In covariance matrix estimation, one of the challenges lies in finding a suitable model and an efficient estimation method. Two commonly used modelling 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和高效的估计方法。文献中常用的两种建模方法涉及对协方差矩阵或其逆矩阵施加线性约束。另一种方法则考虑对协方差矩阵的对数矩阵施加线性约束。本文针对协方差矩阵的不同变换提出了一个包含上述例子的线性约束通用框架。我们提出的估计方法通过求解一个凸问题得到M估计量，从而能够进行相对直接的渐近和有限样本分析。在建立通用理论后，我们重点研究相关系数矩阵的建模与稀疏性问题。基于几何视角的洞察使我们能够扩展协方差矩阵建模领域的若干最新成果，其中包括为相关系数矩阵空间提供无约束参数化方法——这是对近期一项利用矩阵对数研究成果的补充。