Near Linear Lower Bound for Dimension Reduction in L1

Given a set of n points in ℓ₁, 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 ℓ₂ norm, where O((log n)/ϵ²) dimensions suffice to achieve 1+ϵ distortion. In sharp contrast, there is a significant gap between upper and lower bounds for dimension reduction in ℓ₁. A recent result shows that distortion 1+ϵ can be achieved with n/ϵ² 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 ℓ₁. 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 ℓ₁.