Embedding approximately low-dimensional $\ell_2^2$ metrics into $\ell_1$

TL;DR

This paper extends Goemans' embedding of low-dimensional $\, ext{ extlbrackdbl}\, ext{ extlbrackdbl} ext{ extlbrackdbl}$ metrics into $\, ext{ extlbrackdbl}\, ext{ extlbrackdbl}$ with average distortion bounds related to the stable rank, enabling improved algorithms for Sparsest Cut.

Contribution

It introduces an average distortion embedding from approximately low-dimensional $\, ext{ extlbrackdbl}\, ext{ extlbrackdbl} ext{ extlbrackdbl}$ metrics into $\, ext{ extlbrackdbl}\, ext{ extlbrackdbl}$ based on stable rank, with applications to graph partitioning.

Findings

Provides an embedding with average distortion at most the stable rank of the matrix.

Improves approximation algorithms for Sparsest Cut on low threshold-rank graphs.

Offers new insights and simpler proofs related to $\, ext{ extlbrackdbl}\, ext{ extlbrackdbl} ext{ extlbrackdbl}$ metrics and Goemans' theorem.

Abstract

Goemans showed that any $n$ points $x_1, \dotsc x_n$ in $d$-dimensions satisfying $\ell_2^2$ triangle inequalities can be embedded into $\ell_{1}$, with worst-case distortion at most $\sqrt{d}$. We extend this to the case when the points are approximately low-dimensional, albeit with average distortion guarantees. More precisely, we give an $\ell_{2}^{2}$-to-$\ell_{1}$ embedding with average distortion at most the stable rank, $\mathrm{sr}(M)$, of the matrix $M$ consisting of columns $\{x_i-x_j\}_{i<j}$. Average distortion embedding suffices for applications such as the Sparsest Cut problem. Our embedding gives an approximation algorithm for the \sparsestcut problem on low threshold-rank graphs, where earlier work was inspired by Lasserre SDP hierarchy, and improves on a previous result of the first and third author [Deshpande and Venkat, In Proc. 17th APPROX, 2014]. Our ideas give a new perspective on $\ell_{2}^{2}$ metric, an alternate proof of Goemans' theorem, and a simpler proof for average distortion $\sqrt{d}$. Furthermore, while the seminal result of Arora, Rao and Vazirani giving a $O(\sqrt{\log n})$ guarantee for Uniform Sparsest Cut can be seen to imply Goemans' theorem with average distortion, our work opens up the possibility of proving such a result directly via a Goemans'-like theorem.