Goemans showed that any @math points @math in @math -dimensions satisfying @math triangle inequalities can be embedded into @math , with worst-case distortion at most @math . We extend this to the case when the points are approximately low-dimensional, albeit with average distortion guarantees. More precisely, we give an @math -to- @math embedding with average distortion at most the stable rank, @math , of the matrix @math consisting of columns @math . Average distortion embedding suffices for applications such as the Sparsest Cut problem. Our embedding gives an approximation algorithm for the 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 @math metric, an alternate proof of Goemans' theorem, and a simpler proof for average distortion @math . Furthermore, while the seminal result of Arora, Rao and Vazirani giving a @math 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.
Let (G) be the minimum conductance of an undirected graph G, and let 0= = 2, (G) = O(k) , and this performance guarantee is achieved by the spectral partitioning algorithm. This improves Cheeger's inequality, and the bound is optimal up to a constant factor for any k. Our result shows that the spectral partitioning algorithm is a constant factor approximation algorithm for finding a sparse cut if $ is a constant for some constant k. This provides some theoretical justification to its empirical performance in image segmentation and clustering problems. We extend the analysis to other graph partitioning problems, including multi-way partition, balanced separator, and maximum cut.