Analysis on local optimum existence form of K-means-type

With the hypothesis of Gaussian distribution of patterns, K-means and its extensions are good for clustering. As the representative of partitional clustering algorithm, K-means follows rules for running: numbers of clusters to be set, cluster initialization to be specified and certain objective function to be optimized. In general, FCM, ANN, EM share the identical idea with K-means in the beginning of running, and local optimum is the basic perspective of these K-means-type clustering methods. How numbers of clusters and cluster initialization affect local optimum existence is the query of this paper, the analysis will be given. In this paper, K-means-type algorithms are summarized, convergence proof will be shown, local optimum existence form is analyzed in detail and the classical probability expression of the existence is presented.