Shrinking the Covariance Matrix Using Convex Penalties on the Matrix-Log Transformation
Shrinking the Covariance Matrix Using Convex Penalties on the Matrix-Log Transformation
复制标题
使用矩阵-对数变换上的凸惩罚来缩小协方差矩阵
DOI：
10.1080/10618600.2020.1814788
复制
发表时间：
2021
影响因子：
2.4
通讯作者：
Tyler, David E.
中科院分区：
文献类型：
--
作者：
Yi, Mengxi;Tyler, David E.
对于q维数据，当样本量相对于q较小或适中时，样本协方差矩阵的惩罚版本很重要。由于多元正态抽样下的负对数似然是凸的，协方差矩阵的逆，所以通常考虑也是凸的加性惩罚。最近，Deng和Tsui以及Yu等人提出了严格意义上Σ根函数的惩罚，并且是凸的，但不是凸的。然而，由此产生的惩罚优化问题既不是凸的也不是内的。然而，在本文中，我们将在Σ中展示这些不利的优化问题在测地线上是凸的。这允许我们建立相应的惩罚协方差矩阵的存在性和唯一性。更一般地，我们证明了Σ中的测地线凸性等价于惩罚的凸性，惩罚是Σ的根的函数。此外，当使用这种惩罚时，所得到的惩罚优化问题可以简化为Σ根对数上的q维凸优化问题，然后可以很容易地通过牛顿算法求解。本文的补充材料可在网上获得。
Forq-dimensional data, penalized versions of the sample covariance matrix are important when the sample size is small or modest relative toq. Since the negative log-likelihood under multivariate normal sampling is convex in, the inverse of the covariance matrix, it is common to consider additive penalties which are also convex in. More recently, Deng and Tsui and Yu et al. have proposed penalties which are strictly functions of the roots of Σ and are convex in, but not in. The resulting penalized optimization problems, though, are neither convex innor in. In this article, however, we show these penalized optimization problems to be geodesically convex in Σ. This allows us to establish the existence and uniqueness of the corresponding penalized covariance matrices. More generally, we show that geodesic convexity in Σ is equivalent to convexity infor penalties which are functions of the roots of Σ. In addition, when using such penalties, the resulting penalized optimization problem reduces to aq-dimensional convex optimization problem on the logs of the roots of Σ, which can then be readily solved via Newton’s algorithm. Supplementary materials for this article are available online.
影响因子：
1
作者：
L. Skovgaard
通讯作者：
L. Skovgaard
影响因子：
2.7
作者：
David E. Tyler;Mengxi Yi
通讯作者：
David E. Tyler;Mengxi Yi
形状的规范定义
影响因子：
0.8
作者：
D. Paindaveine
通讯作者：
D. Paindaveine