2.4.1 Feature extraction

The principal components are not directly accessible because $\varphi$($\bf x$) is not known. However, projections onto the components can be computed
(Schölkopf et al., 1998b). A projection f of a pattern $\bf z$ in the original space onto a principal component in feature space can be computed as follows:

$$\varphi \bf z \bf w \sum_{{i=1}}^{n} \alpha_{i}^{} \bf z \bf x_{i}^{}$$ (2.30)

The computational load for each projection onto a principal component is high, n evaluations of k($\bf z$,$\bf x_{i}^{}$) are needed. In appendix B.2, a speed-up is described that uses a reduced set of m < n patterns, instead of {$\bf x_{i}^{}$}. This reduces the computation time by the factor m/n (Schölkopf et al., 1998a).