  Given a set of data, one central goal is to group them into clusters based on
some notion of similarity between the individual objects. One of the most
popular and widely-used approaches is k-means despite the computational
hardness to find its global minimum. We study and compare the properties of
different convex relaxations by relating them to corresponding proximity
conditions, an idea originally introduced by Kumar and Kannan. Using conic
duality theory, we present an improved proximity condition under which the
Peng-Wei relaxation of k-means recovers the underlying clusters exactly. Our
proximity condition improves upon Kumar and Kannan, and is comparable to that
of Awashti and Sheffet where proximity conditions are established for
projective k-means. In addition, we provide a necessary proximity condition for
the exactness of the Peng-Wei relaxation. For the special case of equal cluster
sizes, we establish a different and completely localized proximity condition
under which the Amini-Levina relaxation yields exact clustering, thereby having
addressed an open problem by Awasthi and Sheffet in the balanced case. Our
framework is not only deterministic and model-free but also comes with a clear
geometric meaning which allows for further analysis and generalization.
Moreover, it can be conveniently applied to analyzing various data generative
models such as the stochastic ball models and Gaussian mixture models. With
this method, we improve the current minimum separation bound for the stochastic
ball models and achieve the state-of-the-art results of learning Gaussian
mixture models.
