List-Decodable Sparse Mean Estimation

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 {\em sparse polynomials} to filter outliers, which may find more applications.