Intrinsic K-means clustering over homogeneous manifolds

Abstract

The original K-means algorithm is widely applied for clustering in Euclidean spaces. Nevertheless, due to the non-flat characteristics of the Riemannian manifold, standard Euclidean K-means algorithms yield inferior results on such data. To address this issue, this paper presents an intrinsic K-means clustering algorithm on homogeneous manifolds based on the geodesic distance. It allows the development of K-means-based methods for frequently occurring non-vector spaces in robotics, such as directional vector modelling \(\mathbb {S}^2\) and pose estimation \(\mathbb {S}^3\). First, the Riemannian metric of the homogeneous manifold is delivered; on this basis, the intrinsic K-means is proposed using Karcher mean, and its convergence is proved. Then, differences between the proposed algorithm and four projection-based algorithms, such as embedding projection, stereographic projection, central projection and logarithmic projection, are discussed by investigating their distance preservation on manifolds. Finally, to evaluate the effectiveness of the proposed algorithm, it is compared with the projection-based algorithms on \(\mathbb {S}^n\). The results show that the intrinsic K-means achieves better clustering results, where the clustering accuracy of the proposed method is improved by 47% and 27% on average on artificial \(\mathbb {S}^2\) and \(\mathbb {S}^3\) datasets, respectively. Meanwhile, the noise immunity of the proposed algorithm becomes more evident with the noise ratio increase.