Dimension reduction techniques for ℓp (1 ≤ p ≤ 2), with applications

Yair Bartal, Lee Ad Gottlieb

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

5 Scopus citations


For Euclidean space (ℓ2), there exists the powerful dimension reduction transform of Johnson and Lindenstrauss [26], with a host of known applications. Here, we consider the problem of dimension reduction for all ℓp spaces 1 ≤ p ≤ 2. Although strong lower bounds are known for dimension reduction in ℓ1, Ostrovsky and Rabani [40] successfully circumvented these by presenting an ℓ1 embedding that maintains fidelity in only a bounded distance range, with applications to clustering and nearest neighbor search. However, their embedding techniques are specific to ℓ1 and do not naturally extend to other norms. In this paper, we apply a range of advanced techniques and produce bounded range dimension reduction embeddings for all of 1 ≤ p ≤ 2, thereby demonstrating that the approach initiated by Ostrovsky and Rabani for ℓ1 can be extended to a much more general framework. We also obtain improved bounds in terms of the intrinsic dimensionality. As a result we achieve improved bounds for proximity problems including snowflake embeddings and clustering.

Original languageAmerican English
Title of host publication32nd International Symposium on Computational Geometry, SoCG 2016
EditorsSandor Fekete, Anna Lubiw
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959770095
StatePublished - 1 Jun 2016
Event32nd International Symposium on Computational Geometry, SoCG 2016 - Boston, United States
Duration: 14 Jun 201617 Jun 2016

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
ISSN (Print)1868-8969


Conference32nd International Symposium on Computational Geometry, SoCG 2016
Country/TerritoryUnited States

Bibliographical note

Publisher Copyright:
© Yair Bartal and Lee-Ad Gottlieb.


  • Dimension reduction
  • Embeddings


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