标题： 熵协方差模型 显示英文标题

标题： Entropic covariance models

摘要： 在协方差矩阵估计中，一个挑战在于找到合适的模型和高效的估计方法。 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矩阵对数施加线性约束。 在本文中，我们提出了一个关于协方差矩阵不同变换的线性约束的一般框架，包括上述例子。 我们提出的估计方法求解一个凸问题，并产生一个$M$-估计器，允许相对简单的渐近（一般情况下）和有限样本分析（在高斯情况下）。 特别是，我们恢复了标准$\sqrt{n/d}$率，其中$d$是底层模型的维度。 我们的几何见解使得能够扩展协方差矩阵建模中的各种最新结果。 这包括提供相关矩阵空间的无约束参数化，这是对最近利用矩阵对数的结果的一种替代方法。 显示英文摘要

摘要： 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.