On the symmetrical Kullback-Leibler Jeffreys centroids

Due to the success of the bag-of-word modeling paradigm, clustering histograms has become an important ingredient of modern information processing. Clustering histograms can be performed using the celebrated k-means centroid-based algorithm. From the viewpoint of applications, it is usually required to deal with symmetric distances. In this letter, we consider the Jeffreys divergence that symmetrizes the Kullback-Leibler divergence, and investigate the computation of Jeffreys centroids. We first prove that the Jeffreys centroid can be expressed analytically using the Lambert W function for positive histograms. We then show how to obtain a fast guaranteed approximation when dealing with frequency histograms. Finally, we conclude with some remarks on the k-means histogram clustering.