Approximating k-means-type clustering via semidefinite programming

One of the fundamental clustering problems is to assign n points into k clusters based on minimal sum-of-squared distances (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) problem. 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 linear programming and SDP relaxations. Second, we consider the issue of how to extract a feasible solution of the original 0-1 SDP model from the optimal 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 done in O(n(k2 - 2k + 2)) time. In case of biclustering, the running time of our rounding procedure can be reduced to O(n log n). We show that our algorithm provides a 2-approximate solution to the original problem. Promising numerical results for biclustering based on our new method are reported.