# Measured descent: a new embedding method for finite metrics

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## 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## Preview

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