Outlier-Robust Sparse Estimation via Non-Convex Optimization
It addresses outlier-robust high-dimensional statistics for machine learning and data analysis, offering practical improvements over prior methods.
The paper tackles robust sparse mean estimation and robust sparse PCA by developing non-convex optimization formulations, showing that approximate stationary points yield near-optimal solutions and enabling efficient algorithms with broader distributional assumptions.
We explore the connection between outlier-robust high-dimensional statistics and non-convex optimization in the presence of sparsity constraints, with a focus on the fundamental tasks of robust sparse mean estimation and robust sparse PCA. We develop novel and simple optimization formulations for these problems such that any approximate stationary point of the associated optimization problem yields a near-optimal solution for the underlying robust estimation task. As a corollary, we obtain that any first-order method that efficiently converges to stationarity yields an efficient algorithm for these tasks. The obtained algorithms are simple, practical, and succeed under broader distributional assumptions compared to prior work.