  Robust mean estimation is one of the most important problems in statistics:
given a set of samples in $\mathbb{R}^d$ where an $\alpha$ fraction are drawn
from some distribution $D$ and the rest are adversarially corrupted, we aim to
estimate the mean of $D$. A surge of recent research interest has been focusing
on the list-decodable setting where $\alpha \in (0, \frac12]$, and the goal is
to output a finite number of estimates among which at least one approximates
the target mean. In this paper, we consider that the underlying distribution
$D$ is Gaussian with $k$-sparse mean. Our main contribution is the first
polynomial-time algorithm that enjoys sample complexity $O\big(\mathrm{poly}(k,
\log d)\big)$, i.e. poly-logarithmic in the dimension. One of our core
algorithmic ingredients is using low-degree sparse polynomials to filter
outliers, which may find more applications.
