Measured descent: a new embedding method for finite metrics

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).

Authors and Affiliations

1. IBM Almaden Research Center, San Jose, CA, 95120, USA

   R. Krauthgamer

2. Computer Science Division, University of California, Berkeley, CA, 04720, USA

   J. R. Lee

3. Seibel Center for Computer Science, University of Illinois at Urbana-Champaign, Urbana, IL, 61801, USA

   M. Mendel

4. One Microsoft Way, Microsoft Research, Redmond, WA, 98052, USA

   A. Naor

Corresponding author

Correspondence to R. Krauthgamer.

Received: April 2004 Accepted: August 2004 Revision: December 2004

J.R.L. Supported by NSF grant CCR-0121555 and an NSF Graduate Research Fellowship.

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