Sketch-and-solve approaches to k-means clustering by semidefinite programming

We study a sketch-and-solve approach to speed up the Peng–Wei semidefinite relaxation of k-means clustering. When the data are appropriately separated we identify the k-means optimal clustering. Otherwise, our approach provides a high-confidence lower bound on the optimal k-means value. This lower bound is data-driven; it does not make any assumption on the data nor how they are generated. We provide code and an extensive set of numerical experiments where we use this approach to certify approximate optimality of clustering solutions obtained by k-means++.