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

Principal Component Projection and Regression in Nearly Linear Time through Asymmetric SVRG