Distributionally Robust Inverse Covariance Estimation: The Wasserstein Shrinkage Estimator

TL;DR

This paper proposes a novel distributionally robust inverse covariance estimation method using Wasserstein ambiguity sets, resulting in a tractable semidefinite program with an analytical nonlinear shrinkage solution that is well-conditioned and preserves eigenvalue order.

Contribution

It introduces a new Wasserstein-based distributionally robust estimator for inverse covariance, with an analytical solution and properties emerging naturally from the model.

Findings

Estimator is invertible and well-conditioned for p>n

Preserves eigenvalue order and is rotation-equivariant

Develops an efficient algorithm for Gaussian graphical models

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.