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0.28
x
2
^
E
(e)
1
0.27
0.26
0
0.25
−1
x
1
e
0.24
0
1
2
3
4
−2
−1
0
1
2
(a) Epochs=0
^
E
(e)
x
2
1
0.295
0
0.29
0.285
−1
x
1
e
0
1
2
3
4
−2
−1
0
1
(b) Epochs=32
0.8
x
2
^
E
(e)
1
0.6
0
0.4
−1
x
1
e
0.2
0
1
2
3
4
−2
−1
0
1
(c) Epochs=40
Fig. 3.11 The two-class circular uniform datasets at different epochs of the MEE
perceptron. The left graphs show the datasets with the linear decision border (solid
line). The right graphs show the error PDF in the
E
=[
−
2
,
2]
support.
Table 3.1 shows other comparisons between the min
P
e
and
P
ed
(300) val-
ues for the two problem types. The min
P
e
value for the circular uniform
distribution can be obtained by straightforward trigonometric derivations;
amazingly, however, the optimal vertical line separating the classes does not
correspond, in general, to the intersection point of the circles, and there is no
closed-form algebraic expression of its distance from the origin. Its value was
determined by optimization search techniques. For the Gaussian distributions
there is also, in general, no closed-form expression, except for equal covariance
