Adversarial principal component analysis

Abstract

This paper studies the following question: where should an adversary place an outlier of a given magnitude in order to maximize the error of the subspace estimated by PCA? We give the exact location of this worst possible outlier, and the exact expression of the maximum possible error. Equivalently, we determine the information-theoretic bounds on how much an outlier can tilt a subspace in its direction. This in turn provides universal (worst-case) error bounds for PCA under arbitrary noisy settings. Our results also have several implications on adaptive PCA, online PCA, and rank-one updates. We illustrate our results with a subspace tracking experiment.