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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:

f = $\displaystyle \varphi$($\displaystyle \bf z$)T$\displaystyle \bf w$ = $\displaystyle \sum_{{i=1}}^{n}$$\displaystyle \alpha_{i}^{}$k($\displaystyle \bf z$,$\displaystyle \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).

Heiko Hoffmann