Information Technology Reference
In-Depth Information
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0.0
0.2
0.4
0.6
0.8
1.0
X
Fig. 4.
Banana data set (points from class
C
+
),
min-max
normalized to [0
,
1]
2
.Inan
perfect case (error-less detection), the
r
-chunk detectors should cover regions outside
the “banana” shape. The region within the “banana” shape is the generalization region
and should consists of undetectable elements, i.e. holes and self elements.
for both data sets, holes occur as they should in generalization regions when
8
10. This phenomena is discussed and explained in [8]. To summarize
results from [8], a detector matching length which is not at least as long as the
semantical representation of the underlying data — in this case 8 bits for
x
and
y
coordinates — results in incorrect generalization regions.
What is more interesting though, is the observation that a (randomly deter-
mined) permutation mask shatters the semantical representation of the under-
lying data (see Fig. 5-8 (b,d,f,h,j,l,n,p,r,t)) and therefore, holes are randomly
distributed across the space instead of being concentrated inside or close to self
regions. This observation also means that detectors are not covering areas around
the self regions, instead they recognize elements which are also randomly dis-
tributed across the space. Furthermore one can see that the number of holes
— when applying permutation masks (see Fig. 5-8 (b,d,f,h,j,l,n,p,r,t)) — is in
some cases significantly higher than without permutation masks (see Fig. 5-8
(a,c,d,e,g,i,k,m,q,s)). This observation could be explained with the previous ob-
servation, that permutation masks distort the underlying data and therefore
shatter self regions. As a consequence the underlying data is transformed into a
collection of random chunks. For randomly determined self elements, Stibor et
al. [6] showed that the number of holes increase exponentially for
r
:=
l
≤
r
≤
0.
Of course this shattering effect is linked very strongly to the mapping function
employed. However it is clear that each permutation mask — except the identity
permutation — semantically (more or less) distort the data. Furthermore, we
believe that finding a permutation mask which does not significantly distort the
semantical representation of the data may be computational intractable
6
.
→
6
In the worst-case, one have to check all
n
! permutations of
S
n
.
