Approximating K-means-type clustering via semidefinite programming

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 an 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. We show that for bi-clustering, our algorithm can provide a 2-approximate solution to the original problem in $O(n\log n)$ time. Promising numerical results based on our new method are reported.



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