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Neel Shah
NIFTy
Commits
4a44d1e8
Commit
4a44d1e8
authored
Jan 15, 2019
by
Torsten Ensslin
Browse files
documentation done as far as Torsten can do it
parent
3ff172dc
Changes
1
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Inline
Side-by-side
nifty5/operators/energy_operators.py
View file @
4a44d1e8
...
...
@@ -27,31 +27,31 @@ from .simple_linear_operators import VdotOperator
class
EnergyOperator
(
Operator
):
""" Basis class EnergyOperator.
""" Basis class EnergyOperator, an abstract class from which
other specific EnergyOperator subclasses are derived.
Operator which has a scalar domain as target domain.
An EnergyOperator returns a scalar for a field, and a linearized
An EnergyOperator has a scalar domain as target domain.
It turns a field into a scalar and a linearization into a linearization.
It is intended as an objective function for field inference.
Typical usage in IFT:
as an information Hamiltonian ( = negative log probability)
or as a Gibbs free energy ( = averaged Hamiltonian), aka Kullbach-Leibler
divergence.
An EnergyOperator can also provide its gradient as an EndomorphicOperator
that converts a field into a field, the gradient of the Hamiltonian at the
field location.
or as a Gibbs free energy ( = averaged Hamiltonian),
aka Kullbach-Leibler divergence.
"""
_target
=
DomainTuple
.
scalar_domain
()
class
SquaredNormOperator
(
EnergyOperator
):
""" NIFTy class for a squared norm energy.
The NIFTy SquaredNormOperator class derives from the EnergyOperator class.
""" Class for squared field norm energy.
Usage
-----
E = SquaredNormOperator() represents a field energy E that is the L2 norm
of a field f:
A SquaredNormOperator represents a field energy E that is the L2 norm of a
field f: E = f^dagger f
E(f) = f^dagger f
"""
def
__init__
(
self
,
domain
):
self
.
_domain
=
domain
...
...
@@ -66,12 +66,19 @@ class SquaredNormOperator(EnergyOperator):
class
QuadraticFormOperator
(
EnergyOperator
):
""" NIFTy class for quadratic field energies.
""" Class for quadratic field energies.
Parameters
----------
op : EndomorphicOperator
kernel of quadratic form
The NIFTy QuadraticFormOperator derives from the EnergyOperator class.
Usage
-----
E = QuadraticFormOperator(op) represents a field energy that is a
quadratic form in a field f with kernel op:
It represents a field energy E that is a quadratic form of a field f with
kernel op: E = f^dagger op f /2
E(f) = 0.5 f^dagger op f
"""
def
__init__
(
self
,
op
):
from
.endomorphic_operator
import
EndomorphicOperator
...
...
@@ -91,6 +98,28 @@ class QuadraticFormOperator(EnergyOperator):
class
GaussianEnergy
(
EnergyOperator
):
""" Class for energies of fields with Gaussian probability distribution.
Attributes
----------
mean = mean (field) of the Gaussian,
default = 0
covariance = field covariance of the Gaussian,
default = identity operator
domain = domain of field,
default = domain of mean or covariance if specified
One of the attributes has to be specified at instanciation of a GaussianEnergy
to inform about the domain, otherwise an exception is rasied.
Usage
-----
E = GaussianEnergy(mean = m, covariance = D) represents (up to constants)
E(f) = - log G(f-m, D) = 0.5 (f-m)^dagger D^-1 (f-m)
an information energy for a Gaussian distribution with mean m and covariance D.
"""
def
__init__
(
self
,
mean
=
None
,
covariance
=
None
,
domain
=
None
):
self
.
_domain
=
None
if
mean
is
not
None
:
...
...
@@ -127,6 +156,23 @@ class GaussianEnergy(EnergyOperator):
class
PoissonianEnergy
(
EnergyOperator
):
"""Class for likelihood-energies of expected count field constrained by
Poissonian count data.
Parameters
----------
d : Field
data field with counts
Usage
-----
E = GaussianEnergy(d) represents (up to an f-independent term log(d!))
E(f) = -log Poisson(d|f) = sum(f) - d^dagger log(f),
where f is a field in data space (d.domain) with the expectation values for
the counts.
"""
def
__init__
(
self
,
d
):
self
.
_d
=
d
self
.
_domain
=
DomainTuple
.
make
(
d
.
domain
)
...
...
@@ -143,7 +189,11 @@ class PoissonianEnergy(EnergyOperator):
class
InverseGammaLikelihood
(
EnergyOperator
):
def
__init__
(
self
,
d
):
"""Special class for inverse Gamma distributed covariances.
