# User-Friendly Covariance Estimation for Heavy-Tailed Distributions

arXiv: https://arxiv.org/abs/1811.01520 · HTML: https://mlsanity.com/paper/1811.01520/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Nov 2018.

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

- [High Dimensional Statistical Estimation under Uniformly Dithered One-bit Quantization](https://mlsanity.com/paper/2202.13157/) (2022)
- [On Estimating Rank-One Spiked Tensors in the Presence of Heavy Tailed Errors](https://mlsanity.com/paper/2107.09660/) (2021)
- [Provably Robust Temporal Difference Learning for Heavy-Tailed Rewards](https://mlsanity.com/paper/2306.11455/) (2023)
- [Statistically Optimal Robust Mean and Covariance Estimation for Anisotropic Gaussians](https://mlsanity.com/paper/2301.09024/) (2023)
- [Robust convex biclustering with a tuning-free method](https://mlsanity.com/paper/2212.03122/) (2022)
- [Bit Efficient Toeplitz Covariance Estimation](https://mlsanity.com/paper/2412.12678/) (2024)
- [Adaptive posterior distributions for uncertainty analysis of covariance matrices in Bayesian inversion problems for multioutput signals](https://mlsanity.com/paper/2501.01148/) (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)
- [Positive Semidefinite Matrix Supermartingales](https://mlsanity.com/paper/2401.15567/) (2024)
- [Sparse Linear Regression when Noises and Covariates are Heavy-Tailed and Contaminated by Outliers](https://mlsanity.com/paper/2408.01336/) (2024)
- [Asymptotic spectrum of weighted sample covariance: another proof of spectrum convergence](https://mlsanity.com/paper/2410.14408/) (2024)
- [Fast and Robust: Computationally Efficient Covariance Estimation for Sub-Weibull Vectors](https://mlsanity.com/paper/2512.17632/) (2025)
- [Information Hidden in Gradients of Regression with Target Noise](https://mlsanity.com/paper/2601.18546/) (2026)
- [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 offer a survey of recent results on covariance estimation for heavy-tailed distributions. By unifying ideas scattered in the literature, we propose user-friendly methods that facilitate practical implementation. Specifically, we introduce element-wise and spectrum-wise truncation operators, as well as their M-estimator counterparts, to robustify the sample covariance matrix. Different from the classical notion of robustness that is characterized by the breakdown property, we focus on the tail robustness which is evidenced by the connection between nonasymptotic deviation and confidence level. The key observation is that the estimators needs to adapt to the sample size, dimensionality of the data and the noise level to achieve optimal tradeoff between bias and robustness. Furthermore, to facilitate their practical use, we propose data-driven procedures that automatically calibrate the tuning parameters. We demonstrate their applications to a series of structured models in high dimensions, including the bandable and low-rank covariance matrices and sparse precision matrices. Numerical studies lend strong support to the proposed methods.