102
5 Sampling
By the way, by extending the Ising model, we will find the Hamiltonian of the
Hopfield model which appears in the context of machine learning (see Chap. 10).
First, consider the Edwards–Anderson model, which is a slightly extended version
of the Ising model. The coupling constant of the nearest neighbors depends on the
location of the spins,
H [s] = −
i,j
k i,j s i s j − B
i
s i .
(5.85)
When viewed as a spinglass model, the Gaussian mean is typically taken for the
coupling constant after taking the state sum for s. Not only can we consider the
nearest neighbor dependence of the coupling constant, we can also consider a model
with a coordinate-dependent magnetic field,
H [s] = −
i,j
k i,j s i s j −
i
B i s i .
(5.86)
This is called the energy function of the Hopfield model. 30 This is the Hamiltonian
used for Boltzmann machines. This form of Hamiltonian also appears in the
description of spinglass models.
Here we describe the difference between the viewpoints of statistical mechanics
and machine learning. In statistical mechanics, the magnetization M, which is
the spatial average of spins, for a given inverse temperature β and a given external
magnetic field B, is obtained. On the other hand, in machine learning, we have
to find k i,j that reproduces a given s, which is exactly an inverse problem. These
inverse problems are generally described in Chap. 7.
Now, let us double the number of species of spins and call them v and h,
H [v, h] = −
i,j
k
vv
i,j v i v j −
i
B
v
i v i −
i,j
k
hh
i,j h i h j −
i
B
h
i h i −
i,j
k
vh
i,j v i h j .
(5.87)
k vv is the coupling between v spins, k hh is the coupling between h spins, k vh is the
coupling between the spin v and the spin h. B is the coupling of the external field to
each spin. This model also has a name, the Boltzmann machine, and will appear in
the next chapter. To avoid some difficulties, the restricted Boltzmann machine with
k vv
i,j = k hh
i,j = 0 is actually used. The details will be explained in the next chapter.
In this way, although the Ising model is a simple model, it has become a source of
ideas and applications for machine learning.
30 Strictly speaking, it is necessary to impose Hebb’s rule and a symmetry on the coupling constant
k i,j , but we do not get into the details here.
5 Sampling
By the way, by extending the Ising model, we will find the Hamiltonian of the
Hopfield model which appears in the context of machine learning (see Chap. 10).
First, consider the Edwards–Anderson model, which is a slightly extended version
of the Ising model. The coupling constant of the nearest neighbors depends on the
location of the spins,
H [s] = −
i,j
k i,j s i s j − B
i
s i .
(5.85)
When viewed as a spinglass model, the Gaussian mean is typically taken for the
coupling constant after taking the state sum for s. Not only can we consider the
nearest neighbor dependence of the coupling constant, we can also consider a model
with a coordinate-dependent magnetic field,
H [s] = −
i,j
k i,j s i s j −
i
B i s i .
(5.86)
This is called the energy function of the Hopfield model. 30 This is the Hamiltonian
used for Boltzmann machines. This form of Hamiltonian also appears in the
description of spinglass models.
Here we describe the difference between the viewpoints of statistical mechanics
and machine learning. In statistical mechanics, the magnetization M, which is
the spatial average of spins, for a given inverse temperature β and a given external
magnetic field B, is obtained. On the other hand, in machine learning, we have
to find k i,j that reproduces a given s, which is exactly an inverse problem. These
inverse problems are generally described in Chap. 7.
Now, let us double the number of species of spins and call them v and h,
H [v, h] = −
i,j
k
vv
i,j v i v j −
i
B
v
i v i −
i,j
k
hh
i,j h i h j −
i
B
h
i h i −
i,j
k
vh
i,j v i h j .
(5.87)
k vv is the coupling between v spins, k hh is the coupling between h spins, k vh is the
coupling between the spin v and the spin h. B is the coupling of the external field to
each spin. This model also has a name, the Boltzmann machine, and will appear in
the next chapter. To avoid some difficulties, the restricted Boltzmann machine with
k vv
i,j = k hh
i,j = 0 is actually used. The details will be explained in the next chapter.
In this way, although the Ising model is a simple model, it has become a source of
ideas and applications for machine learning.
30 Strictly speaking, it is necessary to impose Hebb’s rule and a symmetry on the coupling constant
k i,j , but we do not get into the details here.