4.1 Convolutional Neural Network
59
Fig. 4.3 Schematic diagram
of the coupling constant (4.5).
The variables here are only
the degrees of freedom J αβ
x ij
d IJ
J IJ,ij
Also, if the feature of the image we want to capture is “cat-like,” that feature
should not depend on the pixel coordinates I, J of the image. This is because
sometimes a cat appears in the upper right corner of the photo, and sometimes in the
middle. Then, it is natural for the purpose of (4.1) not to include the I J dependence
in the coupling constant. This reduces the coupling constant to the form
J I J,ij =
αβ
J αβ δ i,s 1 I +α δ j,s 2 J +β .
(4.5)
See Fig. 4.3. Then, it leads to a Hamiltonian,
H
conv
J,x ({d I J }) = −
I J
d I J
αβ
J αβ x s 1 I +α,s 2 J +β + J
.
(4.6)
Calculating the Boltzmann weights with this Hamiltonian, we find
Q J ({d I J = 1}|x) = σ
αβ
J αβ x s 1 I +α,s 2 J +β + J
.
(4.7)
The operation
x ij →
αβ
J αβ x s 1 I +α,s 2 J +β
(4.8)
is called a convolution, 1 and a neural network that includes a convolution operation
is called a convolutional neural network. It was first introduced in [36]. 2 As
1 This convolution is essentially the same as that used in mathematical physics, although the signs
may be different.
2 It was written in [36] that the author got the idea of the convolutional neural network from the
paper by Hubel and Wiesel [37] which pointed out that there are two types of cells in the visual
cortex, and from the paper by Fukushima and Miyake which implemented those to a neural network
[38].
59
Fig. 4.3 Schematic diagram
of the coupling constant (4.5).
The variables here are only
the degrees of freedom J αβ
x ij
d IJ
J IJ,ij
Also, if the feature of the image we want to capture is “cat-like,” that feature
should not depend on the pixel coordinates I, J of the image. This is because
sometimes a cat appears in the upper right corner of the photo, and sometimes in the
middle. Then, it is natural for the purpose of (4.1) not to include the I J dependence
in the coupling constant. This reduces the coupling constant to the form
J I J,ij =
αβ
J αβ δ i,s 1 I +α δ j,s 2 J +β .
(4.5)
See Fig. 4.3. Then, it leads to a Hamiltonian,
H
conv
J,x ({d I J }) = −
I J
d I J
αβ
J αβ x s 1 I +α,s 2 J +β + J
.
(4.6)
Calculating the Boltzmann weights with this Hamiltonian, we find
Q J ({d I J = 1}|x) = σ
αβ
J αβ x s 1 I +α,s 2 J +β + J
.
(4.7)
The operation
x ij →
αβ
J αβ x s 1 I +α,s 2 J +β
(4.8)
is called a convolution, 1 and a neural network that includes a convolution operation
is called a convolutional neural network. It was first introduced in [36]. 2 As
1 This convolution is essentially the same as that used in mathematical physics, although the signs
may be different.
2 It was written in [36] that the author got the idea of the convolutional neural network from the
paper by Hubel and Wiesel [37] which pointed out that there are two types of cells in the visual
cortex, and from the paper by Fukushima and Miyake which implemented those to a neural network
[38].