AI 中文总结
本文提出RADA-DC，一种结合对偶正则化与DC线性化的黎曼交替下降上升方法，用于求解基数约束的K-means聚类问题，在聚类数较大时优于K-means++等基线。
AI 中文摘要
K-means是信号处理和机器学习中广泛采用的聚类方法。本文通过紧致嵌入子流形上的基数约束形式研究K-means聚类。我们用差分凸（DC）惩罚替代基数约束，并建立全局误差界，证明当惩罚参数超过有限阈值时，惩罚形式和约束形式具有相同的全局最小化器。为求解由此产生的非光滑黎曼DC问题，我们将其重构为极小极大问题，并提出RADA-DC，一种结合对偶正则化与DC线性化的黎曼交替下降上升方法。在标准假设和适当参数选择下，RADA-DC在O(ε^{-3})次迭代内找到ε-黎曼临界点。我们在合成和真实数据集上进行实验，证明当聚类数量较大时，所提方法在解质量上优于包括K-means++在内的测试基线，且计算成本具有竞争力。
英文摘要
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 $ε$-Riemannian critical point within $O(ε^{-3})$ 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.
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