A New Theoretical Framework for K-Means-Type Clustering

Abstract

One of the fundamental clustering problems is to assign n points into k clusters based on the minimal sum-of-squares(MSSC), which is known to be NP-hard. In this paper, by using matrix arguments, we first model MSSC as a so-called 0-1 semidefinite programming (SDP). The classical K-means algorithm can be interpreted as a special heuristics for the underlying 0-1 SDP. Moreover, the 0-1 SDP model can be further approximated by the relaxed and polynomially solvable linear and semidefinite programming. This opens new avenues for solving MSSC. The 0-1 SDP model can be applied not only to MSSC, but also to other scenarios of clustering as well. In particular, we show that the recently proposed normalized k-cut and spectral clustering can also be embedded into the 0-1 SDP model in various kernel spaces.