A discussion on fuzzy c-means++ based on conjugate functions

Abstract

The method of c-means++ (k-means++), which is an improvement of c-means, uses a modified initialization procedure of centroids, which is to place initial centroids so as not to make them biased. In the initialization procedure, the centroids are sequentially placed, and each centroid is selected as follows: for each data point, the distnace of the point to the centroid that is closest among previously selected centroids is calculated, then the next centroid is selected among the data points under the probability reflecting the calculated distances, i.e., points with larger distances are more likely to be selected. In this paper, using the idea of conjugate functions, we extend c-means++ to fuzzy c-means++. Using Legendre-Fenchel transformation, we reformulate the optimization problem of a fuzzy c-means method as minimization of the sum of a function of the data points, which is a soft minimum of squared distances between a data point, as the argument, and the centroids. For evaluation of the distnace between a data point and the current set of centroids, we replace the minimum function used in c-means++ with the soft minimum function obtained as a result of the reformuation.