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3.8 Improving the Output of the SOM Schema 41

map encoding (i.e. node Location in the neural array) are advantageous
when the distribution of stochastic transmission errors is decreasing with
distance to the original data. In case of an error the reconstruction will
restore neighboring features, resulting in a more “faithful” compression.
Ritter showed the strict monotonic relationship between the stimulus
density in the m-dimensional input space and the density of the match-
ing weight vectors. Regions with high input stimulus density P x will be
represented by more specialized neurons than regions with lower stimu-
lus density. For certain conditions the density of weight vectors could be

derived to be proportional to P x , with the exponent m
m
(Ritter 1991).

3.8 Improving the Output of the SOM Schema

As discussed before, many learning applications desire continuous valued
outputs. How can the SOM network learn smooth input–output map-
pings?
Similar to the binning in the hyper-rectangular recursive partitioning
algorithm (CART), the original output learning strategy was the super-
vised teaching of an attached constant y a (or vector y a) for every winning
neuron a
F x y a (3.11)

The next important step to increase the output precision was the intro-
duction of a locally valid mapping around the reference vector. Cleve-
land (1979) introduced the idea of locally weighted linear regression for
uni-variate approximation and later for multivariate regression (Cleve-
land and Devlin 1988). Independently, Ritter and Schulten (1986) devel-
oped the similar idea in the context of neural networks, which was later
coined the Local Linear Map (“LLM”) approach.
Within each subregion, the Voronoi cell (depicted in Fig. 3.5), the output
is defined by a tangent hyper-plane described by the additional vector (or
matrix) B
x w a
F x y a B a (3.12)

By this means, a univariate function is approximated by a set of tangents.
In general, the output F x is discontinuous, since the hyper-planes do not
match at the Voronoi cell borders.