Measured descent: A new embedding method for finite metrics

Robert Krauthgamer, James R. Lee, Manor Mendel, Assaf Naor

Research output: Contribution to journalConference article

33 Scopus citations

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 J. Bourgain and S. Rao. We prove that any n-point metric space (X, d) embeds in Hilbert space with distortion O(√α X · log n), where α X is a geometric estimate on the decomposability of X. An an immediate corollary, we obtain an O(√log λ X · 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(logn)) 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 l O(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 2 n).

Original languageEnglish (US)
Pages (from-to)434-443
Number of pages10
JournalProceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
StatePublished - Dec 1 2004
Externally publishedYes
EventProceedings - 45th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2004 - Rome, Italy
Duration: Oct 17 2004Oct 19 2004

All Science Journal Classification (ASJC) codes

  • Engineering(all)

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