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Given a set of $n$ points in $\ell_{1}$, how many dimensions are needed to represent all pairwise distances within a specific distortion ? This dimension-distortion tradeoff question is well understood for the $ell_{2}$ norm, where $O((\logn)/\epsilon^{2})$ 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 $O(n/\epsilon^{2})$ dimensions. On the other hand, the only lower bounds known are that distortion $\delta$ requires $n^{\Omega(1/\delta^2)}$ dimension 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 $\ell_{1}$.

joint work with Alexandr Andoni, Moses S. Charikar, and Huy L. Nguyen