Fast, Exact Bootstrap Principal Component Analysis for p>1 million

Many have suggested a bootstrap procedure for estimating the sampling variability of principal component analysis (PCA) results. However, when the number of measurements per subject ($p$) is much larger than the number of subjects ($n$), the challenge of calculating and storing the leading principal components from each bootstrap sample can be computationally infeasible. To address this, we outline methods for fast, exact calculation of bootstrap principal components, eigenvalues, and scores. Our methods leverage the fact that all bootstrap samples occupy the same $n$-dimensional subspace as the original sample. As a result, all bootstrap principal components are limited to the same $n$-dimensional subspace and can be efficiently represented by their low dimensional coordinates in that subspace. Several uncertainty metrics can be computed solely based on the bootstrap distribution of these low dimensional coordinates, without calculating or storing the $p$-dimensional bootstrap components. Fast bootstrap PCA is applied to a dataset of sleep electroencephalogram (EEG) recordings ($p=900$, $n=392$), and to a dataset of brain magnetic resonance images (MRIs) ($p\approx$ 3 million, $n=352$). For the brain MRI dataset, our method allows for standard errors for the first 3 principal components based on 1000 bootstrap samples to be calculated on a standard laptop in 47 minutes, as opposed to approximately 4 days with standard methods.