138
7 Unsupervised Deep Learning
Here, |v| 1 is the sum of the absolute values of the components, |v| 1 = |v 1 | + |v 2 | +
· · · for the vector v. This is a regression called LASSO (least absolute shrinkage and
selection operator). This regression is known to have no computational difficulties
and to have better properties than the ridge regression.
The problem of solving simultaneous equations can occur in various situations,
and one example is when two quantities are related by an integral transformation.
This is indeed the case for image sensing and black hole shadows. In the context
of physics, the energy spectrum ρ(ω) and the Green’s function G(τ ) are typically
related by the following equation: the Fourier transform,
G(τ ) =
dωK(τ, ω)ρ(ω).
(7.16)
Discretizing time τ and energy ω and considering the integral as Riemann sum, this
can be regarded as the following equation of a vector G, ρ and a matrix K:
G = Kρ .
(7.17)
We want to know ρ(ω) from G(τ ). In actual calculations, as for the Green function
G(τ ) only about 10 points are known, while as its energy spectrum ρ, we want to
obtain more than 1000 points, for example. So, this is an underdetermined system, 11
and you can use LASSO, which we described above. 12 For more details, we suggest
that readers look at [105].
11 In the case of black hole shadows in the EHT (event horizon telescope) project, we know peaks
at several values of energy as data, while we want an image that is the inverse Fourier transform.
Thus essentially this is the same as the above.
12 For Green’s function, it is necessary to perform a singular value decomposition and move to a
base where essential data is easy to see.
7 Unsupervised Deep Learning
Here, |v| 1 is the sum of the absolute values of the components, |v| 1 = |v 1 | + |v 2 | +
· · · for the vector v. This is a regression called LASSO (least absolute shrinkage and
selection operator). This regression is known to have no computational difficulties
and to have better properties than the ridge regression.
The problem of solving simultaneous equations can occur in various situations,
and one example is when two quantities are related by an integral transformation.
This is indeed the case for image sensing and black hole shadows. In the context
of physics, the energy spectrum ρ(ω) and the Green’s function G(τ ) are typically
related by the following equation: the Fourier transform,
G(τ ) =
dωK(τ, ω)ρ(ω).
(7.16)
Discretizing time τ and energy ω and considering the integral as Riemann sum, this
can be regarded as the following equation of a vector G, ρ and a matrix K:
G = Kρ .
(7.17)
We want to know ρ(ω) from G(τ ). In actual calculations, as for the Green function
G(τ ) only about 10 points are known, while as its energy spectrum ρ, we want to
obtain more than 1000 points, for example. So, this is an underdetermined system, 11
and you can use LASSO, which we described above. 12 For more details, we suggest
that readers look at [105].
11 In the case of black hole shadows in the EHT (event horizon telescope) project, we know peaks
at several values of energy as data, while we want an image that is the inverse Fourier transform.
Thus essentially this is the same as the above.
12 For Green’s function, it is necessary to perform a singular value decomposition and move to a
base where essential data is easy to see.