Linear separation theorem in distributional clustering

Distributional clustering is a method to find the clustering of probability distributions which minimizes a conditional entropy. This method is considered to be a formulation of “summarizing world to know target information”. We show a theorem which shows an important geometrical property of this clustering. This theorem represents a linear separation property between any two clusters. By this theorem, we can reduce the number of evaluations of the conditional entropy to find the clustering which gives the minimum conditional entropy