This paper introduces a regularized projection matrix approximation framework designed to recover cluster information from the affinity matrix. The model is formulated as a projection approximation problem, incorporating an entry-wise penalty function. We investigate three distinct penalty functions, each specifically tailored to address bounded, positive, and sparse scenarios. To solve this problem, we propose direct optimization on the Stiefel manifold, utilizing the Cayley transformation along with the Alternating Direction Method of Multipliers (ADMM) algorithm. Additionally, we provide a theoretical analysis that establishes the convergence properties of ADMM, demonstrating that the convergence point satisfies the KKT conditions of the original problem. Numerical experiments conducted on both synthetic and real-world datasets reveal that our regularized projection matrix approximation approach significantly outperforms state-of-the-art methods in clustering performance.

翻译：本文提出了一种正则化投影矩阵逼近框架，旨在从亲和矩阵中恢复聚类信息。该模型被构建为投影逼近问题，并引入了逐项惩罚函数。我们研究了三种不同的惩罚函数，每种函数分别针对有界、正定和稀疏场景进行专门设计。为解决此问题，我们提出在Stiefel流形上直接进行优化，利用Cayley变换结合交替方向乘子法（ADMM）算法。此外，我们提供了理论分析以确立ADMM的收敛特性，证明其收敛点满足原问题的KKT条件。在合成数据集和真实数据集上进行的数值实验表明，我们的正则化投影矩阵逼近方法在聚类性能上显著优于现有最先进方法。