When only few data samples are accessible, utilizing structural prior knowledge is essential for estimating covariance matrices and their inverses. One prominent example is knowing the covariance matrix to be Toeplitz structured, which occurs when dealing with wide sense stationary (WSS) processes. This work introduces a novel class of positive definiteness ensuring likelihood-based estimators for Toeplitz structured covariance matrices (CMs) and their inverses. In order to accomplish this, we derive positive definiteness enforcing constraint sets for the Gohberg-Semencul (GS) parameterization of inverse symmetric Toeplitz matrices. Motivated by the relationship between the GS parameterization and autoregressive (AR) processes, we propose hyperparameter tuning techniques, which enable our estimators to combine advantages from state-of-the-art likelihood and non-parametric estimators. Moreover, we present a computationally cheap closed-form estimator, which is derived by maximizing an approximate likelihood. Due to the ensured positive definiteness, our estimators perform well for both the estimation of the CM and the inverse covariance matrix (ICM). Extensive simulation results validate the proposed estimators' efficacy for several standard Toeplitz structured CMs commonly employed in a wide range of applications.

翻译：当仅有少量数据样本可用时，利用结构先验知识对于估计协方差矩阵及其逆矩阵至关重要。一个典型例子是已知协方差矩阵具有Toeplitz结构，这出现在处理宽平稳（WSS）过程时。本文提出了一类新颖的、确保正定性的基于似然的估计器，用于估计Toeplitz结构协方差矩阵及其逆矩阵。为实现此目标，我们推导了对称逆Toeplitz矩阵的Gohberg-Semencul参数化中强制正定性的约束集。受GS参数化与自回归（AR）过程之间关系的启发，我们提出了超参数调优技术，使我们的估计器能够结合最先进似然估计器与非参数估计器的优势。此外，我们提出了一种计算成本低廉的闭式估计器，该估计器通过最大化近似似然推导得出。由于确保了正定性，我们的估计器在协方差矩阵和逆协方差矩阵的估计中均表现良好。大量仿真结果验证了所提估计器在多种标准Toeplitz结构协方差矩阵（广泛应用于各类场景）上的有效性。