Exact Recovery of Mangled Clusters with Same-Cluster Queries

We study the cluster recovery problem in the semi-supervised active clustering framework. Given a finite set of input points, and an oracle revealing whether any two points lie in the same cluster, our goal is to recover all clusters exactly using as few queries as possible. To this end, we relax the spherical k -means cluster assumption of Ashtiani et al.\ to allow for arbitrary ellipsoidal clusters with margin. This removes the assumption that the clustering is center-based (i.e., defined through an optimization problem), and includes all those cases where spherical clusters are individually transformed by any combination of rotations, axis scalings, and point deletions. We show that, even in this much more general setting, it is still possible to recover the latent clustering exactly using a number of queries that scales only logarithmically with the number of input points.