List-Decodable Sparse Mean Estimation
This addresses robust statistics for high-dimensional data with adversarial corruptions, offering a novel algorithmic solution with improved efficiency.
The paper tackles the problem of list-decodable sparse mean estimation for Gaussian distributions with k-sparse means, achieving a polynomial-time algorithm with sample complexity O(poly(k, log d)), which is poly-logarithmic in dimension.
Robust mean estimation is one of the most important problems in statistics: given a set of samples in $\mathbb{R}^d$ where an $α$ 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 $α\in (0, \frac12]$, 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.