## Abstract

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 _{1} 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)} 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}.

Original language | English |
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Title of host publication | Proceedings - 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011 |

Pages | 315-323 |

Number of pages | 9 |

DOIs | |

State | Published - 1 Dec 2011 |

Externally published | Yes |

Event | 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011 - Palm Springs, CA, United States Duration: 22 Oct 2011 → 25 Oct 2011 |

### Conference

Conference | 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011 |
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Country/Territory | United States |

City | Palm Springs, CA |

Period | 22/10/11 → 25/10/11 |

## Keywords

- dimension reduction
- metric embedding