As a special class of non-negative matrix factorization, symmetric non-negative matrix factorization (SymNMF) has been widely used in the machine learning field to mine the hidden non-linear structure of data. Due to the non-negative constraint and non-convexity of SymNMF, the efficiency of existing methods is generally unsatisfactory. To tackle this issue, we propose a two-phase algorithm to solve the SymNMF problem efficiently. In the first phase, we drop the non-negative constraint of SymNMF and propose a new model with penalty terms, in order to control the negative component of the factor. Unlike previous methods, the factor sequence in this phase is not required to be non-negative, allowing fast unconstrained optimization algorithms, such as the conjugate gradient method, to be used. In the second phase, we revisit the SymNMF problem, taking the non-negative part of the solution in the first phase as the initial point. To achieve faster convergence, we propose an interpolation projected gradient (IPG) method for SymNMF, which is much more efficient than the classical projected gradient method. Our two-phase algorithm is easy to implement, with convergence guaranteed for both phases. Numerical experiments show that our algorithm performs better than others on synthetic data and unsupervised clustering tasks.

中文翻译：

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鲁棒对称非负矩阵分解的两阶段算法

作为一类特殊的非负矩阵分解，对称非负矩阵分解（SymNMF）已被广泛应用于机器学习领域，以挖掘数据的隐藏非线性结构。由于SymNMF的非负约束和非凸性，现有方法的效率普遍不理想。为了解决这个问题，我们提出了一种两阶段算法来有效地解决 SymNMF 问题。在第一阶段，我们放弃了 SymNMF 的非负约束，并提出了一个带有惩罚项的新模型，以控制因子的负分量。与之前的方法不同，该阶段的因子序列不需要为非负，允许使用快速无约束优化算法，例如共轭梯度法。在第二阶段，我们重新审视 SymNMF 问题，以第一阶段解的非负部分为初始点。为了实现更快的收敛，我们为 SymNMF 提出了一种插值投影梯度（IPG）方法，它比经典的投影梯度方法更有效。我们的两阶段算法很容易实现，两个阶段都保证收敛。数值实验表明，我们的算法在合成数据和无监督聚类任务上的表现优于其他算法。