Restoring Definiteness via Shrinking, with an Application to Correlation Matrices with a Fixed Block
Restoring Definiteness via Shrinking, with an Application to Correlation Matrices with a Fixed Block
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DOI：
10.1137/140996112
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发表时间：
2016-01-01
期刊：
影响因子：
10.2
通讯作者：
Sego, Vedran
中科院分区：
文献类型：
--
作者：
Higham, Nicholas J.;Strabic, Natasa;Sego, Vedran
半正定矩阵的不定近似出现在涉及协方差矩阵和相关矩阵的各种数据分析应用中。提出了一种恢复不定矩阵M-0的半正定性的方法，该方法构造了M-0与半正定目标矩阵M-1的凸线性组合S（alpha）= alpha M-1 +（1 - alpha）M-0.在统计学中，通过将估计值M-0与M-1中的新信息相结合来改进估计值M-0的这种构造被称为收缩。我们对M-0没有统计假设，并将最佳收缩参数定义为a* = min{alpha是[0，1]的元素：S（alpha）是半正定的}。我们描述了三个算法计算alpha*。一种算法是基于二分法，使用Cholesky分解来测试确定性;第二种采用牛顿的方法;和第三个发现最小的特征值的对称确定广义特征值问题。我们表明，权重，反映在M-0的各个条目的信心，可以用来构建一个自然的选择的目标矩阵M-1。我们详细地处理一个问题的变种，其中一个半正定领导主子矩阵的M-0保持固定，显示如何固定块可以被利用来降低成本的二分法和广义特征值方法。数值实验表明，当应用于相关矩阵的不确定近似时，收缩可以比计算最近的相关矩阵快至少一个数量级。
Indefinite approximations of positive semidefinite matrices arise in various data analysis applications involving covariance matrices and correlation matrices. We propose a method for restoring positive semidefiniteness of an indefinite matrix M-0 that constructs a convex linear combination S(alpha) = alpha M-1 + (1 - alpha) M-0 of M-0 and a positive semidefinite target matrix M-1. In statistics, this construction for improving an estimate M-0 by combining it with new information in M-1 is known as shrinking. We make no statistical assumptions about M-0 and define the optimal shrinking parameter as a* = min{alpha is an element of[0, 1] : S(alpha) is positive semidefinite}. We describe three algorithms for computing alpha*. One algorithm is based on the bisection method, with the use of Cholesky factorization to test definiteness; a second employs Newton's method; and a third finds the smallest eigenvalue of a symmetric definite generalized eigenvalue problem. We show that weights that reflect confidence in the individual entries of M-0 can be used to construct a natural choice of the target matrix M-1. We treat in detail a problem variant in which a positive semidefinite leading principal submatrix of M-0 remains fixed, showing how the fixed block can be exploited to reduce the cost of the bisection and generalized eigenvalue methods. Numerical experiments show that when applied to indefinite approximations of correlation matrices shrinking can be at least an order of magnitude faster than computing the nearest correlation matrix.