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Geometric & Functional Analysis GAFA

, Volume 15, Issue 4, pp 839–858 | Cite as

Measured descent: a new embedding method for finite metrics

  • R. Krauthgamer
  • J. R. Lee
  • M. Mendel
  • A. Naor
Original Paper

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 \(O{\left( {{\sqrt {\alpha _{X} \cdot \log n} }} \right)},\) where α X is a geometric estimate on the decomposability of X. As an immediate corollary, we obtain an \(O{\left( {{\sqrt {(\log \lambda _{X} )\log n} }} \right)}\) 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 \({\ell }^{{O(\log n)}}_{\infty } \) 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).

Keywords

Hilbert Space Probability Measure Planar Graph Primary Method Unify Framework 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Verlag, Basel 2005

Authors and Affiliations

  • R. Krauthgamer
    • 1
  • J. R. Lee
    • 2
  • M. Mendel
    • 3
  • A. Naor
    • 4
  1. 1.IBM Almaden Research CenterSan JoseUSA
  2. 2.Computer Science DivisionUniversity of CaliforniaBerkeleyUSA
  3. 3.Seibel Center for Computer ScienceUniversity of Illinois at Urbana-ChampaignUrbanaUSA
  4. 4.One Microsoft WayMicrosoft ResearchRedmondUSA

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