Near Linear Lower Bound for Dimension Reduction in L1
Author(s): Andoni, Alexandr; Charikar, Mosese S; Neiman, Ofer; Nguyen, Huy L
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DC Field | Value | Language |
---|---|---|
dc.contributor.author | Andoni, Alexandr | - |
dc.contributor.author | Charikar, Mosese S | - |
dc.contributor.author | Neiman, Ofer | - |
dc.contributor.author | Nguyen, Huy L | - |
dc.date.accessioned | 2021-10-08T19:44:42Z | - |
dc.date.available | 2021-10-08T19:44:42Z | - |
dc.date.issued | 2011 | en_US |
dc.identifier.citation | Andoni, Alexandr, Moses S. Charikar, Ofer Neiman, and Huy L. Nguyen. "Near Linear Lower Bound for Dimension Reduction in L1." 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science (2011): pp. 315-323. doi:10.1109/FOCS.2011.87 | en_US |
dc.identifier.issn | 0272-5428 | - |
dc.identifier.uri | http://web.mit.edu/andoni/www/papers/dim2-focs.pdf | - |
dc.identifier.uri | http://arks.princeton.edu/ark:/88435/pr1c80j | - |
dc.description.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 2 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/δ 2 ) 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 . | en_US |
dc.format.extent | 315 - 323 | en_US |
dc.language.iso | en_US | en_US |
dc.relation.ispartof | IEEE 52nd Annual Symposium on Foundations of Computer Science | en_US |
dc.rights | Author's manuscript | en_US |
dc.title | Near Linear Lower Bound for Dimension Reduction in L1 | en_US |
dc.type | Conference Article | en_US |
dc.identifier.doi | 10.1109/FOCS.2011.87 | - |
pu.type.symplectic | http://www.symplectic.co.uk/publications/atom-terms/1.0/conference-proceeding | en_US |
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