  In this paper, we propose and study a Nystr\"om 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 Nystr\"om 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.
