In the high-dimensional data setting, the sample covariance matrix is singular. In order to get a numerically stable and positive definite modification of the sample covariance matrix in the high-dimensional data setting, in this paper we consider the condition number constrained covariance matrix approximation problem and present its explicit solution with respect to the Frobenius norm. The condition number constraint guarantees the numerical stability and positive definiteness of the approximation form simultaneously. By exploiting the special structure of the data matrix in the high-dimensional data setting, we also propose some new algorithms based on efficient matrix decomposition techniques. Numerical experiments are also given to show the computational efficiency of the proposed algorithms.

中文翻译：

---

条件数约束协方差矩阵逼近的一种有效数值方法

在高维数据设置中，样本协方差矩阵是奇异的。为了在高维数据设置中获得样本协方差矩阵的数值稳定和正定修正，在本文中，我们考虑了条件数约束协方差矩阵逼近问题，并提出了其关于 Frobenius 范数的显式解。条件数约束同时保证了近似形式的数值稳定性和正定性。通过利用高维数据设置中数据矩阵的特殊结构，我们还提出了一些基于高效矩阵分解技术的新算法。还给出了数值实验以表明所提出算法的计算效率。