Tensor Graphical Model: Non-Convex Optimization and Statistical Inference
Tensor Graphical Model: Non-Convex Optimization and Statistical Inference
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DOI：
10.1109/tpami.2019.2907679
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发表时间：
2016-09
影响因子：
23.6
通讯作者：
Xiang Lyu;W. Sun;Zhaoran Wang;Han Liu;Jian Yang;Guang Cheng
中科院分区：
文献类型：
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作者：
Xiang Lyu;W. Sun;Zhaoran Wang;Han Liu;Jian Yang;Guang Cheng
我们考虑的估计和推理的图形模型，其特征在于高维张量值数据的依赖结构。为了便于估计与张量的每种方式相对应的精度矩阵，我们假设数据遵循张量正态分布，其协方差具有Kronecker乘积结构。在这个模型的估计和推理的一个关键挑战是，它的惩罚最大似然估计涉及最小化非凸目标函数的事实。为了解决这个问题，本文做了两个贡献：（i）尽管这个估计问题的非凸性，我们证明了交替最小化算法，迭代估计每个稀疏精度矩阵，同时固定其他，达到一个最佳的统计收敛速度的估计。(ii)我们提出了一个去偏的统计推断过程中测试的真实支持的稀疏精度矩阵的假设，并采用它来测试越来越多的假发现率（FDR）控制的假设。我们的检验统计量的渐近正态性和FDR控制程序的一致性。我们的理论结果得到了彻底的数值研究的支持，我们在自闭症谱系障碍的神经成像研究和用户广告点击分析方面的真实的应用带来了新的科学发现和商业见解。所提出的方法被编码到一个公开可用的R包Tlasso。
We consider the estimation and inference of graphical models that characterize the dependency structure of high-dimensional tensor-valued data. To facilitate the estimation of the precision matrix corresponding to each way of the tensor, we assume the data follow a tensor normal distribution whose covariance has a Kronecker product structure. A critical challenge in the estimation and inference of this model is the fact that its penalized maximum likelihood estimation involves minimizing a non-convex objective function. To address it, this paper makes two contributions: (i) In spite of the non-convexity of this estimation problem, we prove that an alternating minimization algorithm, which iteratively estimates each sparse precision matrix while fixing the others, attains an estimator with an optimal statistical rate of convergence. (ii) We propose a de-biased statistical inference procedure for testing hypotheses on the true support of the sparse precision matrices, and employ it for testing a growing number of hypothesis with false discovery rate (FDR) control. The asymptotic normality of our test statistic and the consistency of FDR control procedure are established. Our theoretical results are backed up by thorough numerical studies and our real applications on neuroimaging studies of Autism spectrum disorder and users’ advertising click analysis bring new scientific findings and business insights. The proposed methods are encoded into a publicly available R package Tlasso.