Alexandr Andoni, Moses S. Charikar, Ofer Neiman, and Huy L. Nguyen
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 nΩ(1/δ2) dimension and that distortion 1+ε requires n1/2-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 n1-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.
|Published in||Symposium on Foundations of Computer Science (FOCS)|