On the Impossibility of Dimension Reduction for Doubling Subsets

A major open problem in the field of metric embedding is the existence of dimension reduction for n-point subsets of Euclidean space, such that both distortion and dimension depend only on the doubling constant of the pointset, and not on its cardinality. In this paper, we negate this possibility for `p spaces with p > 2. In particular, we introduce an n-point subset of `p with doubling constant O(1), and demonstrate that any embedding of the set into `p with distortionD must haveD ≥ Ω (( c logn d ) 1 2− 1 p ) .