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 Frechet 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 5 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(log² n).