Nonsmooth Penalized Clustering via $\ell _{p}$ Regularized Sparse Regression
提出一种非光滑惩罚聚类模型，通过ℓ_p正则化稀疏回归自动学习聚类数量，并设计平滑信赖域算法求解，在模拟和实际数据上验证了有效性。
Clustering has been widely used in data analysis. A majority of existing clustering approaches assume that the number of clusters is given in advance. Recently, a novel clustering framework is proposed which can automatically learn the number of clusters from training data. Based on these works, we propose a nonsmooth penalized clustering model via ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> (0 <; p <; 1) regularized sparse regression. In particular, this model is formulated as a nonsmooth nonconvex optimization, which is based on over-parameterization and utilizes an ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> -norm-based regularization to control the tradeoff between the model fit and the number of clusters. We theoretically prove that the new model can guarantee the sparseness of cluster centers. To increase its practicality for practical use, we adhere to an easy-to-compute criterion and follow a strategy to narrow down the search interval of cross validation. To address the nonsmoothness and nonconvexness of the cost function, we propose a simple smoothing trust region algorithm and present its convergent and computational complexity analysis. Numerical studies on both simulated and practical data sets provide support to our theoretical results and demonstrate the advantages of our new method.