Dimension reduction for finite trees in L_1

TL;DR

This paper demonstrates that finite trees can be embedded into low-dimensional L_1 spaces with minimal distortion, nearly matching theoretical lower bounds, and provides improved bounds for specific tree structures.

Contribution

It introduces a new embedding technique for finite trees into low-dimensional L_1 spaces with near-optimal distortion bounds, resolving longstanding open questions.

Findings

Every n-point tree metric admits a (1+eps)-embedding into C(eps) log n-dimensional L_1 space.

For complete d-ary trees, the embedding achieves C(eps) = O(1/eps^2).

The results match volume lower bounds up to factors depending only on eps.

Abstract

We show that every n-point tree metric admits a (1+eps)-embedding into a C(eps) log n-dimensional L_1 space, for every eps > 0, where C(eps) = O((1/eps)^4 log(1/eps)). This matches the natural volume lower bound up to a factor depending only on eps. Previously, it was unknown whether even complete binary trees on n nodes could be embedded in O(log n) dimensions with O(1) distortion. For complete d-ary trees, our construction achieves C(eps) = O(1/eps^2).