Distributionally Robust Inverse Covariance Estimation: The Wasserstein Shrinkage Estimator
Distributionally Robust Inverse Covariance Estimation: The Wasserstein Shrinkage Estimator
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DOI：
10.1287/opre.2020.2076
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发表时间：
2021-07-23
影响因子：
2.7
通讯作者：
Esfahani, Peyman Mohajerin
中科院分区：
文献类型：
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作者：
Nguyen, Viet Anh;Kuhn, Daniel;Esfahani, Peyman Mohajerin
我们引入了一个分布鲁棒的最大似然估计模型与Wasserstein模糊集推断逆协方差矩阵的p维高斯随机向量从n个独立的样本。所提出的模型最大限度地减少了最坏的情况下（最大值）的斯坦的损失在所有正常的参考分布在规定的Wasserstein距离的正态分布特征的样本均值和样本协方差矩阵。我们证明，这个估计问题是等价于一个半定的程序，这是易于处理的理论，但超出了通用的解决方案的实际相关的问题尺寸p。在没有任何先验结构信息，估计问题有一个分析的解决方案，自然被解释为一个非线性收缩估计。新的收缩估计不仅是可逆的，而且即使对于p > n也是良好的条件，而且是旋转等变的，并且保持了样本协方差矩阵特征值的阶。这些理想的属性不是强加的特设，但自然出现从底层的分布鲁棒优化模型。最后，我们开发了一个序列二次近似算法，有效地解决了一般的估计问题的条件独立约束通常遇到高斯图形模型。
We introduce a distributionally robust maximum likelihood estimation model with a Wasserstein ambiguity set to infer the inverse covariance matrix of a p-dimensional Gaussian random vector from n independent samples. The proposed model minimizes the worst case (maximum) of Stein's loss across all normal reference distributions within a prescribed Wasserstein distance from the normal distribution characterized by the sample mean and the sample covariance matrix. We prove that this estimation problem is equivalent to a semidefinite program that is tractable in theory but beyond the reach of general-purpose solvers for practically relevant problem dimensions p. In the absence of any prior structural information, the estimation problem has an analytical solution that is naturally interpreted as a nonlinear shrinkage estimator. Besides being invertible and well conditioned even for p > n, the new shrinkage estimator is rotation equivariant and preserves the order of the eigenvalues of the sample covariance matrix. These desirable properties are not imposed ad hoc but emerge naturally from the underlying distributionally robust optimization model. Finally, we develop a sequential quadratic approximation algorithm for efficiently solving the general estimation problem subject to conditional independence constraints typically encountered in Gaussian graphical models.