A lower bound on dimension reduction for trees in \ell_1

TL;DR

This paper establishes a lower bound on the dimension needed for low-distortion embeddings of star metrics into , showing that high-dimensional space is necessary for accurate low-distortion representations.

Contribution

It provides a new lower bound on the dimension required for embedding star metrics into  with low distortion, advancing understanding of dimension reduction limits.

Findings

Lower bound on embedding dimension scales with  and 

Embedding star metrics into  with low distortion requires high dimension

Quantifies limitations of dimension reduction for specific metric spaces.

Abstract

There is a constant c > 0 such that for every $\epsilon \in (0,1)$ and $n \geq 1/\epsilon^2$, the following holds. Any mapping from the $n$-point star metric into $\ell_1^d$ with bi-Lipschitz distortion $1+\epsilon$ requires dimension $$d \geq {c\log n\over \epsilon^2\log (1/\epsilon)}.$$