Computationally Efficient Robust Sparse Estimation in High Dimensions

Many conventional statistical procedures are extremely sensitive to seemingly minor deviations from modeling assumptions. This problem is exacerbated in modern high-dimensional settings, where the problem dimension can grow with and possibly exceed the sample size. We consider the problem of robust estimation of sparse functionals, and provide a computationally and statistically efficient algorithm in the high-dimensional setting. Our theory identifies a unified set of deterministic conditions under which our algorithm guarantees accurate recovery. By further establishing that these deterministic conditions hold with high-probability for a wide range of statistical models, our theory applies to many problems of considerable interest including sparse mean and covariance estimation; sparse linear regression; and sparse generalized linear models. In certain settings, such as the detection and estimation of sparse principal components in the spiked covariance model, our general theory does not yield optimal sample complexity, and we provide a novel algorithm based on the same intuition which is able to take advantage of further structure of the problem to achieve nearly optimal rates.