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 $\mathbb{R}^d$ with corruption rate $\epsilon>0$, our algorithm has sample complexity $(k^2/\epsilon^2)\mathrm{polylog}(d/\epsilon)$, runs in sample polynomial time, and approximates the target mean within $\ell_2$-error $O(\epsilon)$. Previous efficient algorithms inherently incur error $\Omega(\epsilon \sqrt{\log(1/\epsilon)})$. At the technical level, we develop a novel multidimensional filtering method in the sparse regime that may find other applications.

高斯稀疏估计在Huber污染模型中研究，针对均值估计、主成分分析和线性回归三个任务，提出了第一个样本和计算高效的鲁棒估计器，保证了较小的误差，并且在常数因子内达到最优。之前针对这些任务的高效算法都产生了数量上次优的误差。具体而言，对于高斯的鲁棒k稀疏均值估计在具有污染率为ε>0的R^d上，我们的算法具有样本复杂度为 (k^2/ε^2)·polylog(d/ε)，在多项式时间内运行，并且在L2误差为O(ε)的范围内逼近目标均值。之前的高效算法固有地产生了误差Ω(ε√log(1/ε))。在技术层面上，我们开发了一种在稀疏情况下的新型多维过滤方法，可能具有其他应用。

Huber污染下高斯稀疏估计的鲁棒性优化错误