Information Technology Reference
In-Depth Information
According to [19], formulas (2) can be treated as "binding energies" between
"formal proteins"
Z
("antigen") and
X
k
("antibodies").
2.4 Mathematical Properties of SFIN
Proposition 1
. SFIN's projection
P
=
(
p
,...,
p
)
of any training pattern
Z
=
A
,
1
q
i
=
1
,...,
m
, lies within unit cube:
max{|
p
|,...,
|
p
|}
≤
1
.
1
q
'
'
Let
Z
=
A
,
i
=
{
,...,
m
}
. Then, according to SVD (1):
'
1
'
2
'
3
'
Z
=
s
[
Y
]
X
+
s
[
Y
]
X
+
s
[
Y
]
X
+
...
+
s
[
Y
]
X
,
i
i
i
r
r
i
r
1
1
2
2
3
3
where
[
Y
]
is
i
-th coordinate of left singular vector
Y
. Multiply both parts of this
i
'
'
equation by
X
:
Z
X
=
s
[
Y
]
, because
X
k
X
=
0
,
i
≠
k
. Substitution of this
k
k
k
k
i
i
'
result to (2) leads to
p
=
[
Y
]
. Thus,
Y
k
Y
=
1
proves the proposition.
k
k
i
k
Proposition 2
. SFIN
W
without Apoptosis and Immunization recognizes any
training pattern by zero Euclidian distance.
Let
W
=
(
1
V
,...,
V
)
be SFIN corresponded to training patterns
A
,...,
1
A
. Let
0
m
'
'
P
=
(
p
,...,
p
)
. Let
Z
=
A
,
i
=
{
,...,
m
}
. Then, according to the proof of
i
1
i
qi
Proposition 1,
p
=
[
Y
]
=
p
,
k
=
1
,...,
q
. Thus,
d
=
0
, which proves given
k
k
i
ki
ii
Proposition 2.
3 Software Implementation
Based on the above mathematical model of SFIN, consider a description (in a
pseudocode) of the IC algorithm of pattern recognition:
Training
{
1st stage training // map data to SFIN
{
Get training patterns;
Form training matrix;
Compute SVD of the training matrix;
Store q singular values // "binding energies"
Store q right singular vectors; // "antibody-probes"
Store left singular vectors; // cells of SFIN
}
2nd stage training // compress data by SFIN
{ // compute for all cells of SFIN:
Apoptosis;
Immunization;
}
}
