# Faster Principal Component Regression and Stable Matrix Chebyshev Approximation

arXiv: https://arxiv.org/abs/1608.04773 · HTML: https://mlsanity.com/paper/1608.04773/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Aug 2016.

## Built on by (most cited first, 3 of 3)

- [Zeroth-Order Negative Curvature Finding: Escaping Saddle Points without Gradients](https://mlsanity.com/paper/2210.01496/) (2022)
- [Minimax Optimal Regression over Sobolev Spaces via Laplacian Eigenmaps on Neighborhood Graphs](https://mlsanity.com/paper/2111.07394/) (2021)
- [De-floored Principal Component Regression: When Rank Selection Alone Is Insufficient for Prediction](https://mlsanity.com/paper/2607.16638/) (2026)

## Abstract (arXiv)

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.