A Sub-Quadratic 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 \emph{corrupted} set of samples from $\mathcal{N}(\mu,\mathbf{I}_d)$, where the unknown mean $\mu \in \mathbb{R}^d$ is constrained to be $k$-sparse. A series of prior works has developed efficient algorithms for robust sparse mean estimation with sample complexity $\mathrm{poly}(k,\log d, 1/\epsilon)$ and runtime $d² \mathrm{poly}(k,\log d,1/\epsilon)$, where $\epsilon$ is the fraction of contamination. In particular, the fastest runtime of existing algorithms is quadratic ($\Omega(d^2)$), 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 $d^2$. Our main contribution is an algorithm for robust sparse mean estimation which runs in \emph{subquadratic} time using $\mathrm{poly}(k,\log d,1/\epsilon)$ samples. We also provide analogous 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.