Abstract (100 words or less): |  | 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). We show that our 0-1 SDP model provides a unified framework for several clustering approaches such as normalized k-cut and spectral clustering. Moreover, the 0-1 SDP model allows us to solve the underlying problem approximately via the relaxed linear and semidefinite programming.
Secondly, we consider the issue of how to extract a feasible solution of the original MSSC model from the approximate solution of the relaxed SDP problem. By using principal component analysis, we develop a rounding procedure to construct a feasible partitioning from a solution of the relaxed problem. In our rounding procedure, we need to solve a k-means clustering problem in Rk-1, which can be solved in (n ^{k^2(k-1)/2)) time. In case of bi-clustering, the running time of our rounding procedure can be reduced to O(n log n). We show that our algorithm can provide a 2-approximate solution to the original problem. Promising numerical results based on our new method will be reported. |