The state-of-the-art methods for estimating high-dimensional covariance matrices all shrink the eigenvalues of the sample covariance matrix towards a data-insensitive shrinkage target. The underlying shrinkage transformation is either chosen heuristically - without compelling theoretical justification - or optimally in view of restrictive distributional assumptions. In this paper, we propose a principled approach to construct covariance estimators without imposing restrictive assumptions. That is, we study distributionally robust covariance estimation problems that minimize the worst-case Frobenius error with respect to all data distributions close to a nominal distribution, where the proximity of distributions is measured via a divergence on the space of covariance matrices. We identify mild conditions on this divergence under which the resulting minimizers represent shrinkage estimators. We show that the corresponding shrinkage transformations are intimately related to the geometrical properties of the underlying divergence. We also prove that our robust estimators are efficiently computable and asymptotically consistent and that they enjoy finite-sample performance guarantees. We exemplify our general methodology by synthesizing explicit estimators induced by the Kullback-Leibler, Fisher-Rao, and Wasserstein divergences. Numerical experiments based on synthetic and real data show that our robust estimators are competitive with state-of-the-art estimators.

翻译：当前估计高维协方差矩阵的最先进方法均将样本协方差矩阵的特征值向对数据不敏感的收缩目标收缩。其背后的收缩变换要么基于启发式选择——缺乏令人信服的理论依据——要么在限制性分布假设下追求最优性。本文提出一种无需强加限制性假设的原则性协方差估计器构建方法。具体而言，我们研究分布鲁棒协方差估计问题，该问题在协方差矩阵空间上通过散度度量分布邻近性，最小化所有接近名义分布的数据分布下的最坏情况Frobenius误差。我们确定了该散度的温和条件，使得所得极小化子表现为收缩估计器。我们证明相应的收缩变换与底层散度的几何性质密切相关，并验证所提鲁棒估计器具有可高效计算性、渐近一致性及有限样本性能保证。通过整合Kullback-Leibler散度、Fisher-Rao散度和Wasserstein散度诱导的显式估计器，我们例证了该通用方法论。基于合成数据与真实数据的数值实验表明，所提鲁棒估计器与当前最先进估计器相比具有竞争力。