Near Linear Lower Bound for Dimension Reduction in L_1

Alexandr Andoni, Moses S. Charikar, Ofer Neiman, and Huy L. Nguyen

2011

## Abstract

Given a set of *n* points in *ℓ*_{1}, how many dimensions are needed to represent all pairwise 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/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 *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}.

## Details

Publication type | Proceedings |

Published in | Symposium on Foundations of Computer Science (FOCS) |

Publisher | IEEE |

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