A Subquadratic Time Algorithm for Robust Sparse Mean Estimation

We study the algorithmic problem of sparse mean estimation in the presence of adversarial outliers. Specifically, the algorithm observes a corrupted set of samples from N(µ, Id), where the unknown mean µ ∈ Rd is constrained to be k-sparse. A series of prior works has developed efficient algorithms for robust sparse mean estimation with sample complexity poly(k, log d, 1/ϵ) and runtime d2poly(k, log d, 1/ϵ), where ϵ is the fraction of contamination. In particular, the fastest runtime of existing algorithms is quadratic in the dimension, which can be prohibitive in high dimensions. This quadratic barrier in the runtime stems from the reliance of these algorithms on the sample covariance matrix, which is of size d2. Our main contribution is an algorithm for robust sparse mean estimation which runs in subquadratic time using poly(k, log d, 1/ϵ) samples, with similar results for robust sparse PCA. Our results build on algorithmic advances in detecting weak correlations, a generalized version of the light-bulb problem by Valiant (Valiant, 2015).