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 (in general) and finite sample analysis (in the Gaussian case). In particular, we recover standard $\sqrt{n/d}$ rates, where $d$ is the dimension of the underlying model. 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，寻找合适的模型与高效的估计方法是一项核心挑战。文献中两种常用的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形式，提出一个包含上述例子的通用线性约束框架。我们提出的估计方法通过求解凸优化问题得到$M$估计量，可进行相对直接的渐近分析（一般情形）与有限样本分析（高斯情形）。特别地，我们恢复了标准的$\sqrt{n/d}$收敛速率（其中$d$为基础模型维度）。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，包括为相关矩阵空间提供无约束参数化表示——这形成了对近期基于矩阵对数方法研究成果的替代方案。