Graphical models have exhibited their performance in numerous tasks ranging from biological analysis to recommender systems. However, graphical models with hub nodes are computationally difficult to fit, particularly when the dimension of the data is large. To efficiently estimate the hub graphical models, we introduce a two-phase algorithm. The proposed algorithm first generates a good initial point via a dual alternating direction method of multipliers (ADMM), and then warm starts a semismooth Newton (SSN) based augmented Lagrangian method (ALM) to compute a solution that is accurate enough for practical tasks. The sparsity structure of the generalized Jacobian ensures that the algorithm can obtain a nice solution very efficiently. Comprehensive experiments on both synthetic data and real data show that it obviously outperforms the existing state-of-the-art algorithms. In particular, in some high dimensional tasks, it can save more than 70\% of the execution time, meanwhile still achieves a high-quality estimation.

翻译：图模型在从生物分析到推荐系统的众多任务中展现了其优越性能。然而，包含枢纽节点的图模型在拟合时计算困难，尤其当数据维度很高时。为高效估计枢纽图模型，我们提出一种两阶段算法。该算法首先通过交替方向乘子对偶法（ADMM）生成优质初始点，随后利用基于半光滑牛顿（SSN）的增广拉格朗日方法（ALM）进行热启动，求解出满足实际任务精度需求的解。广义雅可比矩阵的稀疏结构保证了算法能以极高效率获得优质解。在合成数据和真实数据上的综合实验表明，该算法明显优于现有最先进方法。特别是在某些高维任务中，该算法可节省70%以上的执行时间，同时仍能实现高质量的估计。