ASYMPTOTIC NORMALITY AND OPTIMALITIES IN ESTIMATION OF LARGE GAUSSIAN GRAPHICAL MODELS
ASYMPTOTIC NORMALITY AND OPTIMALITIES IN ESTIMATION OF LARGE GAUSSIAN GRAPHICAL MODELS
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DOI：
10.1214/14-aos1286
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发表时间：
2015-06-01
影响因子：
4.5
通讯作者：
Zhou, Harrison H.
中科院分区：
文献类型：
--
作者：
Ren, Zhao;Sun, Tingni;Zhou, Harrison H.
高斯图模型是近年来研究变量间关系的一种流行范式，具有广泛的应用前景。本文考虑了一个基本问题：在一个大的高斯图模型中，什么时候可以用参数平方根率估计低维参数？在相对于样本容量的稀疏性条件下，提出了一种新的回归方法，以获得精度矩阵中每个元素的渐近有效估计。当精度矩阵不够稀疏，或相当于样本量不够大时，建立下限以表明在估计每个条目时不再可能达到参数率。这个下界的结果，它提供了一个答案的微妙的样本量的问题，建立了一个新的构造的一个子集的稀疏精度矩阵的应用乐凸轮引理。此外，在最小样本要求下，证明了该估计器在参数率不能达到的情况下具有最优的收敛速度，并将该估计器应用于高斯图模型中边的存在性检验或整个模型的支持度恢复，为了获得由矩阵L（q）测量的整个精度矩阵的自适应速率最优估计，算子范数和对图模型中的潜变量进行推理。所有这些都是在精度矩阵的稀疏性条件和其频谱范围的边条件下实现的。这显著地放松了通常对精度矩阵施加的均匀信号强度条件、协方差矩阵的Hessian张量算子的不可表示性条件或精度矩阵的l（1）约束。数值结果证实了我们的理论研究结果。该算法的ROC曲线，渐近正态分布保持（ANT），支持恢复显着优于流行的GLasso算法。
The Gaussian graphical model, a popular paradigm for studying relationship among variables in a wide range of applications, has attracted great attention in recent years. This paper considers a fundamental question: When is it possible to estimate low-dimensional parameters at parametric square-root rate in a large Gaussian graphical model? A novel regression approach is proposed to obtain asymptotically efficient estimation of each entry of a precision matrix under a sparseness condition relative to the sample size. When the precision matrix is not sufficiently sparse, or equivalently the sample size is not sufficiently large, a lower bound is established to show that it is no longer possible to achieve the parametric rate in the estimation of each entry. This lower bound result, which provides an answer to the delicate sample size question, is established with a novel construction of a subset of sparse precision matrices in an application of Le Cam's lemma. Moreover, the proposed estimator is proven to have optimal convergence rate when the parametric rate cannot be achieved, under a minimal sample requirement.The proposed estimator is applied to test the presence of an edge in the Gaussian graphical model or to recover the support of the entire model, to obtain adaptive rate-optimal estimation of the entire precision matrix as measured by the matrix l(q) operator norm and to make inference in latent variables in the graphical model. All of this is achieved under a sparsity condition on the precision matrix and a side condition on the range of its spectrum. This significantly relaxes the commonly imposed uniform signal strength condition on the precision matrix, irrepresentability condition on the Hessian tensor operator of the covariance matrix or the l(1) constraint on the precision matrix. Numerical results confirm our theoretical findings. The ROC curve of the proposed algorithm, Asymptotic Normal Thresholding (ANT), for support recovery significantly outperforms that of the popular GLasso algorithm.