  We introduce primed-PCA (pPCA), a two-step algorithm for speeding up the
approximation of principal components. This algorithm first runs any
approximate-PCA method to get an initial estimate of the principal components
(priming), and then applies an exact PCA in the subspace they span. Since this
subspace is of small dimension in any practical use, the second step is
extremely cheap computationally. Nonetheless, it improves accuracy
significantly for a given computational budget across datasets. In this setup,
the purpose of the priming is to narrow down the search space, and prepare the
data for the second step, an exact calculation. We show formally that pPCA
improves upon the priming algorithm under very mild conditions, and we provide
experimental validation on both synthetic and real large-scale datasets showing
that it systematically translates to improved performance. In our experiments
we prime pPCA by several approximate algorithms and report an average speedup
by a factor of 7.2 over Oja's rule, and a factor of 10.5 over EigenGame.
