Embedding metric spaces in their intrinsic dimension

Abstract

A fundamental question of metric embedding is whether the metric dimension of a metric space is related to its intrinsic dimension. That is whether the dimension in which it can be embedded in some real normed space is implied by the intrinsic dimension which is reflected by the inherent geometry of the space. The existence of such an embedding was conjectured by Assouad and was later posed as an open problem by others. This question is tightly related to a major goal of many practical application fields: developing tools to represent intrinsically low dimensional metric data sets in a succinct manner. In this paper we give the first algorithmic technique with formal guarantees for finding faithful and low dimensional representations of data lying in high dimensional space. Our main theorem states that every finite metric space X embeds into Euclidean space with dimension O(dim(X)/ϵ) and distortion O(log^(1+ϵ) n), where dim(X) is the doubling dimension of the space X. Moreover, we show that X can be embedded into dimension Õ(dim(X)) with constant average distortion and ℓ_q-distortion for any q < ∞. Our technique also provides a dimension-distortion tradeoff and an extension of Assouad’s theorem, providing distance oracles that improve known construction when dim(X) = o(log |X|).