Distributionally Robust Inverse Covariance Estimation: The Wasserstein Shrinkage Estimator
This work addresses the challenge of robust covariance estimation in high-dimensional statistics, particularly for Gaussian graphical models, but it is incremental as it builds on existing shrinkage and distributionally robust optimization methods.
The authors tackled the problem of estimating the inverse covariance matrix from limited samples by introducing a distributionally robust maximum likelihood estimator with a Wasserstein ambiguity set, which yields a nonlinear shrinkage estimator that remains invertible and well-conditioned even when the number of dimensions exceeds the sample size.
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.