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
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
.