoff question is well understood for the ℓ 1 norm, where O((log n)/ε 2) dimensions suffice to achieve 1+ε distortion. In sharp contrast, there is a significant gap between upper and lower bounds for dimension reduction in ℓ 1. A recent result shows that distortion 1+ε can be achieved with n/ε 2 dimensions. On the other hand, the only lower bounds known are that distortion δ requires n Ω(1/δ2) dimensions and that distortion 1+ε requires n 1/2-O(ε log(1/ε)) dimensions. In this work, we show the first near linear lower bounds for dimension reduction in ℓ 1. In particular, we show that 1+ε distortion requires at least n 1-O(1/log(1/ε)) dimensions. Our proofs are combinatorial, but inspired by linear programming. In fact, our techniques lead to a simple combinatorial argument that is equivalent to the LP based proof of Brinkman-Charikar for lower bounds on dimension reduction in ℓ 1.

Original languageEnglish
Title of host publicationProceedings - 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011
Pages315-323
Number of pages9
DOIs
StatePublished - 1 Dec 2011
Externally publishedYes
Event2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011 - Palm Springs, CA, United States
Duration: 22 Oct 201125 Oct 2011

Conference

Conference2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011
Country/TerritoryUnited States
CityPalm Springs, CA
Period22/10/1125/10/11

Keywords

  • dimension reduction
  • metric embedding

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