Gain with no Pain: Efficient Kernel-PCA by Nyström Sampling
This addresses scalability issues in kernel PCA for machine learning practitioners, offering an incremental improvement over existing methods.
The paper tackles the computational inefficiency of kernel PCA for large datasets by proposing a Nyström sampling approach, showing that it maintains statistical accuracy while significantly improving speed, as supported by theoretical and numerical results.
In this paper, we propose and study a Nyström based approach to efficient large scale kernel principal component analysis (PCA). The latter is a natural nonlinear extension of classical PCA based on considering a nonlinear feature map or the corresponding kernel. Like other kernel approaches, kernel PCA enjoys good mathematical and statistical properties but, numerically, it scales poorly with the sample size. Our analysis shows that Nyström sampling greatly improves computational efficiency without incurring any loss of statistical accuracy. While similar effects have been observed in supervised learning, this is the first such result for PCA. Our theoretical findings, which are also illustrated by numerical results, are based on a combination of analytic and concentration of measure techniques. Our study is more broadly motivated by the question of understanding the interplay between statistical and computational requirements for learning.