  The clustering of data is one of the most important and challenging topics in
data science. The minimum sum-of-squares clustering (MSSC) problem asks to
cluster the data points into $k$ clusters such that the sum of squared
distances between the data points and their cluster centers (centroids) is
minimized. This problem is NP-hard, but there exist exact solvers that can
solve such problem to optimality for small or medium size instances.
  In this paper, we use a branch-and-bound solver based on semidefinite
programming relaxations called SOS-SDP to compute the optimum solutions of the
MSSC problem for various $k$ and for multiple datasets, with real and
artificial data, for which the data provider has provided ground truth
clustering.
  Next, we use several extrinsic and intrinsic measures to evaluate how the
optimum clustering and ground truth clustering matches, and how well these
clusterings perform with respect to the criteria underlying the intrinsic
measures. Our calculations show that the ground truth clusterings are generally
far from the optimum solution to the MSSC problem. Moreover, the intrinsic
measures evaluated on the ground truth clusterings are generally significantly
worse compared to the optimum clusterings. However, when the ground truth
clustering is in the form of convex sets, e.g., ellipsoids, that are well
separated from each other, the ground truth clustering comes very close to the
optimum clustering.
