IEEE Trans Pattern Anal Mach Intell. 2023 May;45(5):5355-5369. doi: 10.1109/TPAMI.2022.3206465. Epub 2023 Apr 3.
In this article, we study the symmetric nonnegative matrix factorization (SNMF) which is a powerful tool in data mining for dimension reduction and clustering. The main contributions of the present work include: (i) a new descent direction for the rank-one SNMF is derived and a strategy for choosing the step size along this descent direction is established; (ii) a progressive hierarchical alternating least squares (PHALS) method for SNMF is developed, which is parameter-free and updates the variables column by column. Moreover, every column is updated by solving a rank-one SNMF subproblem; and (iii) the convergence to the Karush-Kuhn-Tucker (KKT) point set (or the stationary point set) is proved for PHALS. Several synthetical and real data sets are tested to demonstrate the effectiveness and efficiency of the proposed method. Our PHALS provides better performance in terms of the computational accuracy, the optimality gap, and the CPU time, compared with a number of state-of-the-art SNMF methods.
在本文中，我们研究了对称非负矩阵分解（SNMF），这是数据挖掘中用于降维和聚类的强大工具。本工作的主要贡献包括：（i）推导出了秩一 SNMF 的新下降方向，并建立了沿着这个下降方向选择步长的策略；（ii）提出了一种无参数的 SNMF 渐进分层交替最小二乘（PHALS）方法，该方法逐列更新变量，并且每一列通过求解秩一 SNMF 子问题进行更新；（iii）证明了 PHALS 收敛到 Karush-Kuhn-Tucker（KKT）点集（或驻留点集）。通过几个综合数据集和真实数据集进行测试，以验证所提出方法的有效性和效率。与许多先进的 SNMF 方法相比，我们的 PHALS 在计算精度、最优性差距和 CPU 时间方面提供了更好的性能。
