For Euclidean space (ℓ2), there exists the powerful dimension reduction transform of Johnson and Lindenstrauss , 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  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 language||American English|
|Title of host publication||32nd International Symposium on Computational Geometry, SoCG 2016|
|Editors||Sandor Fekete, Anna Lubiw|
|Publisher||Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing|
|State||Published - 1 Jun 2016|
|Event||32nd International Symposium on Computational Geometry, SoCG 2016 - Boston, United States|
Duration: 14 Jun 2016 → 17 Jun 2016
|Name||Leibniz International Proceedings in Informatics, LIPIcs|
|Conference||32nd International Symposium on Computational Geometry, SoCG 2016|
|Period||14/06/16 → 17/06/16|
Bibliographical notePublisher Copyright:
© Yair Bartal and Lee-Ad Gottlieb.
- Dimension reduction