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.
Bio: Viet Anh Nguyen is currently a Ph.D. student in Management of Technology at Ecole Polytechnique Federale de Lausanne. He received a Bachelor of Engineering and a Master of Engineering in Industrial and Systems Engineering from the National University of Singapore in 2011 and 2013 respectively. He also holds a Diplôme d'Ingénieur (promotion Gustave Eiffel) from Ecole Centrale des Arts et Manufactures (Ecole Centrale de Paris). He graduated from the Swiss Program for Beginning Doctoral Students in Economics at the Study Center Gerzensee in 2014. Viet Anh is interested in very large-scale decision making under uncertainty, statistical optimization and machine learning with applications in energy systems, operations management, and data/policy analytics.
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