Hierarchical 3-D von Mises-Fisher Mixture Model Md. Abul Hasnat MDABUL.HASNAT@UNIV-ST-ETIENNE.FR Université Jean Monnet, Saint Etienne, France. Olivier Alata OLIVIER.ALATA@UNIV-ST-ETIENNE.FR Université Jean Monnet, Saint Etienne, France. Alain Trémeau ALAIN.TREMEAU@UNIV-ST-ETIENNE.FR Université Jean Monnet, Saint Etienne, France. Abstract In this paper, we propose a complete method for clustering data, which are in the form of unit vectors. The solution consists of a distribution based clustering algorithm with the assumption of a generative model. In the model, the data is generated from a finite statistical mixture model based on the von Mises-Fisher (vMF) distribution. Initially, Bregman soft clustering algorithm is applied to obtain the parameters of the vMF mixture model (vMF-MM) for certain maximum number of components. Then, a hierarchy of mixture models is generated from the parameters. The hierarchy is generated by appropriately using Bregman divergence to compute dissimilarity among distributions as well as fuse/merge the centroids of the clusters. After constructing the hierarchy, Kullback Leibler divergence (KLD) is used to compute the distance between statistical mixture models with different number of components. Finally, a threshold (KLD value) is used to select number of components of the mixture model. The proposed method is called Hierarchical 3-D von Mises-Fisher mixture model. We validated the method by applying it on simulated data. Additionally, we applied the proposed method to cluster image normal, which are computed from the depth image. As an outcome of the clustering, we obtained a bottom-up segmentation of the depth image. Obtained results confirmed our assumption that the proposed method can be a potential tool to analyze depth images. Proceedings of the 1st Workshop on Divergences and Divergence Learning (at ICML 2013), Atlanta, Georgia, USA, 2013. Copyright 2013 by the author(s). 1. Introduction Data/features in the form of a unit vector exhibits directional behavior. For this type of features, directional distributions (Mardia & Jupp 2000) are the standard choice to construct a statistical mixture model (Murphy 2012). Such data frequently appear in varieties of domains in order to analyze image, speech signals, text documents, gene expressions (Banerjee et al. 2005), treatment beam (Bangert et al. 2010) etc. The sample space for directional distributions is circle (  ), sphere (  ) and hypersphere (  ). Most prominent distributions in directional statistics (von Mises-Fisher, Kent, Watson, Bingham etc.) belong to exponential family of distributions (Mardia & Jupp 2000). Directional distributions are associated with complicated normalizing constants. For this reason, analytical solution to obtain maximum likelihood estimate (MLE) of the parameters even for a single distribution is difficult (Sra 2012). The minimal set of parameters in a directional distribution is the mean and concentration. Satisfactory approximation is available (Mardia & Jupp 2000) for lower dimensional data (). However, for higher dimensional data, estimation of the concentration parameters is non-trivial since it involves functional inversion of ratios of special function such as Bessel functions (Banerjee et al. 2005). The fundamental directional distribution is the von Mises- Fisher (vMF) distribution, which is also called Fisher distribution for   () (Mardia & Jupp 2000). It models data concentrated around a mean-direction. Heuristic approximation of the parameters for higher dimensional data distributed according to mixture of vMF distribution is obtained (Banerjee et al. 2005). In data clustering problem, statistical mixture model is one of the most prominent and widely used tools. It consists of a base probability distribution for the observed data and prior probability for the clusters (Murphy 2012) (that generates the data samples). Therefore, a mixture