Differential graphical models are designed to represent the difference between the conditional dependence structures of two groups, thus are of particular interest for scientific investigation. Motivated by modern applications, this manuscript considers an extended setting where each group is generated by a latent variable Gaussian graphical model. Due to the existence of latent factors, the differential network is decomposed into sparse and low-rank components, both of which are symmetric indefinite matrices. We estimate these two components simultaneously using a two-stage procedure: (i) an initialization stage, which computes a simple, consistent estimator, and (ii) a convergence stage, implemented using a projected alternating gradient descent algorithm applied to a nonconvex objective, initialized using the output of the first stage. We prove that given the initialization, the estimator converges linearly with a nontrivial, minimax optimal statistical error. Experiments on synthetic and real data illustrate that the proposed nonconvex procedure outperforms existing methods.
翻译:不同的图形模型旨在代表两个组的有条件依赖结构之间的差别,因此对于科学研究来说特别有意义。在现代应用的推动下,本手稿考虑了一个扩大的设置,其中每个组由潜伏变量高斯图形模型产生。由于存在潜在因素,差分网络被分解成稀疏和低级别的组成部分,两者都是对称的无限期矩阵。我们同时用一个两阶段程序来估计这两个组成部分:(一) 初始化阶段,该阶段计算一个简单、一致的估测器,和(二) 趋同阶段,该阶段采用一个预测的交替梯度下行算法,用于非电离子目标,初始时使用第一阶段的输出。我们证明,鉴于初始化,估计器与一个非对称、微量最佳统计错误的线性组合。对合成和真实数据的实验表明,拟议的非电解式程序比现有方法要好。