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This paper advances K-means clustering by introducing a Riemannian difference-of-convex optimization framework that ensures global optimality under cardinality constraints. The proposed RADA-DC algorithm achieves competitive computational costs while improving solution quality on large-scale datasets compared to standard baselines like K-means++. This work is significant for researchers in machine learning seeking robust, theoretically grounded methods for complex manifold-based data structures.
Riemannian Difference-of-Convex Optimization for K-Means Clustering
arXiv:2609.34310v1 Announce Type: new Abstract: K-means is a widely adopted clustering approach in signal processing and machine learning. In this paper, we study K-means clustering through a cardinality-constrained formulation on a compact embedded submanifold. We replace the cardinality constraint with a difference-of-convex (DC) penalty and establish a global error bound to prove that the penalized and constrained formulations share the same global minimizers whenever the penalty parameter exceeds a finite threshold. To solve the resulting nonsmooth Riemannian DC problem, we reformulate it as a minimax problem and propose RADA-DC, a Riemannian alternating descent ascent method combining dual regularization with DC linearization. Under standard assumptions and suitable parameter choices, RADA-DC finds an $\epsilon$-Riemannian critical point within $O(\epsilon^{-3})$ iterations. We conduct experiments on synthetic and real-world datasets to demonstrate that the proposed method outperforms the tested baselines, including K-means++, in solution quality at competitive computational cost when the number of clusters is large.
Abstract:K-means is a widely adopted clustering approach in signal processing and machine learning. In this paper, we study K-means clustering through a cardinality-constrained formulation on a compact embedded submanifold. We replace the cardinality constraint with a difference-of-convex (DC) penalty and establish a global error bound to prove that the penalized and constrained formulations share the same global minimizers whenever the penalty parameter exceeds a finite threshold. To solve the resulting nonsmooth Riemannian DC problem, we reformulate it as a minimax problem and propose RADA-DC, a Riemannian alternating descent ascent method combining dual regularization with DC linearization. Under standard assumptions and suitable parameter choices, RADA-DC finds an $\epsilon$-Riemannian critical point within $O(\epsilon^{-3})$ iterations. We conduct experiments on synthetic and real-world datasets to demonstrate that the proposed method outperforms the tested baselines, including K-means++, in solution quality at competitive computational cost when the number of clusters is large.
Machine Learning (cs.LG); Optimization and Control (math.OC)
arXiv:2609.34310 [cs.LG]
(or arXiv:2609.34310v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2609.34310 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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