Dimension Reduction in $L_p$, $0<p<2$

TL;DR

This paper demonstrates that for all 0<p<2, any set of k points in L_p space can be embedded into finite-dimensional l_p space with arbitrarily small distortion, with dimension linear in k and polynomial in 1/ε.

Contribution

It extends previous results by showing that for all 0<p<2, embeddings with low distortion are possible into finite-dimensional l_p spaces with dimension linear in the number of points.

Findings

Embedding dimension is linear in the number of points k.

Distortion can be made arbitrarily close to 1.

Applicable for all 0<p<2, generalizing prior work for p=1.

Abstract

Complementing a recent observation of Newman and Rabinovich for $p=1$ we observe here that for all $0<p<2$ any $k$ points in $L_p$ embeds with distortion $(1+\e)$ into $\ell_p^n$ where $n$ is linear in $k$ (and polynomial in $\e^{-1}$).