A new perspective on centroid algorithms for unsupervised discrete clustering in Riemannian manifolds: an approach via optimization

Abstract

Clustering is an unsupervised learning technique used to discover hidden structures in data. By identifying patterns and similarities among objects, this methodology enables the organization of datasets into homogeneous groups. This work presents a new perspective on a specific class of partition-based algorithms for unsupervised discrete clustering in complete finite-dimensional Riemannian manifolds by introducing statistical shape analyses to classify clusters and improve label selection. To achieve this aim, we extend the statistical concepts of skewness and kurtosis to Riemannian settings, based on vector operations in the tangent plane at each point of the manifold. Assumptions regarding the injectivity radius and the boundedness of the sectional curvature are initially adopted to enable local convex analysis. Nevertheless, the well-posedness of Riemannian weighted centroids is ensured by analyzing the coercivity and quasiconvexity properties of the distance function’s powers. In addition, we demonstrate that well-posedness and continuity extend to any positive power of the distance function, rather than being limited to powers greater than or equal to 1. Computational experiments involving diffusion tensor imaging and hyperspectral image segmentation are performed. We also present a statistical analysis to demonstrate the practical applicability and computational performance of our technique compared with leading clustering approaches. Ultimately, our methodology is applicable to any field where data modeling resides in a complete finite-dimensional Riemannian manifold.