Measured descent: A new embedding method for finite metrics

R. Krauthgamer, J. R. Lee, M. Mendel, A. Naor

Research output: Contribution to journalArticlepeer-review

Abstract

We devise a new embedding technique, which we call measured descent, based on decomposing a metric space locally, at varying speeds, according to the density of some probability measure. This provides a refined and unified framework for the two primary methods of constructing Fréchet embeddings for finite metrics, due to Bourgain (1985) and Rao (1999). We prove that any n-point metric space (X, d) embeds in Hilbert space with distortion O(√αX · log n), where α X is a geometric estimate on the decomposability of X. As an immediate corollary, we obtain an O(√αX · log n) distortion embedding, where λ X is the doubling constant of X. Since λ X ≤ n, this result recovers Bourgain's theorem, but when the metric X is, in a sense, "low-dimensional," improved bounds are achieved. Our embeddings are volume-respecting for subsets of arbitrary size. One consequence is the existence of (k, O(log n)) volume-respecting embeddings for all 1 ≤ k ≤ n, which is the best possible, and answers positively a question posed by U. Feige. Our techniques are also used to answer positively a question of Y. Rabinovich, showing that any weighted n-point planar graph embeds in lO(log n) with O(1) distortion. The O(log n) bound on the dimension is optimal, and improves upon the previously known bound of O((log n)2).

Original languageEnglish (US)
Pages (from-to)839-858
Number of pages20
JournalGeometric and Functional Analysis
Volume15
Issue number4
DOIs
StatePublished - Aug 2005

ASJC Scopus subject areas

  • Analysis
  • Geometry and Topology

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