  Given a data matrix $\mathbf{A} \in \mathbb{R}^{n \times d}$, principal
component projection (PCP) and principal component regression (PCR), i.e.
projection and regression restricted to the top-eigenspace of $\mathbf{A}$, are
fundamental problems in machine learning, optimization, and numerical analysis.
In this paper we provide the first algorithms that solve these problems in
nearly linear time for fixed eigenvalue distribution and large n. This improves
upon previous methods which have superlinear running times when both the number
of top eigenvalues and inverse gap between eigenspaces is large. We achieve our
results by applying rational approximations to reduce PCP and PCR to solving
asymmetric linear systems which we solve by a variant of SVRG. We corroborate
these findings with preliminary empirical experiments.
