Near Linear Lower Bound for Dimension Reduction in L1

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 .