# Statistically Optimal Robust Mean and Covariance Estimation for Anisotropic Gaussians

arXiv: https://arxiv.org/abs/2301.09024 · HTML: https://mlsanity.com/paper/2301.09024/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 75 references that are in the map.
- First posted on arXiv 21 Jan 2023.

## Builds on (most cited first, 15 of 15)

- [Mean estimation and regression under heavy-tailed distributions--a survey](https://mlsanity.com/paper/1906.04280/) (2019)
- [Agnostic Estimation of Mean and Covariance](https://mlsanity.com/paper/1604.06968/) (2016)
- [Statistical Query Lower Bounds for Robust Estimation of High-dimensional Gaussians and Gaussian Mixtures](https://mlsanity.com/paper/1611.03473/) (2016)
- [Recent Advances in Algorithmic High-Dimensional Robust Statistics](https://mlsanity.com/paper/1911.05911/) (2019)
- [Being Robust (in High Dimensions) Can Be Practical](https://mlsanity.com/paper/1703.00893/) (2017)
- [Outlier Robust Mean Estimation with Subgaussian Rates via Stability](https://mlsanity.com/paper/2007.15618/) (2020)
- [Robust Covariance and Scatter Matrix Estimation under Huber's Contamination Model](https://mlsanity.com/paper/1506.00691/) (2015)
- [High-Dimensional Robust Mean Estimation in Nearly-Linear Time](https://mlsanity.com/paper/1811.09380/) (2018)
- [Best Arm Identification for Contaminated Bandits](https://mlsanity.com/paper/1802.09514/) (2018)
- [User-Friendly Covariance Estimation for Heavy-Tailed Distributions](https://mlsanity.com/paper/1811.01520/) (2018)
- [How Hard Is Robust Mean Estimation?](https://mlsanity.com/paper/1903.07870/) (2019)
- [Covariance Estimation: Optimal Dimension-free Guarantees for Adversarial Corruption and Heavy Tails](https://mlsanity.com/paper/2205.08494/) (2022)
- [Outlier-Robust Sparse Mean Estimation for Heavy-Tailed Distributions](https://mlsanity.com/paper/2211.16333/) (2022)
- [Non-asymptotic bounds for percentiles of independent non-identical random variables](https://mlsanity.com/paper/1808.07997/) (2018)
- [Nearly minimax robust estimator of the mean vector by iterative spectral dimension reduction](https://mlsanity.com/paper/2204.02323/) (2022)

## Built on by (most cited first, 3 of 3)

- [Dimension-free Private Mean Estimation for Anisotropic Distributions](https://mlsanity.com/paper/2411.00775/) (2024)
- [Robust Mean Estimation Without Moments for Symmetric Distributions](https://mlsanity.com/paper/2302.10844/) (2023)
- [Precise sample covariance spectral norm error -- an RDT view](https://mlsanity.com/paper/2607.14460/) (2026)

## Abstract (arXiv)

Assume that X1, ldots, XN is an varepsilon-contaminated sample of N independent Gaussian vectors in ℝd with mean μ and covariance Sigma. In the strong varepsilon-contamination model we assume that the adversary replaced an varepsilon fraction of vectors in the original Gaussian sample by any other vectors. We show that there is an estimator widehat μ of the mean satisfying, with probability at least 1 - δ, a bound of the form [ |widehatμ - μ|2 ≤ cleft(√fracoperatornameTr(Sigma)N + √frac|Sigma|log(1/δ)N + varepsilon√|Sigma|right), ] where c > 0 is an absolute constant and |Sigma| denotes the operator norm of Sigma. In the same contaminated Gaussian setup, we construct an estimator widehat Sigma of the covariance matrix Sigma that satisfies, with probability at least 1 - δ, [ left|widehatSigma - Sigmaright| ≤ cleft(√frac|Sigma|operatornameTr(Sigma)N + |Sigma|√fraclog(1/δ)N + varepsilon|Sigma|right). ] Both results are optimal up to multiplicative constant factors. Despite the recent significant interest in robust statistics, achieving both dimension-free bounds in the canonical Gaussian case remained open. In fact, several previously known results were either dimension-dependent and required Sigma to be close to identity, or had a sub-optimal dependence on the contamination level varepsilon. As a part of the analysis, we derive sharp concentration inequalities for central order statistics of Gaussian, folded normal, and chi-squared distributions.