Clustering subgaussian mixtures with k-means

We introduce a model-free, parameter-free relaxand-round algorithm for k-means clustering, based on a semidefinite programming relaxation (SDP) due to Peng and Wei [1]. The algorithm interprets the SDP output as a denoised version of the original data and then rounds this output to a hard clustering. We analyze the performance of this algorithm in the setting where the data is drawn from a subgaussian mixture model. We also study the fundamental limits of estimating subgaussian centers with k-means clustering in order to compare our approximation guarantee to the theoretically optimal k-means clustering solution. In particular, our guarantee has no dependence on the number of points, and for equidistant clusters with O (k) separation, our guarantee is optimal up to a factor of k.