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 pairwise 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) dimension 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.