Statistical initialization of intrinsic K-means clustering on homogeneous manifolds

• Published in: Applied intelligence (Dordrecht, Netherlands) Vol. 53; no. 5; pp. 4959 - 4978
• Main Authors: Tan, Chao, Zhao, Huan, Ding, Han
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.03.2023; Springer Nature B.V
• Subjects: Accuracy; Algorithms; Artificial Intelligence; Cluster analysis; Clustering; Computer Science; Differential geometry; Euclidean space; Learning; Machines; Manifolds (mathematics); Manufacturing; Mechanical Engineering; Orthogonality; Processes; Riemann manifold; Topological manifolds; Vector quantization; means initialization; Dimensionality reduction; means; Homogeneous manifolds
• ISSN: 0924-669X; 1573-7497
• DOI: 10.1007/s10489-022-03698-8

The K -means algorithm is widely applied for clustering, and its clustering effect is influenced by its initialization. However, most existing works focus on the initialization of K and centers in Euclidean spaces, but few works in the literature deal with the initialization of K -means clustering on Riemannian manifolds. In this paper, we propose a unified scheme for learning K and selecting the initial centers for intrinsic K -means clustering on homogeneous manifolds, which can also be generalized to other types of manifolds. First, geodesic verticality is presented based on the geometric properties abstracted from the definition of orthogonality in Euclidean spaces. Then, geodesic projection on Riemannian manifolds is proposed for learning K , which achieves nonlinear dimensionality reduction and improves the computing efficiency. Additionally, the Riemannian metric of S n is derived for the statistical initialization of the centers to improve the clustering accuracy. Finally, the intrinsic K -means algorithm for clustering on homogeneous manifolds based on the Karcher mean is given by applying the proposed manifold initialization, which improves the clustering effect. Simulations and experimental studies are conducted to show the effectiveness and accuracy of the proposed K -means scheme on manifolds.