Part of Advances in Neural Information Processing Systems 35 (NeurIPS 2022) Main Conference Track
Shiwei Zeng, Jie Shen
Robust mean estimation is one of the most important problems in statistics: given a set of samples in Rd 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∈(0,12], 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(poly(k,logd)), 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.