Faster Principal Component Regression and Stable Matrix Chebyshev Approximation

Zeyuan Allen-Zhu; Yuanzhi Li

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arXiv · 1608.04773

Faster Principal Component Regression and Stable Matrix Chebyshev Approximation

Zeyuan Allen-Zhu·Yuanzhi Li

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Abstract

We solve principal component regression (PCR), up to a multiplicative accuracy $1+\gamma$, by reducing the problem to $\tilde{O}(\gamma^{-1})$ 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 $\tilde{O}(\gamma^{-2})$ 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.

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BibTeX RIS

Zeyuan Allen-Zhu, Yuanzhi Li. 2016-08-16. Faster Principal Component Regression and Stable Matrix Chebyshev Approximation. https://arxiv.org/abs/1608.04773

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