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## 5.2.2 Cylindrical potential

We use the reconstruction error (Diamantaras and Kung, 1996, p. 45) as a potential field in feature space,

 E() = -  . (5.5)

is a vector originating in the center of the distribution in feature space, () = () - . The matrix contains the q row vectors (q is the number of principal components)5.2.

We need to eliminate in (5.5), and write the potential as a function of a vector taken from the original space. The projection fl() of () onto the eigenvectors = () can be readily evaluated using the kernel function k,

 fl() = ()T = () - ()() - () = k(,) - k(,) - k(,) + k(,) . (5.6)

The second equality uses (5.1). As a result, E() can be expressed as

 E() = - fl()2 . (5.7)

The scalar product equals the potential field of a sphere (5.3). Thus, the expression of the potential E() can be further simplified to

 E() = ES() - fl()2 , (5.8)

which is the desired form of the cylindrical potential.

The above computation of fl() requires n evaluations of the kernel function for each . Since, for each component l, the same kernel can be used, the total number of kernel evaluations is also n. With the speed-up described in appendix B.2, this number can be reduced to m. Here, the expression () is estimated by (), and 1/n() by (). Doing these replacements, the equation for fl() (5.6) can be approximated by

 fl() = -   k(,) - k(,) + k(,) . (5.9)

The last two terms and do not need to be evaluated for each , but can be computed beforehand. Therefore, the computation is dominated by the m evaluations of the kernel function, which need to be carried out only once for all eigenvectors l.

Subsections

Next: 5.2.2.1 Examples Up: 5.2 Pattern association algorithm Previous: 5.2.1 Spherical potential
Heiko Hoffmann
2005-03-22