In this paper, we study the problem of robust sparse mean estimation, where the goal is to estimate a $k$-sparse mean from a collection of partially corrupted samples drawn from a heavy-tailed distribution. Existing estimators face two critical challenges in this setting. First, they are limited by a conjectured computational-statistical tradeoff, implying that any computationally efficient algorithm needs $\tilde\Omega(k^2)$ samples, while its statistically-optimal counterpart only requires $\tilde O(k)$ samples. Second, the existing estimators fall short of practical use as they scale poorly with the ambient dimension. This paper presents a simple mean estimator that overcomes both challenges under moderate conditions: it runs in near-linear time and memory (both with respect to the ambient dimension) while requiring only $\tilde O(k)$ samples to recover the true mean. At the core of our method lies an incremental learning phenomenon: we introduce a simple nonconvex framework that can incrementally learn the top-$k$ nonzero elements of the mean while keeping the zero elements arbitrarily small. Unlike existing estimators, our method does not need any prior knowledge of the sparsity level $k$. We prove the optimality of our estimator by providing a matching information-theoretic lower bound. Finally, we conduct a series of simulations to corroborate our theoretical findings. Our code is available at https://github.com/huihui0902/Robust_mean_estimation.

翻译：本文研究鲁棒稀疏均值估计问题，目标是从一组部分受污染的、来自重尾分布的样本中估计一个$k$-稀疏均值。现有估计量在此场景下面临两个关键挑战：首先，它们受限于推测性的计算-统计权衡，即任何计算高效的算法都需要$\tilde\Omega(k^2)$个样本，而其统计最优的对应方法仅需$\tilde O(k)$个样本；其次，现有估计量因随环境维度的扩展性能差而难以实际应用。本文提出一种简单均值估计量，在适度条件下克服了两大挑战：它能在近线性时间与内存（均针对环境维度）内运行，同时仅需$\tilde O(k)$个样本即可恢复真实均值。我们方法的核心在于一种增量学习现象：引入一个简单的非凸框架，可逐步学习均值的前$k$个非零元素，同时使零元素保持任意小。与现有估计量不同，我们的方法无需事先已知稀疏度$k$。通过提供匹配的信息论下界，我们证明了估计量的最优性。最后，通过一系列仿真实验验证了理论结果。代码见https://github.com/huihui0902/Robust_mean_estimation。