In covariance matrix estimation, one of the challenges lies in finding a suitable model and an efficient estimation method. Two commonly used approaches in the literature involve imposing linear restrictions on the covariance matrix or its inverse. Another approach considers linear restrictions on the matrix logarithm of the covariance matrix. In this paper, we present a general framework for linear restrictions on different transformations of the covariance matrix, including the mentioned examples. Our proposed estimation method solves a convex problem and yields an M-estimator, allowing for relatively straightforward asymptotic and finite sample analysis. After developing the general theory, we focus on modelling correlation matrices and on sparsity. Our geometric insights allow to extend various recent results in covariance matrix modelling. This includes providing unrestricted parametrizations of the space of correlation matrices, which is alternative to a recent result utilizing the matrix logarithm.

翻译：在协方差矩阵估计中，一个关键挑战在于寻找合适的模型与高效的估计方法。文献中常用的两类方法涉及对协方差矩阵或其逆矩阵施加线性约束。另一类方法则考虑对协方差矩阵的矩阵对数施加线性约束。本文提出一个通用框架，用于对协方差矩阵的不同变换（包括上述实例）施加线性约束。我们提出的估计方法通过求解凸问题获得M估计量，从而能够进行相对直接的渐近与有限样本分析。在建立通用理论后，我们聚焦于相关矩阵建模与稀疏性问题。基于几何视角的见解使我们能够扩展协方差矩阵建模领域多项最新成果，其中包括提供相关矩阵空间的无约束参数化——这是对近期基于矩阵对数的研究成果的另一种替代方案。