  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.
