On the symmetrical Kullback-Leibler Jeffreys centroids

Clustering histograms became an important ingredient of modern information processing thanks to the success of the bag-of-word modeling paradigm. Histogram clustering 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. We consider the Jeffreys divergence that symmetrizes the Kullback-Leibler divergence, and investigate the computation of centroids with respect to that distance. We first prove that the Jeffreys centroid can be expressed analytically in closed form using the Lambert $W$ function for {\em positive} histograms. We then show how to obtain a fast guaranteed tight approximation when dealing with {\em frequency} histograms. Finally, we conclude with some remarks on the $k$-means histogram clustering.