Ultra-low-dimensional embeddings for doubling metrics

Abstract

We consider the problem of embedding a metric into low-dimensional Euclidean space. The classical theorems of Bourgain, and of Johnson and Lindenstrauss say that any metric onnpoints embeds into anO(logn)-dimensional Euclidean space withO(logn) distortion. Moreover, a simple “volume” argument shows that this bound is nearly tight: a uniform metric onnpoints requires nearly logarithmic number of dimensions to embed with logarithmic distortion. It is natural to ask whether such a volume restriction is the only hurdle to low-dimensional embeddings. In other words, dodoublingmetrics, that do not have large uniform submetrics, and thus no volume hurdles to low dimensional embeddings, embed in low dimensional Euclidean spaces with small distortion?In this article, we give a positive answer to this question. We show how to embed any doubling metrics intoO(log logn) dimensions withO(logn) distortion. This is the first embedding for doubling metrics into fewer than logarithmic number of dimensions, even allowing for logarithmic distortion.This result is one extreme point of our general trade-off between distortion and dimension: given ann-point metric(V,d)with doubling dimension dimD, and any target dimensionTin the range Ω(dimDlog logn) ≤T≤O(logn), we show that the metric embeds into Euclidean space RTwithO(logn√ dimD/T) distortion.