Optimal (Euclidean) Metric Compression

TL;DR

This paper establishes tight bounds for compressing all pairwise distances among n points in Euclidean space with minimal distortion, surpassing traditional dimension reduction limits and demonstrating the feasibility of more efficient metric compression schemes.

Contribution

It provides the first asymptotically tight bounds for Euclidean metric compression, improving over classical dimension reduction methods and showing such compression is possible beyond dimension reduction.

Findings

Tight bounds for Euclidean metric compression established

Compression schemes outperform classical dimension reduction

Results apply to  and general metrics

Abstract

We study the problem of representing all distances between $n$ points in $\mathbb R^d$, with arbitrarily small distortion, using as few bits as possible. We give asymptotically tight bounds for this problem, for Euclidean metrics, for $\ell_1$ (a.k.a.~Manhattan) metrics, and for general metrics. Our bounds for Euclidean metrics mark the first improvement over compression schemes based on discretizing the classical dimensionality reduction theorem of Johnson and Lindenstrauss (Contemp.~Math.~1984). Since it is known that no better dimension reduction is possible, our results establish that Euclidean metric compression is possible beyond dimension reduction.