Ultra-lowdimensional embeddings for doubling metrics

We consider the problem of embedding a metric into low-dimensional Euclidean space. The classical theo-rems of Bourgain, and of Johnson and Lindenstrauss say that any metric on n points embeds into an O(log n)-dimensional Euclidean space with O(log n) distortion. Moreover, a simple “volume ” argument shows that this bound is nearly tight: a uniform metric on n points re-quires nearly logarithmic number of dimensions to em-bed with logarithmic distortion. It is natural to ask whether such a volume restriction is the only hurdle to low-dimensional embeddings. In other words, do dou-blingmetrics, that do not have large uniform submetrics, and thus no volume hurdles to low dimensional embed-dings, embed in low dimensional Euclidean spaces with small distortion? In this paper, we give a positive answer to this question. We show how to embed any doubling metrics into O(log log n) dimensions with o(log n) 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 an n-point metric (V, d) with doubling dimension dimD, and any target dimension T in the range Ω(dimD log log n) ≤ T ≤ O(log n), we show that the metric embeds into Euclidean space RT with O(log n dimD /T) distortion