We study principal component analysis (PCA), where given a dataset in $\mathbb{R}^d$ from a distribution, the task is to find a unit vector $v$ that approximately maximizes the variance of the distribution after being projected along $v$. Despite being a classical task, standard estimators fail drastically if the data contains even a small fraction of outliers, motivating the problem of robust PCA. Recent work has developed computationally-efficient algorithms for robust PCA that either take super-linear time or have sub-optimal error guarantees. Our main contribution is to develop a nearly-linear time algorithm for robust PCA with near-optimal error guarantees. We also develop a single-pass streaming algorithm for robust PCA with memory usage nearly-linear in the dimension.

翻译：我们研究主成分分析（PCA）问题：给定来自某分布的数据集$\mathbb{R}^d$，目标是找到一个单位向量$v$，使得数据沿$v$方向投影后的方差近似最大化。尽管这是一项经典任务，但若数据中存在少量离群值，标准估计器将彻底失效，这催生了鲁棒PCA问题的研究。近期工作已开发出计算高效的鲁棒PCA算法，但这些算法要么需要超线性时间，要么具有次优的误差保证。我们的主要贡献是提出一种近乎线性时间的鲁棒PCA算法，并实现近最优的误差保证。此外，我们还开发了一种单遍流式鲁棒PCA算法，其内存使用量与维度近乎线性相关。