Minimax estimation of large precision matrices with bandable Cholesky factor
Minimax estimation of large precision matrices with bandable Cholesky factor
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DOI：
10.1214/19-aos1893
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发表时间：
2017-12
期刊：
影响因子：
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通讯作者：
Yu Liu;Zhao Ren
中科院分区：
文献类型：
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作者：
Yu Liu;Zhao Ren
过去十年见证了重大的方法和理论进展，估计大型精度矩阵。特别是，有一些科学应用，如纵向数据，气象学和光谱学，其中变量的排序可以通过精度矩阵的Cholesky因子上的可带结构来解释。然而，极大极小理论仍然在很大程度上是未知的，而不是在相应的可带协方差矩阵上建立的极大极小结果。在这篇论文中，我们主要研究两种常用的参数空间，并在算子范数和Frobenius范数下得到最优收敛速度。发现了一个引人注目的现象：两种类型的参数空间在算子范数下有根本不同，但在Frobenius范数下具有相同的速率最优性，这与相应的两种类型的可带协方差矩阵在两种范数下的等价性形成鲜明对比。这个基本区别是通过仔细构造相应的极大极小下界来建立的。开发了两个新的估计过程：对于算子范数，我们的最佳过程基于一种针对精度矩阵所有主要子矩阵的新型局部裁剪估计器，而对于弗罗贝纽斯范数，我们的最佳过程依赖于一种微妙的基于回归的块阈值规则。Lepski的方法被认为是实现最佳适应。我们进一步建立率最优的nonparanormal模型，通过应用我们的本地裁剪程序的秩为基础的估计。数值研究证实了我们的理论研究结果。
Last decade witnesses significant methodological and theoretical advances in estimating large precision matrices. In particular, there are scientific applications such as longitudinal data, meteorology and spectroscopy in which the ordering of the variables can be interpreted through a bandable structure on the Cholesky factor of the precision matrix. However, the minimax theory has still been largely unknown, as opposed to the well established minimax results over the corresponding bandable covariance matrices. In this thesis, we focus on two commonly used types of parameter spaces, and develop the optimal rates of convergence under both the operator norm and the Frobenius norm. A striking phenomenon is found: two types of parameter spaces are fundamentally different under the operator norm but enjoy the same rate optimality under the Frobenius norm, which is in sharp contrast to the equivalence of corresponding two types of bandable covariance matrices under both norms. This fundamental difference is established by carefully constructing the corresponding minimax lower bounds. Two new estimation procedures are developed: for the operator norm, our optimal procedure is based on a novel local cropping estimator targeting on all principle submatrices of the precision matrix while for the Frobenius norm, our optimal procedure relies on a delicate regression-based block-thresholding rule. Lepski's method is considered to achieve optimal adaptation. We further establish rate optimality in the nonparanormal model, by applying our local cropping procedure to the rank-based estimators. Numerical studies are carried out to confirm our theoretical findings.