# Near-Optimal Algorithms for Gaussians with Huber Contamination: Mean Estimation and Linear Regression

arXiv: https://arxiv.org/abs/2312.01547 · HTML: https://mlsanity.com/paper/2312.01547/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 42 of its 60 references that are in the map.
- First posted on arXiv 4 Dec 2023.

## Builds on (most cited first, 30 of 42)

- [Robust Estimators in High Dimensions without the Computational Intractability](https://mlsanity.com/paper/1604.06443/) (2016)
- [Agnostic Estimation of Mean and Covariance](https://mlsanity.com/paper/1604.06968/) (2016)
- [Sever: A Robust Meta-Algorithm for Stochastic Optimization](https://mlsanity.com/paper/1803.02815/) (2018)
- [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)
- [Efficient Algorithms and Lower Bounds for Robust Linear Regression](https://mlsanity.com/paper/1806.00040/) (2018)
- [Efficient Algorithms for Outlier-Robust Regression](https://mlsanity.com/paper/1803.03241/) (2018)
- [Randomized Block Krylov Methods for Stronger and Faster Approximate Singular Value Decomposition](https://mlsanity.com/paper/1504.05477/) (2015)
- [Statistical Query Algorithms and Low-Degree Tests Are Almost Equivalent](https://mlsanity.com/paper/2009.06107/) (2020)
- [Outlier Robust Mean Estimation with Subgaussian Rates via Stability](https://mlsanity.com/paper/2007.15618/) (2020)
- [Robust regression with covariate filtering: Heavy tails and adversarial contamination](https://mlsanity.com/paper/2009.12976/) (2020)
- [Robust Covariance and Scatter Matrix Estimation under Huber's Contamination Model](https://mlsanity.com/paper/1506.00691/) (2015)
- [Robust and Heavy-Tailed Mean Estimation Made Simple, via Regret Minimization](https://mlsanity.com/paper/2007.15839/) (2020)
- [Robust Linear Regression: Optimal Rates in Polynomial Time](https://mlsanity.com/paper/2007.01394/) (2020)
- [Quantum Entropy Scoring for Fast Robust Mean Estimation and Improved Outlier Detection](https://mlsanity.com/paper/1906.11366/) (2019)
- [Robust Meta-learning for Mixed Linear Regression with Small Batches](https://mlsanity.com/paper/2006.09702/) (2020)
- [Robustly Learning a Gaussian: Getting Optimal Error, Efficiently](https://mlsanity.com/paper/1704.03866/) (2017)
- [High-Dimensional Robust Mean Estimation in Nearly-Linear Time](https://mlsanity.com/paper/1811.09380/) (2018)
- [Robust subgaussian estimation of a mean vector in nearly linear time](https://mlsanity.com/paper/1906.03058/) (2019)
- [Robust Sub-Gaussian Principal Component Analysis and Width-Independent Schatten Packing](https://mlsanity.com/paper/2006.06980/) (2020)
- [Optimal Robust Linear Regression in Nearly Linear Time](https://mlsanity.com/paper/2007.08137/) (2020)
- [Faster Algorithms for High-Dimensional Robust Covariance Estimation](https://mlsanity.com/paper/1906.04661/) (2019)
- [Concentration Inequalities for Statistical Inference](https://mlsanity.com/paper/2011.02258/) (2020)
- [Robust Sparse Mean Estimation via Sum of Squares](https://mlsanity.com/paper/2206.03441/) (2022)
- [Outlier-Robust High-Dimensional Sparse Estimation via Iterative Filtering](https://mlsanity.com/paper/1911.08085/) (2019)
- [Clustering Mixture Models in Almost-Linear Time via List-Decodable Mean Estimation](https://mlsanity.com/paper/2106.08537/) (2021)
- [Streaming Algorithms for High-Dimensional Robust Statistics](https://mlsanity.com/paper/2204.12399/) (2022)
- [List Decodable Mean Estimation in Nearly Linear Time](https://mlsanity.com/paper/2005.09796/) (2020)
- [Robust estimation via generalized quasi-gradients](https://mlsanity.com/paper/2005.14073/) (2020)

## Built on by (most cited first, 7 of 7)

- [Efficient Multivariate Robust Mean Estimation Under Mean-Shift Contamination](https://mlsanity.com/paper/2502.14772/) (2025)
- [Robust Sparse Estimation for Gaussians with Optimal Error under Huber Contamination](https://mlsanity.com/paper/2403.10416/) (2024)
- [Linear Regression under Missing or Corrupted Coordinates](https://mlsanity.com/paper/2509.19242/) (2025)
- [RINS-T: Robust Implicit Neural Solvers for Time Series Linear Inverse Problems](https://mlsanity.com/paper/2510.17396/) (2025)
- [Robust Regression with Adaptive Contamination in Response: Optimal Rates and Computational Barriers](https://mlsanity.com/paper/2604.04228/) (2026)
- [Information-Computation Tradeoffs for Noiseless Linear Regression with Oblivious Contamination](https://mlsanity.com/paper/2510.10665/) (2025)
- [Beyond Heuristic Prompting: A Concept-Guided Bayesian Framework for Zero-Shot Image Recognition](https://mlsanity.com/paper/2603.07911/) (2026)

## Abstract (arXiv)

We study the fundamental problems of Gaussian mean estimation and linear regression with Gaussian covariates in the presence of Huber contamination. Our main contribution is the design of the first sample near-optimal and almost linear-time algorithms with optimal error guarantees for both of these problems. Specifically, for Gaussian robust mean estimation on ℝd with contamination parameter ε ∈ (0, ε0) for a small absolute constant ε0, we give an algorithm with sample complexity n = tildeO(d/ε2) and almost linear runtime that approximates the target mean within ℓ2-error O(ε). This improves on prior work that achieved this error guarantee with polynomially suboptimal sample and time complexity. For robust linear regression, we give the first algorithm with sample complexity n = tildeO(d/ε2) and almost linear runtime that approximates the target regressor within ℓ2-error O(ε). This is the first polynomial sample and time algorithm achieving the optimal error guarantee, answering an open question in the literature. At the technical level, we develop a methodology that yields almost-linear time algorithms for multi-directional filtering that may be of broader interest.