Near linear lower bound for dimension reduction in ℓ1

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 ₁ 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/δ2) 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.