Measured descent: a new embedding method for finite metrics

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).