2011 IEEE 52nd Annual Symposium on Foundations of Computer Science (2011)

Palm Springs, California USA

Oct. 22, 2011 to Oct. 25, 2011

ISSN: 0272-5428

ISBN: 978-0-7695-4571-4

pp: 315-323

DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/FOCS.2011.87

ABSTRACT

Given a set of $n$ points in $\ell_{1}$, 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 $\ell_{2}$ norm, where $O((\log n)/\epsilon^{2})$ dimensions suffice to achieve $1+\epsilon$ distortion. In sharp contrast, there is a significant gap between upper and lower bounds for dimension reduction in $\ell_{1}$. A recent result shows that distortion $1+\epsilon$ can be achieved with $n/\epsilon^{2}$ dimensions. On the other hand, the only lower bounds known are that distortion $\delta$ requires $n^{\Omega(1/\delta^2)}$ dimensions and that distortion $1+\epsilon$ requires $n^{1/2-O(\epsilon \log(1/\epsilon))}$ dimensions. In this work, we show the first near linear lower bounds for dimension reduction in $\ell_{1}$. In particular, we show that $1+\epsilon$ distortion requires at least $n^{1-O(1/\log(1/\epsilon))}$ 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 $\ell_{1}$.

INDEX TERMS

dimension reduction, metric embedding

CITATION

A. Andoni, O. Neiman, M. S. Charikar and H. L. Nguyen, "Near Linear Lower Bound for Dimension Reduction in L1,"

*2011 IEEE 52nd Annual Symposium on Foundations of Computer Science(FOCS)*, Palm Springs, California USA, 2011, pp. 315-323.

doi:10.1109/FOCS.2011.87

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