Embedding the diamond graph in $L_p$ and dimension reduction in $L_1$

TL;DR

This paper establishes lower bounds on embedding the diamond graph into $L_p$ spaces and demonstrates limitations of dimension reduction in $L_1$, resolving a long-standing open problem.

Contribution

It provides new lower bounds for embeddings of diamond graphs into $L_p$ spaces and proves the impossibility of Johnson-Lindenstrauss type dimension reduction in $L_1$.

Findings

Embedding of diamond graphs into $L_p$ requires high distortion.

Any $L_1$ embedding of large point sets into low dimensions incurs significant distortion.

No Johnson-Lindenstrauss style dimension reduction exists in $L_1$.

Abstract

We show that any embedding of the level-k diamond graph of Newman and Rabinovich into $L_p$, $1 < p \le 2$, requires distortion at least $\sqrt{k(p-1) + 1}$. An immediate consequence is that there exist arbitrarily large n-point sets $X \subseteq L_1$ such that any D-embedding of X into $\ell_1^d$ requires $d \geq n^{\Omega(1/D^2)}$. This gives a simple proof of the recent result of Brinkman and Charikar which settles the long standing question of whether there is an $L_1$ analogue of the Johnson-Lindenstrauss dimension reduction lemma.