Information Technology Reference
In-Depth Information
a
b
2D Hermite basis function
2D Hermite basis function
1
1
0.5
0.5
0
0
−0.5
−0.5
−1
−1
10
10
5
10
5
10
5
5
0
0
0
0
y
−5
y
−5
−5
−5
x
x
−10
−10
−10
−10
Fig. 12.3
2D Neural Network based on the QHO eigenstates: (
a
) basis function B
3;3
.x;˛/ (
b
)
basis function B
4;1
.x;˛/
These two-dimensional basis functions are again orthonormal, i.e. it holds
Z
d
2
xB
n
.x;˛/B
m
.x;˛/ D ı
n
1
m
1
ı
n
2
m
2
(12.17)
The basis functions B
k
1
;k
2
.x/ are the eigenstates of the two-dimensional harmonic
oscillator, which is a generalization of Eq. (
12.5
). These basis functions form
a complete basis for integrable functions of two variables. A two-dimensional
function f.x/can thus be written in the series expansion:
1
X
f.x/D
c
k
B
k
1
;k
2
.x;˛/
(12.18)
k
1
;k
2
The choice of an appropriate scale coefficient ˛ and of the maximum order
k
max
1
;k
ma
2
is a practical issue. Indicative basis functions B
1;2
.x;˛/, B
3;3
.x;˛/ and
B
4;1
.x;˛/ of a 2D feed-forward quantum neural network are depicted in Fig.
12.2
b,
and Fig.
12.3
a,b, respectively.
Remark.
The significance of the results of Sect.
12.4
is summarized in the sequel:
(i) Orthogonality of the basis functions and invariance under the Fourier trans-
form, (subject only to a change of scale): this means that the energy distribution
in the proposed neural network can be estimated without moving to the
frequency domain. The values of the weights of the neural network provide
a measure of how energy is distributed in the various modes ˇ
k
.x;˛/ of the
signal that is approximated by the neural network.