Information Technology Reference
In-Depth Information
♦
⊗
A
=[
∩
,
∩
,
{
h
t
u
,v
w
}]
,
(f)
F
K
P
L
Q
where
a
k
i
,
l
j
,...,
a
k
i
,
l
j
),
h
t
u
,v
w
=
f
p
r
,
q
s
(
for
t
u
=
=
∈
∩
v
w
=
=
q
s
∈
∩
R
.
The so defined IMFEs generate some new ideas. For example, when we have
a sequence of real (complex) functions
f
1
,
k
i
p
r
K
P
and
l
j
L
Q
with elements of
x
f
2
,...,
f
n
∈
F
and sequence of real
(complex) numbers
a
1
,
a
2
,...,
a
n
, then we can calculate sequentially the values
. But, as we saw above, when these functions and these
numbers are elements of IMs, we can calculate them in parallel.
The
open questions
are:
f
1
(
a
1
),
f
2
(
a
2
),...
f
n
(
a
n
)
1. Will we obtain some new (additional) possibilities, if we use IMFEs?
2. To develop a theory of IMFEswith respect to the theory of functions and functional
analysis.
5.5 An Example: IM-Interpretation of a Multilayer Perceptron
The artificial neural networks represent a mathematical model inspired by the bio-
logical neural networks. Its functions are borrowed from the functions of human
brain. There is not yet an uniform opinion on the definition of neural networks, yet
increasingly more specialists share the view that neural networks are a number of
simple connected items, each featuring a rather limited local memory. These items are
connected with connections, transferring numerical data, coded with various tools.
The classical three-layered neural network, in abbreviated notation, has the form
In multilayered networks, the exits of one layer become entries for the next one.
The equations describing this operation are:
a
m
+
1
f
m
+
1
m
+
1
b
m
+
1
a
m
=
(w
.
+
)
for
m
=
0
,
1
,
2
,...,
M
−
1, where:
•
m
is the current number of the layers in the network;
•
M
is the number of the layers in the network;
•
P
is an entry networks vector;
a
m
•
is the exit of the
m
-th layer of the neural network;
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