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.

翻译：我们探索外部-有机燃烧高维统计与在宽度制约下非冷却优化之间的关联,重点是强势稀薄平均估计和强势稀薄的五氯苯甲醚的基本任务。我们为这些问题开发了新颖和简单的优化配方,使相关优化问题的任何近似固定点都能为潜在的稳健估算任务找到近乎最佳的解决方案。作为推论,我们了解到,任何一阶方法,只要与稳态有效结合,就能为这些任务找到高效的算法。获得的算法是简单、实用的,在比以往工作更广泛的分配假设下取得成功。