Clustering has been one of the most basic and essential problems in unsupervised learning due to various applications in many critical fields. The recently proposed sum-of-nums (SON) model by Pelckmans et al. (2005), Lindsten et al. (2011) and Hocking et al. (2011) has received a lot of attention. The advantage of the SON model is the theoretical guarantee in terms of perfect recovery, established by Sun et al. (2018). It also provides great opportunities for designing efficient algorithms for solving the SON model. The semismooth Newton based augmented Lagrangian method by Sun et al. (2018) has demonstrated its superior performance over the alternating direction method of multipliers (ADMM) and the alternating minimization algorithm (AMA). In this paper, we propose a Euclidean distance matrix model based on the SON model. An efficient majorization penalty algorithm is proposed to solve the resulting model. Extensive numerical experiments are conducted to demonstrate the efficiency of the proposed model and the majorization penalty algorithm.

翻译：聚类作为无监督学习中最基础且重要的问题之一，已在众多关键领域的各类应用中展现出核心价值。由Pelckmans等（2005）、Lindsten等（2011）与Hocking等（2011）提出的求和范数（SON）模型近年来备受关注。该模型的核心优势在于具有理论可恢复性保证——Sun等（2018）已建立完美的理论框架，同时为设计高效求解算法提供了广阔空间。Sun等（2018）提出的半光滑牛顿增广拉格朗日方法在处理该模型时，展现出优于交替方向乘子法（ADMM）和交替最小化算法（AMA）的性能。本文基于SON模型提出了一种新的欧氏距离矩阵模型，并设计了高效的主化惩罚算法进行求解。通过大量数值实验验证了所提模型与主化惩罚算法的有效性。