Information Technology Reference
In-Depth Information
the time derivatives can be expressed, except for a constant 2, as
1
w (
t
)
!
2
2
−
2
k
for LUO
4
1
w (
t
)
dV
(w (
t
))
2
2
−
2
k
=
for OJAn
(2.95)
6
"
dt
1
w (
2
2
−
2
k
for EXIN
8
t
)
Integrating the equations with respect to the time gives the same results as in the
first proof.
Remark 63 (OJA's Time Constant)
OJA has the same time constant as OJAn
[
195, App., p. 455
]
.
Remark 64 (Low Initial Weights)
From Theorem 62 it may seem that OJAn
and LUO can give better results than MCA EXIN by choosing large initial con-
ditions. Unfortunately, this choice is not good because of the flatness of the RQ
landscape
(
seeRemark56
)
, the too large fluctuations for FENG
(
see below
)
and
the fact that it is to difficult to stop the algorithm reliably
(
see the divergence
analysis
)
for all the other learning laws.
According to the analysis above, the best choice for MCA EXIN would be
null initial conditions. However, as can easily be checked, this is not possible.
Not only, but too low weights may generate too high oscillations of the weights
[see eq. (2.35)]. It is shown in [24] and in the next chapters that the MCA neuron
can be endowed with a particular scheduling (DLS scheduling) which allows the
neuron to start with infinitesimal initial conditions and remain for a certain time
with low weights, just following a stable path in the weight phase diagram. This
improved version has been called
MCA EXIN
+
.
Remark 65 (Computation of
λ
n
)
It has been noted experimentally that all
MCA learning laws yield the correct value of
λ
n
well before an accurate estimate
of z
n
.
2.6 DYNAMICS OF THE MCA NEURONS
The purpose of this section is the analysis of the temporal behavior of all MCA
neurons by using not only the ODE approximation but, above all, the stochastic
discrete laws. Indeed, using only the ODE approximation does not reveal some
of the most important features of these algorithms. For example, it will be
shown that the constancy of the weight modulus [see eq. (2.27)] for OJAn, LUO,
and MCA EXIN, which is a consequence of the use of the ODE, is not valid
except, as a very first approximation, in approaching the minor component (see
Search WWH ::
Custom Search