# Robust Sparse Estimation for Gaussians with Optimal Error under Huber Contamination

arXiv: https://arxiv.org/abs/2403.10416 · HTML: https://mlsanity.com/paper/2403.10416/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 41 references that are in the map.
- First posted on arXiv 15 Mar 2024.

## Builds on (most cited first, 20 of 20)

- [Statistical Query Lower Bounds for Robust Estimation of High-dimensional Gaussians and Gaussian Mixtures](https://mlsanity.com/paper/1611.03473/) (2016)
- [Reducibility and Statistical-Computational Gaps from Secret Leakage](https://mlsanity.com/paper/2005.08099/) (2020)
- [Statistical Query Algorithms and Low-Degree Tests Are Almost Equivalent](https://mlsanity.com/paper/2009.06107/) (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)
- [Robust Sub-Gaussian Principal Component Analysis and Width-Independent Schatten Packing](https://mlsanity.com/paper/2006.06980/) (2020)
- [High Dimensional Robust Sparse Regression](https://mlsanity.com/paper/1805.11643/) (2018)
- [Faster Algorithms for High-Dimensional Robust Covariance Estimation](https://mlsanity.com/paper/1906.04661/) (2019)
- [Statistical and computational trade-offs in estimation of sparse principal components](https://mlsanity.com/paper/1408.5369/) (2014)
- [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)
- [Robust estimation via generalized quasi-gradients](https://mlsanity.com/paper/2005.14073/) (2020)
- [Outlier-Robust Sparse Estimation via Non-Convex Optimization](https://mlsanity.com/paper/2109.11515/) (2021)
- [List-Decodable Sparse Mean Estimation via Difference-of-Pairs Filtering](https://mlsanity.com/paper/2206.05245/) (2022)
- [Outlier-Robust Sparse Mean Estimation for Heavy-Tailed Distributions](https://mlsanity.com/paper/2211.16333/) (2022)
- [List-Decodable Sparse Mean Estimation](https://mlsanity.com/paper/2205.14337/) (2022)
- [Nearly-Linear Time and Streaming Algorithms for Outlier-Robust PCA](https://mlsanity.com/paper/2305.02544/) (2023)
- [The Full Landscape of Robust Mean Testing: Sharp Separations between Oblivious and Adaptive Contamination](https://mlsanity.com/paper/2307.10273/) (2023)
- [Near-Optimal Algorithms for Gaussians with Huber Contamination: Mean Estimation and Linear Regression](https://mlsanity.com/paper/2312.01547/) (2023)

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

- [Testing Noise Assumptions of Learning Algorithms](https://mlsanity.com/paper/2501.09189/) (2025)
- [RobustLDS: Learning linear dynamical systems under adversarial corruptions](https://mlsanity.com/paper/2610.11906/) (2026)

## Abstract (arXiv)

We study Gaussian sparse estimation tasks in Huber's contamination model with a focus on mean estimation, PCA, and linear regression. For each of these tasks, we give the first sample and computationally efficient robust estimators with optimal error guarantees, within constant factors. All prior efficient algorithms for these tasks incur quantitatively suboptimal error. Concretely, for Gaussian robust k-sparse mean estimation on ℝd with corruption rate ε>0, our algorithm has sample complexity (k2/ε2)polylog(d/ε), runs in sample polynomial time, and approximates the target mean within ℓ2-error O(ε). Previous efficient algorithms inherently incur error Omega(ε √log(1/ε)). At the technical level, we develop a novel multidimensional filtering method in the sparse regime that may find other applications.