Statistical Optimality and Computational Efficiency of Nystr\"om Kernel PCA

TL;DR

This paper analyzes the trade-off between computational efficiency and statistical accuracy in Nyström kernel PCA, demonstrating it matches the full KPCA's performance and outperforms random feature approximations.

Contribution

It provides a theoretical analysis showing Nyström KPCA retains statistical performance while reducing computational costs, and compares it favorably against random feature methods.

Findings

Nyström KPCA matches the statistical performance of full KPCA.

Nyström KPCA is computationally more efficient than full KPCA.

Nyström KPCA outperforms random feature approximation in statistical behavior.

Abstract

Kernel methods provide an elegant framework for developing nonlinear learning algorithms from simple linear methods. Though these methods have superior empirical performance in several real data applications, their usefulness is inhibited by the significant computational burden incurred in large sample situations. Various approximation schemes have been proposed in the literature to alleviate these computational issues, and the approximate kernel machines are shown to retain the empirical performance. However, the theoretical properties of these approximate kernel machines are less well understood. In this work, we theoretically study the trade-off between computational complexity and statistical accuracy in Nystr\"om approximate kernel principal component analysis (KPCA), wherein we show that the Nystr\"om approximate KPCA matches the statistical performance of (non-approximate) KPCA while remaining computationally beneficial. Additionally, we show that Nystr\"om approximate KPCA outperforms the statistical behavior of another popular approximation scheme, the random feature approximation, when applied to KPCA.