Estimating covariance matrices with high-dimensional complex data presents significant challenges, particularly concerning positive definiteness, sparsity, and numerical stability. Existing robust sparse estimators often fail to guarantee positive definiteness in finite samples, while subsequent positive-definite correction can degrade sparsity and lack explicit control over the condition number. To address these limitations, we propose a novel robust and well-conditioned sparse covariance matrix estimator. Our key innovation is the direct incorporation of a condition number constraint within a robust adaptive thresholding framework. This constraint simultaneously ensures positive definiteness, enforces a controllable level of numerical stability, and preserves the desired sparse structure without resorting to post-hoc modifications that compromise sparsity. We formulate the estimation as a convex optimization problem and develop an efficient alternating direction algorithm with guaranteed convergence. Theoretically, we establish that the proposed estimator achieves the minimax optimal convergence rate under the Frobenius norm. Comprehensive simulations and real-data applications demonstrate that our method consistently produces positive definite, well-conditioned, and sparse estimates, and achieves comparable or superior numerical stability to eigenvalue-bound methods while requiring less tuning parameters.

翻译：针对高维复杂数据的协方差矩阵估计面临显著挑战，尤其在正定性、稀疏性和数值稳定性方面。现有鲁棒稀疏估计器通常无法保证有限样本下的正定性，而后续的正定性修正可能破坏稀疏性，且缺乏对条件数的显式控制。为克服这些局限，我们提出一种新颖的鲁棒且良条件稀疏协方差矩阵估计器。我们的核心创新在于将条件数约束直接纳入鲁棒自适应阈值框架。该约束同时确保正定性、强制执行可控的数值稳定性水平，并保持期望的稀疏结构，无需采用可能损害稀疏性的后验修正。我们将该估计问题构建为凸优化问题，并开发了具有收敛保证的高效交替方向算法。理论上，我们证明所提估计器在Frobenius范数下达到极小极大最优收敛速率。综合仿真与真实数据应用表明，我们的方法始终生成正定、良条件且稀疏的估计结果，在需要更少调参的同时，其数值稳定性与特征值边界方法相当或更优。