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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
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 $\ell_{1}$.
dimension reduction, metric embedding

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.
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