In this paper, we propose a new kernel function that makes use of Riemannian geodesic distance s among data points, and present a Geometric median shift algorithm over Riemannian Manifolds. Relying on the geometric median shift, together with geodesic distances, our approach is able to effectively cluster data points distributed on Riemannian manifolds. In addition to improving the clustering results, Using both Riemannian Manifolds and Euclidean spaces, We compare the geometric median shift and mean shift algorithms on synthetic and real data sets for the tasks of clustering.
"On the contrary, a median is more robust since the median of a set is a point in the set. Therefore, a large body of algorithms in the literature utilize the median to perform the process of mode seeking       . For example, Shapira et al.  show how median updating manages to find a natural classification for high-dimensional data set in Euclidean space. "
[Show abstract][Hide abstract] ABSTRACT: The mean shift algorithms have been successfully applied to many areas, such as data clustering, feature analysis, and image segmentation. However, they still have two limitations. One is that they are ineffective in clustering data with low dimensional manifolds because of the use of the Euclidean distance for calculating distances. The other is that they sometimes produce poor results for data clustering and image segmentation. This is because a mean may not be a point in a data set. In order to overcome the two limitations, we propose a novel approach for the median shift over Riemannian manifolds that uses the geometric median and geodesic distances. Unlike the mean, the geometric median of a data set is one of points in the set. Compared to the Euclidean distance, the geodesic distances can better describe data points distributed on Riemannian manifolds. Based on these two facts, we first present a novel density function that characterizes points on a manifold with the geodesic distance. The shift of the geometric median over the Riemannian manifold is derived from maximizing this density function. After this, we present an algorithm for geometric median shift over Riemannian manifolds, together with theoretical proofs of its correctness. Extensive experiments have demonstrated that our method outperforms the state-of-the-art algorithms in data clustering, image segmentation, and noise filtering on both synthetic data sets and real image databases.
Information Sciences 01/2013; 220:292-305. DOI:10.1016/j.ins.2012.07.009 · 4.04 Impact Factor
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