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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,
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
f_{l}() of
() onto the eigenvectors
= () can be readily evaluated using the kernel function k,
f_{l}() 
= 
()^{T} 


= 
()  ()()  () 


= 
k(,)  k(,)  k(,) 


+ 
k(,) . 
(5.6) 
The second equality uses (5.1). As a result,
E() can be expressed as
E() =  f_{l}()^{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() = E_{S}()  f_{l}()^{2} , 
(5.8) 
which is the desired form of the cylindrical potential.
The above computation of
f_{l}() 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 speedup 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
f_{l}() (5.6) can be approximated by
f_{l}() 
= 
 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
20050322