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

Goemans showed that any n points x1, . . . xn in d-dimensions satisfying l 2 2 triangle inequalities can be embedded into l1, with worst-case distortion at most √ d. We extend this to the case when the points are approximately low-dimensional, albeit with average distortion guarantees. More precisely, we give an l2-to-l1 embedding with average distortion at most the stable rank, sr (M), of the matrix M consisting of columns {xi − xj}i<j. Average distortion embedding suffices for applications such as the SPARSEST CUT problem. Our embedding gives an approximation algorithm for the SPARSEST CUT 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 l2 metric, an alternate proof of Goemans’ theorem, and a simpler proof for average distortion √ d. Furthermore, while the seminal result of Arora, Rao and Vazirani giving a O( √ 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. ∗Microsoft Research India, [email protected] †Tata Institute of Fundamental Research, [email protected] ‡Tata Institute of Fundamental Research, [email protected] 1