### Abstract

We study the metric properties of finite subsets of L _{1}. The analysis of such metrics is central to a number of important algorithmic problems involving the cut structure of weighted graphs, including the Sparsest Cut Problem, one of the most compelling open problems in the field of approximation. Additionally, many open questions in geometric non-linear functional analysis involve the properties of finite subsets of L _{1}. We present some new observations concerning the relation of L _{1} to dimension, topology, and Euclidean distortion. We show that every n-point subset of L _{1} embeds into L _{2} with average distortion O(√log n), yielding the first evidence that the conjectured worst-case bound of O(√log n) is valid. We also address the issue of dimension reduction in L _{p} for p ∈(1,2). We resolve a question left open in [1] about the impossibility of linear dimension reduction in the above cases, and we show that the example of [2,3] cannot be used to prove a lower bound for the non-linear case. This is accomplished by exhibiting constant-distortion embeddings of snowflaked planar metrics into Euclidean space.

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

Pages (from-to) | 401-412 |

Number of pages | 12 |

Journal | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |

Volume | 2976 |

State | Published - 2004 |

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### ASJC Scopus subject areas

- Computer Science(all)
- Biochemistry, Genetics and Molecular Biology(all)
- Theoretical Computer Science

### Cite this

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)*,

*2976*, 401-412.

**Metric structures in L 1 : Dimension, snowflakes, and average distortion.** / Lee, James R.; Mendel, Manor; Naor, Assaf.

Research output: Contribution to journal › Article

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)*, vol. 2976, pp. 401-412.

}

TY - JOUR

T1 - Metric structures in L 1

T2 - Dimension, snowflakes, and average distortion

AU - Lee, James R.

AU - Mendel, Manor

AU - Naor, Assaf

PY - 2004

Y1 - 2004

N2 - We study the metric properties of finite subsets of L 1. The analysis of such metrics is central to a number of important algorithmic problems involving the cut structure of weighted graphs, including the Sparsest Cut Problem, one of the most compelling open problems in the field of approximation. Additionally, many open questions in geometric non-linear functional analysis involve the properties of finite subsets of L 1. We present some new observations concerning the relation of L 1 to dimension, topology, and Euclidean distortion. We show that every n-point subset of L 1 embeds into L 2 with average distortion O(√log n), yielding the first evidence that the conjectured worst-case bound of O(√log n) is valid. We also address the issue of dimension reduction in L p for p ∈(1,2). We resolve a question left open in [1] about the impossibility of linear dimension reduction in the above cases, and we show that the example of [2,3] cannot be used to prove a lower bound for the non-linear case. This is accomplished by exhibiting constant-distortion embeddings of snowflaked planar metrics into Euclidean space.

AB - We study the metric properties of finite subsets of L 1. The analysis of such metrics is central to a number of important algorithmic problems involving the cut structure of weighted graphs, including the Sparsest Cut Problem, one of the most compelling open problems in the field of approximation. Additionally, many open questions in geometric non-linear functional analysis involve the properties of finite subsets of L 1. We present some new observations concerning the relation of L 1 to dimension, topology, and Euclidean distortion. We show that every n-point subset of L 1 embeds into L 2 with average distortion O(√log n), yielding the first evidence that the conjectured worst-case bound of O(√log n) is valid. We also address the issue of dimension reduction in L p for p ∈(1,2). We resolve a question left open in [1] about the impossibility of linear dimension reduction in the above cases, and we show that the example of [2,3] cannot be used to prove a lower bound for the non-linear case. This is accomplished by exhibiting constant-distortion embeddings of snowflaked planar metrics into Euclidean space.

UR - http://www.scopus.com/inward/record.url?scp=35048840167&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=35048840167&partnerID=8YFLogxK

M3 - Article

VL - 2976

SP - 401

EP - 412

JO - Lecture Notes in Computer Science

JF - Lecture Notes in Computer Science

SN - 0302-9743

ER -