  We study the problem of list-decodable sparse mean estimation. Specifically,
for a parameter $\alpha \in (0, 1/2)$, we are given $m$ points in
$\mathbb{R}^n$, $\lfloor \alpha m \rfloor$ of which are i.i.d. samples from a
distribution $D$ with unknown $k$-sparse mean $\mu$. No assumptions are made on
the remaining points, which form the majority of the dataset. The goal is to
return a small list of candidates containing a vector $\widehat \mu$ such that
$\| \widehat \mu - \mu \|_2$ is small. Prior work had studied the problem of
list-decodable mean estimation in the dense setting. In this work, we develop a
novel, conceptually simpler technique for list-decodable mean estimation. As
the main application of our approach, we provide the first sample and
computationally efficient algorithm for list-decodable sparse mean estimation.
In particular, for distributions with "certifiably bounded" $t$-th moments in
$k$-sparse directions and sufficiently light tails, our algorithm achieves
error of $(1/\alpha)^{O(1/t)}$ with sample complexity $m =
(k\log(n))^{O(t)}/\alpha$ and running time $\mathrm{poly}(mn^t)$. For the
special case of Gaussian inliers, our algorithm achieves the optimal error
guarantee of $\Theta (\sqrt{\log(1/\alpha)})$ with quasi-polynomial sample and
computational complexity. We complement our upper bounds with nearly-matching
statistical query and low-degree polynomial testing lower bounds.
