### Abstract

We show that for every large enough integer N, there exists an N-point subset of L _{1} such that for every D > 1, embedding it into ℓ _{1} ^{d} with distortion D requires dimension d at least N^{Ω} (1/D^{2}), and that for every e{open} > 0 and large enough integer N, there exists an N-point subset of L _{1} such that embedding it into ℓ _{1} ^{d} with distortion 1 + e{open} requires dimension d at least N^{Ω} (1/D^{2})). These results were previously proven by Brinkman and Charikar [JACM, 2005] and by Andoni, Charikar, Neiman and Nguyen [FOCS 2011]. We provide an alternative and arguably more intuitive proof based on an entropy argument.

Original language | English (US) |
---|---|

Pages (from-to) | 825-832 |

Number of pages | 8 |

Journal | Israel Journal of Mathematics |

Volume | 195 |

Issue number | 2 |

DOIs | |

State | Published - Jun 2013 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**Entropy-based bounds on dimension reduction in L1
.** / Regev, Oded.

Research output: Contribution to journal › Article

*Israel Journal of Mathematics*, vol. 195, no. 2, pp. 825-832. https://doi.org/10.1007/s11856-012-0137-6

}

TY - JOUR

T1 - Entropy-based bounds on dimension reduction in L1

AU - Regev, Oded

PY - 2013/6

Y1 - 2013/6

N2 - We show that for every large enough integer N, there exists an N-point subset of L 1 such that for every D > 1, embedding it into ℓ 1 d with distortion D requires dimension d at least NΩ (1/D2), and that for every e{open} > 0 and large enough integer N, there exists an N-point subset of L 1 such that embedding it into ℓ 1 d with distortion 1 + e{open} requires dimension d at least NΩ (1/D2)). These results were previously proven by Brinkman and Charikar [JACM, 2005] and by Andoni, Charikar, Neiman and Nguyen [FOCS 2011]. We provide an alternative and arguably more intuitive proof based on an entropy argument.

AB - We show that for every large enough integer N, there exists an N-point subset of L 1 such that for every D > 1, embedding it into ℓ 1 d with distortion D requires dimension d at least NΩ (1/D2), and that for every e{open} > 0 and large enough integer N, there exists an N-point subset of L 1 such that embedding it into ℓ 1 d with distortion 1 + e{open} requires dimension d at least NΩ (1/D2)). These results were previously proven by Brinkman and Charikar [JACM, 2005] and by Andoni, Charikar, Neiman and Nguyen [FOCS 2011]. We provide an alternative and arguably more intuitive proof based on an entropy argument.

UR - http://www.scopus.com/inward/record.url?scp=84883797119&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84883797119&partnerID=8YFLogxK

U2 - 10.1007/s11856-012-0137-6

DO - 10.1007/s11856-012-0137-6

M3 - Article

AN - SCOPUS:84883797119

VL - 195

SP - 825

EP - 832

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

SN - 0021-2172

IS - 2

ER -