In this paper, we consider approximations of principal component projection (PCP) without explicitly computing principal components. This problem has been studied in several recent works. The main feature of existing approaches is viewing the PCP matrix as a matrix function. This underlying function is the composition of a step function with a rational function. To find an approximate PCP, the step function is approximated by a polynomial while the rational function is evaluated by a fast ridge regression solver. In this work, we further improve this process by replacing the rational function with carefully constructed polynomials of low degree. We characterize the properties of polynomials that are suitable for approximating PCP, and then establish an optimization problem to select the optimal one from those polynomials. We show theoretically and confirm numerically that the resulting approximate PCP approach with optimal polynomials is indeed effective for approximations of principal component projection.

在本文中，我们考虑了主成分投影（PCP）的近似值，而未明确计算主成分。在最近的几篇著作中已经研究了这个问题。现有方法的主要特征是将PCP矩阵视为矩阵函数。该基本功能是具有理性功能的阶跃功能的组合。为了找到一个近似的PCP，用多项式对阶跃函数进行近似，而用快速岭回归求解器评估有理函数。在这项工作中，我们通过用精心构造的低次多项式代替有理函数来进一步改善此过程。我们表征适合于逼近PCP的多项式的性质，然后建立一个优化问题，从那些多项式中选择最佳的。