Nearly-Linear Time and Streaming Algorithms for Outlier-Robust PCA

05/04/2023

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by   Ilias Diakonikolas, et al.

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We study principal component analysis (PCA), where given a dataset in 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.