136
7 Unsupervised Deep Learning
the potential function V (x) can be constructed by solving a certain integral equation
from these data. 7
In the inverse scattering method, an unknown potential, or Hamiltonian, was
constructed from asymptotic data on scattering and bound states. Then, what about
a system that has only bound states? In this case, the Hamiltonian eigenvalues will
be the known data. Can we reconstruct the Hamiltonian from energy eigenvalues?
There has been a wide variety of such motivated studies for many years. For
example, when exploring superconducting materials, the interaction Hamiltonian
that can produce a superconducting state even at high temperatures is being sought.
In a broad sense, drug discovery is the same type of problem. As can be seen
from these motivations, we need to model the Hamiltonian to some extent and
determine its coefficients from the data. Recent advances in machine learning
research have shown that machine learning has great potential for determining
models and determining coefficients.
The holographic principle in string theory (AdS/CFT correspondence), which
will be described as an example in Chap. 12, is also an inverse problem as a system
determination problem. It can also be a boundary value problem in the sense of
determining gravity theory, the inside (bulk), from the quantum field theory “living”
on the boundary. In the following chapters, we will also introduce how to view the
neural network itself as the time axis of the time evolution of dynamical systems,
and how to view it as an emergent space in the AdS/CFT correspondence.
A closer look at the relationship between deep learning and physics may reveal
solutions to the “bottleneck” of inverse problems in various physics problems. Formulating the inverse problem is essentially the way of discovering the fundamentals
of science. So it is natural to expect that the discovery of new laws will come from
revealing the relationship between machine learning and physics.
Column: Sparse Modeling
In this column, let us take a look at a technique called sparse modeling. Sparse
modeling is a research field of image sensing such as the one for medical purposes 8
and has been used for nearly 20 years. Some readers may have heard that recently it
was used for the “method that captures the shadow of a black hole” [103]. By using
this method, one can extract beautiful images from noisy data. 9
We take a closer look at Tikhonov’s regularization method described in this
chapter. Generally, an equation system in which the number of unknowns is larger
than the number of equations is called an underdetermined system. Consider the
7 It is known that V (x) cannot be constructed only with an S matrix.
8 For example, it is used for improving MRI (magnetic resonance imaging) with higher resolution.
9 In the case of the observation that captured the “shape” of a black hole (called “black hole
shadow”), the Fourier components were the observed value and the image data was the output.
7 Unsupervised Deep Learning
the potential function V (x) can be constructed by solving a certain integral equation
from these data. 7
In the inverse scattering method, an unknown potential, or Hamiltonian, was
constructed from asymptotic data on scattering and bound states. Then, what about
a system that has only bound states? In this case, the Hamiltonian eigenvalues will
be the known data. Can we reconstruct the Hamiltonian from energy eigenvalues?
There has been a wide variety of such motivated studies for many years. For
example, when exploring superconducting materials, the interaction Hamiltonian
that can produce a superconducting state even at high temperatures is being sought.
In a broad sense, drug discovery is the same type of problem. As can be seen
from these motivations, we need to model the Hamiltonian to some extent and
determine its coefficients from the data. Recent advances in machine learning
research have shown that machine learning has great potential for determining
models and determining coefficients.
The holographic principle in string theory (AdS/CFT correspondence), which
will be described as an example in Chap. 12, is also an inverse problem as a system
determination problem. It can also be a boundary value problem in the sense of
determining gravity theory, the inside (bulk), from the quantum field theory “living”
on the boundary. In the following chapters, we will also introduce how to view the
neural network itself as the time axis of the time evolution of dynamical systems,
and how to view it as an emergent space in the AdS/CFT correspondence.
A closer look at the relationship between deep learning and physics may reveal
solutions to the “bottleneck” of inverse problems in various physics problems. Formulating the inverse problem is essentially the way of discovering the fundamentals
of science. So it is natural to expect that the discovery of new laws will come from
revealing the relationship between machine learning and physics.
Column: Sparse Modeling
In this column, let us take a look at a technique called sparse modeling. Sparse
modeling is a research field of image sensing such as the one for medical purposes 8
and has been used for nearly 20 years. Some readers may have heard that recently it
was used for the “method that captures the shadow of a black hole” [103]. By using
this method, one can extract beautiful images from noisy data. 9
We take a closer look at Tikhonov’s regularization method described in this
chapter. Generally, an equation system in which the number of unknowns is larger
than the number of equations is called an underdetermined system. Consider the
7 It is known that V (x) cannot be constructed only with an S matrix.
8 For example, it is used for improving MRI (magnetic resonance imaging) with higher resolution.
9 In the case of the observation that captured the “shape” of a black hole (called “black hole
shadow”), the Fourier components were the observed value and the image data was the output.