Nonsmooth Penalized Clustering via $\ell _{p}$ Regularized Sparse Regression.

Area of Science:

• Computational statistics and machine learning.
• Data mining and pattern recognition.

Background:

• Traditional clustering algorithms often require pre-specifying the number of clusters.
• Recent advancements enable automatic determination of cluster numbers from data.
• Existing methods face challenges in handling nonsmooth and nonconvex optimization problems.

Purpose of the Study:

• To propose a novel nonsmooth penalized clustering model using L_p-norm regularized sparse regression.
• To automatically learn the number of clusters from training data.
• To guarantee the sparseness of cluster centers and enhance practical applicability.

Main Methods:

• Formulation of a nonsmooth, nonconvex optimization problem based on over-parameterization.
• Utilizing L_p-norm regularization to balance model fit and cluster count.
• Development of a smoothing trust region algorithm to address optimization challenges.

Main Results:

• Theoretical proof of guaranteed sparseness for cluster centers.
• Demonstration of an easy-to-compute criterion and cross-validation interval reduction strategy.
• Validation of theoretical findings and method advantages through numerical studies on diverse datasets.

Conclusions:

• The proposed penalized clustering model effectively determines the number of clusters automatically.
• The method offers theoretical guarantees on cluster center sparseness and practical computational efficiency.
• Numerical results confirm the model's superiority over existing approaches for various data types.