On the Impossibility of Dimension Reduction for Doubling Subsets of ${\ell }_p$, $p>2$

Abstract

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 {\em doubling constant} of the pointset, and not on its cardinality. In this paper, we negate this possibility for ${\ell }_p$ spaces with $p>2$. In particular, we introduce an $n$-point subset of ${\ell }_p$ with doubling constant O(1), and demonstrate that any embedding of the set into ${\ell }_p^d$ with distortion $D$ must have $D\ge \Omega \left({\left(\frac{c\log n}{d}\right)}^{\frac{1}{2}-\frac{1}{p}}\right)$.