Distributionally Robust Inverse Covariance Estimation: The Wasserstein Shrinkage Estimator

Viet Anh Nguyen (), Daniel Kuhn () and Peyman Mohajerin Esfahani ()
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Viet Anh Nguyen: Department of Management Science and Engineering, Stanford University, Stanford, California 94305
Daniel Kuhn: Ecole Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland
Peyman Mohajerin Esfahani: Delft Center for Systems and Control, Delft University of Technology, 2628 CD Delft, Netherlands

Operations Research, 2022, vol. 70, issue 1, 490-515

Abstract: 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.

Keywords: Machine Learning and Data Science; distributionally robust optimization; data-driven optimization; Wasserstein distance; shrinkage estimator; maximum likelihood estimation (search for similar items in EconPapers)
Date: 2022
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Persistent link: https://EconPapers.repec.org/RePEc:inm:oropre:v:70:y:2022:i:1:p:490-515

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