Large-Scale Clustering Based on Data Compression

This paper considers the clustering problem for large data sets. We propose an approach based on distributed optimization. The clustering problem is formulated as an optimization problem of maximizing the classification gain. We show that the optimization problem can be reformulated and decomposed into small-scale sub optimization problems by using the Dantzig-Wolfe decomposition method. Generally speaking, the Dantzig-Wolfe method can only be used for convex optimization problems, where the duality gaps are zero. Although, the considered optimization problem in this paper is non-convex, we prove that the duality gap goes to zero, as the problem size goes to infinity. Therefore, the Dantzig-Wolfe method can be applied here. In the proposed approach, the clustering problem is iteratively solved by a group of computers coordinated by one center processor, where each computer solves one independent small-scale sub optimization problem during each iteration, and only a small amount of data communication is needed between the computers and center processor. Numerical results show that the proposed approach is effective and efficient.