TY - JOUR

T1 - Measured descent

T2 - A new embedding method for finite metrics

AU - Krauthgamer, R.

AU - Lee, J. R.

AU - Mendel, M.

AU - Naor, A.

N1 - Funding Information:
J.R.L. Supported by NSF grant CCR-0121555 and an NSF Graduate Research Fellowship.

PY - 2005/8

Y1 - 2005/8

N2 - 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 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 n)2).

AB - 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 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 n)2).

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U2 - 10.1007/s00039-005-0527-6

DO - 10.1007/s00039-005-0527-6

M3 - Article

AN - SCOPUS:26844509176

VL - 15

SP - 839

EP - 858

JO - Geometric and Functional Analysis

JF - Geometric and Functional Analysis

SN - 1016-443X

IS - 4

ER -