On the Impossibility of Dimension Reduction for Doubling Subsets of $\ell_p$, $p>2$

TL;DR

This paper proves that for $ ext{ell}_p$ spaces with $p>2$, it is impossible to reduce dimensions of doubling subsets without incurring significant distortion, challenging the existence of such embeddings.

Contribution

It introduces a specific $n$-point subset of $ ext{ell}_p$ with bounded doubling constant and proves lower bounds on embedding distortion relative to dimension.

Findings

No dimension reduction with bounded distortion for $ ext{ell}_p$ with $p>2$

Constructs a subset with constant doubling constant

Establishes lower bounds on embedding distortion

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 $D\ge\Omega\left(\left(\frac{c\log n}{d}\right)^{\frac{1}{2}-\frac{1}{p}}\right)$.