# Covariance Estimation: Optimal Dimension-free Guarantees for Adversarial Corruption and Heavy Tails

arXiv: https://arxiv.org/abs/2205.08494 · HTML: https://mlsanity.com/paper/2205.08494/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 May 2022.

## Built on by (most cited first, 14 of 14)

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- [A Parameter-Free Two-Bit Covariance Estimator with Improved Operator Norm Error Rate](https://mlsanity.com/paper/2308.16059/) (2023)
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- [Statistically Optimal Robust Mean and Covariance Estimation for Anisotropic Gaussians](https://mlsanity.com/paper/2301.09024/) (2023)
- [Minimax Linear Regression under the Quantile Risk](https://mlsanity.com/paper/2406.12145/) (2024)
- [Robust Gradient Descent for Phase Retrieval](https://mlsanity.com/paper/2410.10623/) (2024)
- [Do we really need the Rademacher complexities?](https://mlsanity.com/paper/2502.15118/) (2025)
- [Robust Estimation in metric spaces: Achieving Exponential Concentration with a Fr'echet Median](https://mlsanity.com/paper/2504.14161/) (2025)
- [The Decision Geometry of Covariance Estimation for the Global Minimum-Variance Portfolio under Heavy Tails](https://mlsanity.com/paper/2606.27462/) (2026)
- [Robustifying Markowitz](https://mlsanity.com/paper/2212.13996/) (2022)
- [Computable Bernstein Certificates for Cross-Fitted Clipped Covariance Estimation](https://mlsanity.com/paper/2602.14020/) (2026)
- [Precise sample covariance spectral norm error -- an RDT view](https://mlsanity.com/paper/2607.14460/) (2026)

## Abstract (arXiv)

We provide an estimator of the covariance matrix that achieves the optimal rate of convergence (up to constant factors) in the operator norm under two standard notions of data contamination: We allow the adversary to corrupt an η-fraction of the sample arbitrarily, while the distribution of the remaining data points only satisfies that the Lp-marginal moment with some p ≥ 4 is equivalent to the corresponding L2-marginal moment. Despite requiring the existence of only a few moments, our estimator achieves the same tail estimates as if the underlying distribution were Gaussian. As a part of our analysis, we prove a dimension-free Bai-Yin type theorem in the regime p > 4.