Mean-Shift PCA by Knockoff Mean
Provides a theoretically grounded, simple method for robust PCA under mean-shift contamination, a common issue in high-dimensional data analysis.
Mean-shift contamination in PCA causes large deviations in leading components. The authors prove spectral separability of mean-shift spikes and propose a two-stage PCA algorithm that removes them using knockoff mean perturbation, achieving robust recovery without specialized robust PCA methods.
Removing noise is difficult, but adding noise is easy. In this work, we show how to eliminate mean-shift noisy components from PCA by deliberately introducing knockoff mean-shift perturbation. Standard PCA is highly sensitive to shifts in the sample mean: a small fraction of samples from a shifted distribution can cause large deviations in the leading principal components. In high-dimensional regimes, existing Robust PCA approaches cannot handle the mean-shift contamination structure inherent in the mixture model. Using tools from Random Matrix Theory, we prove that the mean-shift spikes are spectrally separable from the stable eigenvalues of the original covariance. Furthermore, the original eigenspace remains asymptotically invariant to the contamination, independent of the mixture weight. Exploiting this spectral stability, we propose a simple, two-stage PCA algorithm by adding knockoff mean that identifies and removes the mean-shift component using only standard PCA operations.