Nonsmooth Optimization Based Algorithms in Cluster Analysis

Abstract

Cluster analysis is an important task in data mining. It deals with the problem of organization of a collection of objects into clusters based on a similarity measure. Various distance functions can be used to define the similarity measure. Cluster analysis problems with the similarity measure defined by the squared Euclidean distance, which is also known as the minimum sum-of-squares clustering, has been studied extensively over the last five decades. However, problems with the L 1 and L ∞ norms have attracted less attention. In this chapter, we consider a nonsmooth nonconvex optimization formulation of the cluster analysis problems. This formulation allows one to easily apply similarity measures defined using different distance functions. Moreover, an efficient incremental algorithm can be designed based on this formulation to solve the clustering problems. We develop incremental algorithms for solving clustering problems where the similarity measure is defined using the L 1, L 2 and L ∞ norms. We also consider different algorithms for solving nonsmooth nonconvex optimization problems in cluster analysis. The proposed algorithms are tested using several real world data sets and compared with other similar algorithms.