Embedding approximately low-dimensional ℓ22 metrics into ℓ1

Goemans showed that any n points x1, . . . xn in d-dimensions satisfying `2 triangle inequalities can be embedded into `1, with worst-case distortion at most √ d. We consider an extension of this theorem to the case when the points are approximately low-dimensional as opposed to exactly low-dimensional, and prove the following analogous theorem, albeit with average distortion guarantees: There exists an `2-to-`1 embedding with average distortion at most the stable rank, sr(M), of the matrixM 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 `2 metric, an alternate proof of Goemans’ theorem, and a simpler proof for average distortion √ d. 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems