从相依数据中学习稀疏高维矩阵值图模型
Learning Sparse High-Dimensional Matrix-Valued Graphical Models From Dependent Data
AI总结：
本文针对相依观测下的稀疏高维矩阵值高斯时间序列，提出基于稀疏组套索和ADMM的频域方法估计条件独立图，并给出局部收敛条件与收敛速率。
摘要：
我们考虑推断一个稀疏、高维、平稳的矩阵值高斯时间序列的条件独立图（CIG）的问题。以往所有关于高维矩阵图模型的工作都假设矩阵变量的观测是独立同分布（i.i.d.）的。本文允许相依观测。我们考虑基于稀疏组套索（sparse-group lasso）的频域问题表述，其功率谱密度（PSD）具有克罗内克可分解结构，并通过交替方向乘子法（ADMM）求解。该问题是双凸的，通过flip-flop优化求解。我们给出了逆PSD估计量在Frobenius范数下局部收敛到真值的充分条件。该结果同时给出了收敛速率。我们利用合成数据和真实数据的数值例子说明了所提方法。
英文摘要：
We consider the problem of inferring the conditional independence graph (CIG) of a sparse, high-dimensional, stationary matrix-variate Gaussian time series. All past work on high-dimensional matrix graphical models assumes that independent and identically distributed (i.i.d.) observations of the matrix-variate are available. Here we allow dependent observations. We consider a sparse-group lasso-based frequency-domain formulation of the problem with a Kronecker-decomposable power spectral density (PSD), and solve it via an alternating direction method of multipliers (ADMM) approach. The problem is bi-convex which is solved via flip-flop optimization. We provide sufficient conditions for local convergence in the Frobenius norm of the inverse PSD estimators to the true value. This result also yields a rate of convergence. We illustrate our approach using numerical examples utilizing both synthetic and real data.
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