Measured descent: A new embedding method for finite metrics

TL;DR

This paper introduces measured descent, a new metric embedding technique that improves bounds for embedding finite metric spaces into Hilbert space and  spaces, especially for low-dimensional or volume-respecting subsets.

Contribution

The paper presents a novel measured descent method that refines embedding bounds for finite metrics, unifies previous approaches, and answers open questions on volume-respecting embeddings and planar graph embeddings.

Findings

Embeds any n-point metric into Hilbert space with O(rtX log n) distortion.

Achieves volume-respecting embeddings with optimal (k, O(log n)) bounds.

Embeds planar graphs into  space with O(1) distortion and O(log n) dimension.

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 Frechet 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(sqrt{alpha_X log n}), where alpha_X is a geometric estimate on the decomposability of X. As an immediate corollary, we obtain an O(sqrt{(log lambda_X) \log n}) distortion embedding, where \lambda_X is the doubling constant of X. Since \lambda_X\le 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 \leq k \leq 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_\infty^{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).