Approximating sparsest cut in low rank graphs via embeddings from approximately low dimensional spaces

We consider the problem of embedding a finite set of points {x1, . . . , xn} 2 Rd that satisfy 22 triangle inequalities into 1, when the points are approximately low-dimensional. Goemans (unpublished, appears in [20]) showed that such points residing in exactly d dimensions can be embedded into 1 with distortion at most p d. We prove the following robust analogue of this statement: if there exists a r-dimensional subspacesuch that the projections onto this subspace satisfy P i,j2[n] kxi - xjk 22 (1) P i,j2[n] kxi - xjk 22 , then there is an embedding of the points into 1 with O(p r) average distortion. A consequence of this result is that the integrality gap of the well-known Goemans-Linial SDP relaxation for the Uniform Sparsest Cut problem is O(p r) on graphs G whose r-th smallest normalized eigenvalue of the Laplacian satisfies r(G)/n(1)SDP (G). Our result improves upon the previously known bound of O(r) on the average distortion, and the integrality gap of the Goemans-Linial SDP under the same preconditions, proven in [7, 6].