Efficient estimation of high-dimensional matrices-including covariance and precision matrices-is a cornerstone of modern multivariate statistics. Most existing studies have focused primarily on the theoretical properties of the estimators (e.g., consistency and sparsity), while largely overlooking the computational challenges inherent in high-dimensional settings. Motivated by recent advances in learning-based optimization method-which integrate data-driven structures with classical optimization algorithms-we explore high-dimensional matrix estimation assisted by machine learning. Specifically, for the optimization problem of high-dimensional matrix estimation, we first present a solution procedure based on the Linearized Alternating Direction Method of Multipliers (LADMM). We then introduce learnable parameters and model the proximal operators in the iterative scheme with neural networks, thereby improving estimation accuracy and accelerating convergence. Theoretically, we first prove the convergence of LADMM, and then establish the convergence, convergence rate, and monotonicity of its reparameterized counterpart; importantly, we show that the reparameterized LADMM enjoys a faster convergence rate. Notably, the proposed reparameterization theory and methodology are applicable to the estimation of both high-dimensional covariance and precision matrices. We validate the effectiveness of our method by comparing it with several classical optimization algorithms across different structures and dimensions of high-dimensional matrices.

翻译：高效估计高维矩阵（包括协方差矩阵和精度矩阵）是现代多元统计学的基石。现有研究主要侧重于估计量的理论性质（如一致性和稀疏性），而很大程度上忽略了高维设置中固有的计算挑战。受近期基于学习的优化方法——将数据驱动结构与经典优化算法相结合——进展的启发，我们探索了机器学习辅助的高维矩阵估计。具体而言，针对高维矩阵估计的优化问题，我们首先提出了一种基于线性化交替方向乘子法（LADMM）的求解流程。然后，我们引入可学习参数，并用神经网络对迭代方案中的近端算子进行建模，从而提升估计精度并加速收敛。理论上，我们首先证明了LADMM的收敛性，然后建立了其重参数化版本的收敛性、收敛速率和单调性；重要的是，我们证明了重参数化LADMM具有更快的收敛速率。值得注意的是，所提出的重参数化理论和方法可同时适用于高维协方差矩阵和精度矩阵的估计。通过将我们的方法与几种经典优化算法在不同结构和维度的高维矩阵上进行对比，验证了其有效性。