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4.2 The Continuous Associative Completion 47

Then ii use the surface point w s as the output of the PSOM in response
to the input x.
To build an input-output mapping, the standard SOM is often extended
by attaching a second vector w out to each formal neuron. Here, we gener-
alize this and view the embedding space X as the Cartesian product of the
input subspace X in and the output subspace X out

d
X X in
X out IR (4.5)

Then, w s can be viewed as an associative completion of the input space
component of x if the distance function dist (in Eq. 4.4) is chosen as the
Euclidean norm applied only to the input components of x (belonging to
in
in
X ). Thus, the function dist actually selects the input subspace X ,

since for the determination of s (Eq. 4.4) and, as a consequence, of w s ,
only those components of x matter, that are regarded in the distance met-
ric dist . The mathematical formulation is the definition of a diagonal
projection matrix

P diag p p d p (4.6)
with diagonal element p k k I, and all other elements zero. The set

I is the subset of components of X (f
g) belonging to the ddesired

in
X . Then, the distance function can be written as
d

T

dist x x x x P x x X p k x k x X p k x k x (4.7)
k
k
k I k

For example, consider a d dimensional embedding space X, where
the components I f
g belong to the input space. Only those must
be specified as inputs to the PSOM:

x
missing
B C
B C
B C
x B C components (4.8)
B x
C
B C
desired
B C
x
A
output
The next step is the costly part in the PSOM operation: the iterative

“best-match” search for the parameter space location s , Eq. 4.4 (see next
section.) In our example Eq. 4.8, the distance metric Eq. 4.7 is specified