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.
Search WWH ::




Custom Search