Convexity-based clustering criteria: theory, algorithms, and applications in statistics

Abstract

This paper deals with the construction of optimum partitions B = (B 1 , ..., B m ) of I R p for a clustering criterion which is based on a convex function of the class centroids E[X|X ∈ B i ] as a generalization of the classical SSQ clustering criterion for n data points. We formulate a dual optimality problem involving two sets of variables and derive a maximum-support-plane (MSP) algorithm for constructing a (sub-)optimum partition as a generalized k-means algorithm. We present various modifications of the basic criterion and describe the corresponding MSP algorithm. It is shown that the method can also be used for solving optimality problems in classical statistics (maximizing Csiszár's φ-divergence) and for simultaneous classification of the rows and columns of a contingency table.

Two-mode clustering methods: astructuredoverview

“…Govaert 38,40 proposed an iterative dynamic clustering procedure to maximize Equation (6). From his part, Bock [50][51][52] suggested to solve the problem by means of a maximum-tangent-plane algorithm. As a matter of fact, Bock has shown that loss function ( 6) is a member of a broad family of loss functions that measure the discrepancy between the actual aggregated frequency distribution obtained from X and the corresponding frequency distribution in the case of independent row and column clusters.…”