# Distributionally Robust Inverse Covariance Estimation: The Wasserstein Shrinkage Estimator

arXiv: https://arxiv.org/abs/1805.07194 · HTML: https://mlsanity.com/paper/1805.07194/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 May 2018.

## Built on by (most cited first, 16 of 16)

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- [Nash Equilibria, Regularization and Computation in Optimal Transport-Based Distributionally Robust Optimization](https://mlsanity.com/paper/2303.03900/) (2023)
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- [Wasserstein Distributionally Robust Optimization with Wasserstein Barycenters](https://mlsanity.com/paper/2203.12136/) (2022)
- [A Distributionally Robust Approach to Regret Optimal Control using the Wasserstein Distance](https://mlsanity.com/paper/2304.06783/) (2023)
- [Distributional Robustness Bounds Generalization Errors](https://mlsanity.com/paper/2212.09962/) (2022)
- [A Geometric Unification of Distributionally Robust Covariance Estimators: Shrinking the Spectrum by Inflating the Ambiguity Set](https://mlsanity.com/paper/2405.20124/) (2024)
- [Adjusted Wasserstein Distributionally Robust Estimator in Statistical Learning](https://mlsanity.com/paper/2303.15579/) (2023)
- [Robust Model Selection of Gaussian Graphical Models](https://mlsanity.com/paper/2211.05690/) (2022)
- [Testing Sparsity Assumptions in Bayesian Networks](https://mlsanity.com/paper/2307.06406/) (2023)
- [Distributionally Robust Skeleton Learning of Discrete Bayesian Networks](https://mlsanity.com/paper/2311.06117/) (2023)
- [Robust Processing and Learning: Principles, Methods, and Wireless Applications](https://mlsanity.com/paper/2602.09848/) (2026)

## Abstract (arXiv)

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.