Faster Principal Component Regression and Stable Matrix Chebyshev Approximation

We solve principal component regression (PCR), up to a multiplicative accuracy 1+γ, by reducing the problem to tildeO(γ-1) black-box calls of ridge regression. Therefore, our algorithm does not require any explicit construction of the top principal components, and is suitable for large-scale PCR instances. In contrast, previous result requires tildeO(γ-2) such black-box calls. We obtain this result by developing a general stable recurrence formula for matrix Chebyshev polynomials, and a degree-optimal polynomial approximation to the matrix sign function. Our techniques may be of independent interests, especially when designing iterative methods.