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0
.
25]
T
.TheBayes
optimal min
P
e
values were computed with the formulas provided by [191].
The experimental error rate results obtained with 2000 instances per class
and 65 epochs training are again in close agreement with the theoretical
values.
Similar experiments found the arctan perceptron with Shannon
EE
risk
having an analogous behavior to that of the arctan perceptron with qua-
dratic Rényi
EE
risk. The arctan perceptrons also behave properly when
applied to real-world datasets; performance figures of MLPs using these per-
ceptrons (and of other types as well) applied to classification tasks of real-
world datasets are presented later, in Sect. 6.1.3.
In this case the optimal
w
1
-normalized weight vector is [1
−
Results of the
H
R
2
Tabl e 3 . 3
arctan perceptron applied to Gaussian data with
unit covariance (see text).
μw
1
w
2
w
0
P
e
min
P
e
0.2 1 -0.0049 -0.1639 0.4323 0.4207
0.5 1 -0.0074 0.0043 0.2990 0.3085
1 1 -0.0433 -0.0231 0.1570 0.1597
1.5 1 -0.0100 0.0029 0.0673 0.0668
2
1 -0.0408 -0.0180 0.0220 0.0228
Tabl e 3 . 4
Results of the
H
R
2
arctan perceptron applied to Gaussian data with
covariance (3.69) (see text).
μw
1
w
2
w
0
P
e
min
P
e
0.2 1 -0.2592 -0.0165 0.4000 0.4253
0.5 1 -0.2629 0.0642 0.3040 0.2965
1 1 -0.2387 -0.0853 0.1470 0.1425
1.5 1 -0.2445 0.0081 0.0517 0.0544
2
1 -0.2117 -0.0155 0.0143 0.0163
We conclude the present section with the presentation of Table 3.5, provid-
ing a synoptic comparison of perceptrons with the three activation functions
that we analyzed, regarding the relation of min
P
e
to critical points of the
error entropy, and this for both the theoretical and empirical settings. For
the empirical setting, besides the experiments described in this and preced-
ing sections, we also took into consideration a large amount of experimental
evidence derived from the application of MLPs to solving real-world classifi-
cation tasks, part of which are reported in Chap. 6.
