# A Geometric Unification of Distributionally Robust Covariance Estimators: Shrinking the Spectrum by Inflating the Ambiguity Set

arXiv: https://arxiv.org/abs/2405.20124 · HTML: https://mlsanity.com/paper/2405.20124/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 72 references that are in the map.
- First posted on arXiv 30 May 2024.

## Builds on (most cited first, 22 of 22)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Data-driven Distributionally Robust Optimization Using the Wasserstein Metric: Performance Guarantees and Tractable Reformulations](https://mlsanity.com/paper/1505.05116/) (2015)
- [Wasserstein Distributionally Robust Optimization: Theory and Applications in Machine Learning](https://mlsanity.com/paper/1908.08729/) (2019)
- [Regularization via Mass Transportation](https://mlsanity.com/paper/1710.10016/) (2017)
- [Finite-Sample Guarantees for Wasserstein Distributionally Robust Optimization: Breaking the Curse of Dimensionality](https://mlsanity.com/paper/2009.04382/) (2020)
- [Statistical Analysis of Wasserstein Distributionally Robust Estimators](https://mlsanity.com/paper/2108.02120/) (2021)
- [Confidence Regions in Wasserstein Distributionally Robust Estimation](https://mlsanity.com/paper/1906.01614/) (2019)
- [Wasserstein Distributionally Robust Kalman Filtering](https://mlsanity.com/paper/1809.08830/) (2018)
- [Lecture notes on ridge regression](https://mlsanity.com/paper/1509.09169/) (2015)
- [Distributionally Robust Inverse Covariance Estimation: The Wasserstein Shrinkage Estimator](https://mlsanity.com/paper/1805.07194/) (2018)
- [Sequential Domain Adaptation by Synthesizing Distributionally Robust Experts](https://mlsanity.com/paper/2106.00322/) (2021)
- [Optimistic Distributionally Robust Optimization for Nonparametric Likelihood Approximation](https://mlsanity.com/paper/1910.10583/) (2019)
- [Bridging Bayesian and Minimax Mean Square Error Estimation via Wasserstein Distributionally Robust Optimization](https://mlsanity.com/paper/1911.03539/) (2019)
- [Mean-Covariance Robust Risk Measurement](https://mlsanity.com/paper/2112.09959/) (2021)
- [Distributionally Robust Fair Principal Components via Geodesic Descents](https://mlsanity.com/paper/2202.03071/) (2022)
- [Sion's Minimax Theorem in Geodesic Metric Spaces and a Riemannian Extragradient Algorithm](https://mlsanity.com/paper/2202.06950/) (2022)
- [Calculating Optimistic Likelihoods Using (Geodesically) Convex Optimization](https://mlsanity.com/paper/1910.07817/) (2019)
- [Distributionally Robust Linear Quadratic Control](https://mlsanity.com/paper/2305.17037/) (2023)
- [Nonparametric Stein-type Shrinkage Covariance Matrix Estimators in High-Dimensional Settings](https://mlsanity.com/paper/1410.4726/) (2014)
- [Robust Kalman Filtering under Model Perturbations](https://mlsanity.com/paper/1508.01906/) (2015)
- [Coverage-Validity-Aware Algorithmic Recourse](https://mlsanity.com/paper/2311.11349/) (2023)
- [Gradient Method for Optimization on Riemannian Manifolds with Lower Bounded Curvature](https://mlsanity.com/paper/1806.02694/) (2018)

## Built on by (most cited first, 2 of 2)

- [Robust Processing and Learning: Principles, Methods, and Wireless Applications](https://mlsanity.com/paper/2602.09848/) (2026)
- [Covariance Shrinkage via Stochastic Interpolation](https://mlsanity.com/paper/2606.07382/) (2026)

## Abstract (arXiv)

The state-of-the-art methods for estimating high-dimensional covariance matrices all shrink the eigenvalues of the sample covariance matrix towards a data-insensitive shrinkage target. The underlying shrinkage transformation is either chosen heuristically - without compelling theoretical justification - or optimally in view of restrictive distributional assumptions. In this paper, we propose a principled approach to construct covariance estimators without imposing restrictive assumptions. That is, we study distributionally robust covariance estimation problems that minimize the worst-case Frobenius error with respect to all data distributions close to a nominal distribution, where the proximity of distributions is measured via a divergence on the space of covariance matrices. We identify mild conditions on this divergence under which the resulting minimizers represent shrinkage estimators. We show that the corresponding shrinkage transformations are intimately related to the geometrical properties of the underlying divergence. We also prove that our robust estimators are efficiently computable and asymptotically consistent and that they enjoy finite-sample performance guarantees. We exemplify our general methodology by synthesizing explicit estimators induced by the Kullback-Leibler, Fisher-Rao, and Wasserstein divergences. Numerical experiments based on synthetic and real data show that our robust estimators are competitive with state-of-the-art estimators.