RL FIXME: To be documented.
"""
def
__init__
(
self
,
d
):
self
.
_d
=
d
self
.
_domain
=
DomainTuple
.
make
(
d
.
domain
)
...
...
@@ -159,6 +209,23 @@ class InverseGammaLikelihood(EnergyOperator):
class
BernoulliEnergy
(
EnergyOperator
):
"""Class for likelihood-energies of expected event frequency constrained by
event data.
Parameters
----------
d : Field
data field with events (=1) or non-events (=0)
Usage
-----
E = BernoulliEnergy(d) represents
E(f) = -log Bernoulli(d|f) = -d^dagger log(f) - (1-d)^dagger log(1-f),
where f is a field in data space (d.domain) with the expected frequencies of
events.
"""
def
__init__
(
self
,
d
):
self
.
_d
=
d
self
.
_domain
=
DomainTuple
.
make
(
d
.
domain
)
...
...
@@ -176,6 +243,41 @@ class BernoulliEnergy(EnergyOperator):
class
Hamiltonian
(
EnergyOperator
):
"""Class for information Hamiltonians.
Parameters
----------
lh : EnergyOperator
a likelihood energy
ic_samp : IterationController
is passed to SamplingEnabler to draw Gaussian distributed samples
with covariance = metric of Hamiltonian
(= Hessian without terms that generate negative eigenvalues)
default = None
Usage
-----
H = Hamiltonian(E_lh) represents
H(f) = 0.5 f^dagger f + E_lh(f)
an information Hamiltonian for a field f with a white Gaussian prior
(unit covariance) and the likelihood energy E_lh.
Tip
---
Other field priors can be represented via transformations of a white
Gaussian field into a field with the desired prior probability structure.
By implementing prior information this way, the field prior is represented
by a generative model, from which NIFTy can draw samples and infer a field
using the Maximum a Posteriori (MAP) or the Variational Bayes (VB) method.
For more details see:
"Encoding prior knowledge in the structure of the likelihood"
Jakob Knollmüller, Torsten A. Ensslin, submitted, arXiv:1812.04403
https://arxiv.org/abs/1812.04403
"""
def
__init__
(
self
,
lh
,
ic_samp
=
None
):
self
.
_lh
=
lh
self
.
_prior
=
GaussianEnergy
(
domain
=
lh
.
domain
)
...
...
@@ -200,12 +302,53 @@ class Hamiltonian(EnergyOperator):
class
SampledKullbachLeiblerDivergence
(
EnergyOperator
):
"""Class for Kullbach Leibler (KL) Divergence or Gibbs free energies
Precisely a sample averaged Hamiltonian (or other energy) that represents
approximatively the relevant part of a KL to be used in Variational Bayes
inference if the samples are drawn from the approximating Gaussian.
Let Q(f) = G(f-m,D) Gaussian used to approximate
P(f|d), the correct posterior with information Hamiltonian
H(d,f) = - log P(d,f) = - log P(f|d) + const.
The KL divergence between those should then be optimized for m. It is
KL(Q,P) = int Df Q(f) log Q(f)/P(f)
= < log Q(f) >_Q(f) - < log P(f) >_Q(f)
= const + < H(f) >_G(f-m,D)
in essence the information Hamiltonian averaged over a Gaussian distribution
centered on the mean m.
SampledKullbachLeiblerDivergence(H) approximates < H(f) >_G(f-m,D) if the
residuals f-m are drawn from covariance D.
Parameters
----------
h: Hamiltonian
the Hamiltonian/energy to be averaged
res_samples : iterable Field
set of residual sample points to be added to mean field
for approximate estimation of the KL
Usage:
------
KL = SampledKullbachLeiblerDivergence(H, samples) represents
KL(m) = sum_i H(m+v_i) / N,
where v_i are the residual samples, N is their number, and m is the mean field
around which the samples are drawn.
Tip:
----
Having symmetrized residual samples, with both, v_i and -v_i being present,
ensures that the distribution mean is exactly represented. This reduces sampling
noise and helps the numerics of the KL minimization process in the variational
Bayes inference.
"""
def
__init__
(
self
,
h
,
res_samples
):
"""
# MR FIXME: does h have to be a Hamiltonian? Couldn't it be any energy?
h: Hamiltonian
N: Number of samples to be used
"""
self
.
_h
=
h
self
.
_domain
=
h
.
domain
self
.
_res_samples
=
tuple
(
res_samples
)
...
...
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