Commit 550aee99 authored by Martin Reinecke's avatar Martin Reinecke
Browse files

Merge remote-tracking branch 'origin/NIFTy_5' into fft_tweaks

parents f60c348b 3996cb92
......@@ -19,7 +19,7 @@ There is a full toolbox of methods that can be used, like the classical approxim
.. [3] T.A. Enßlin (2014), "Astrophysical data analysis with information field theory", AIP Conference Proceedings, Volume 1636, Issue 1, p.49; `arXiv:1405.7701 <http://arxiv.org/abs/1405.7701>`_
.. [4] Wikipedia contributors (2018), `"Information field theory" <https://en.wikipedia.org/w/index.php?title=Information_field_theory&oldid=876731720>`_, Wikipedia, The Free Encyclopedia.
.. [4] Wikipedia contributors (2018), `"Information field theory" <https://en.wikipedia.org/w/index.php?title=Information_field_theory&oldid=876731720>`_, Wikipedia, The Free Encyclopedia.
.. [5] T.A. Enßlin (2019), "Information theory for fields", accepted by Annalen der Physik; `arXiv:1804.03350 <http://arxiv.org/abs/1804.03350>`_
......@@ -85,7 +85,7 @@ The above line of argumentation analogously applies to the discretization of ope
The proper discretization of spaces, fields, and operators, as well as the normalization of position integrals, is essential for the conservation of the continuum limit. Their consistent implementation in NIFTY allows a pixelization independent coding of algorithms.
Free Theory & Implicit Operators
Free Theory & Implicit Operators
--------------------------------
A free IFT appears when the signal field :math:`{s}` and the noise :math:`{n}` of the data :math:`{d}` are independent, zero-centered Gaussian processes of kown covariances :math:`{S}` and :math:`{N}`, respectively,
......@@ -94,7 +94,7 @@ A free IFT appears when the signal field :math:`{s}` and the noise :math:`{n}` o
\mathcal{P}(s,n) = \mathcal{G}(s,S)\,\mathcal{G}(n,N),
and the measurement equation is linear in both, signal and noise,
and the measurement equation is linear in both signal and noise,
.. math::
......@@ -109,15 +109,15 @@ associate professor
\mathcal{H}(d,s)= -\log \mathcal{P}(d,s)= \frac{1}{2} s^\dagger S^{-1} s + \frac{1}{2} (d-R\,s)^\dagger N^{-1} (d-R\,s) + \mathrm{const}
is only of quadratic order in :math:`{s}`, which leads to a linear relation between the data and the posterior mean field.
is only of quadratic order in :math:`{s}`, which leads to a linear relation between the data and the posterior mean field.
In this case, the posterior is
In this case, the posterior is
.. math::
\mathcal{P}(s|d) = \mathcal{G}(s-m,D)
with
with
.. math::
......@@ -129,7 +129,7 @@ the posterior mean field,
D = \left( S^{-1} + R^\dagger N^{-1} R\right)^{-1}
the posterior covariance operator, and
the posterior covariance operator, and
.. math::
......@@ -137,7 +137,7 @@ the posterior covariance operator, and
the information source. The operation in :math:`{d= D\,R^\dagger N^{-1} d}` is also called the generalized Wiener filter.
NIFTy permits to define the involved operators :math:`{R}`, :math:`{R^\dagger}`, :math:`{S}`, and :math:`{N}` implicitely, as routines that can be applied to vectors, but which do not require the explicit storage of the matrix elements of the operators.
NIFTy permits to define the involved operators :math:`{R}`, :math:`{R^\dagger}`, :math:`{S}`, and :math:`{N}` implicitely, as routines that can be applied to vectors, but which do not require the explicit storage of the matrix elements of the operators.
Some of these operators are diagonal in harmonic (Fourier) basis, and therefore only require the specification of a (power) spectrum and :math:`{S= F\,\widehat{P_s} F^\dagger}`. Here :math:`{F = \mathrm{HarmonicTransformOperator}}`, :math:`{\widehat{P_s} = \mathrm{DiagonalOperator}(P_s)}`, and :math:`{P_s(k)}` is the power spectrum of the process that generated :math:`{s}` as a function of the (absolute value of the) harmonic (Fourier) space koordinate :math:`{k}`. For those, NIFTy can easily also provide inverse operators, as :math:`{S^{-1}= F\,\widehat{\frac{1}{P_s}} F^\dagger}` in case :math:`{F}` is unitary, :math:`{F^\dagger=F^{-1}}`.
......@@ -170,7 +170,7 @@ The joint information Hamiltonian for the whitened signal field :math:`{\xi}` re
\mathcal{H}(d,\xi)= -\log \mathcal{P}(d,s)= \frac{1}{2} \xi^\dagger \xi + \frac{1}{2} (d-R\,A\,\xi)^\dagger N^{-1} (d-R\,A\,\xi) + \mathrm{const}.
NIFTy takes advantage of this formulation in several ways:
NIFTy takes advantage of this formulation in several ways:
1) All prior degrees of freedom have unit covariance which improves the condition number of operators which need to be inverted.
2) The amplitude operator can be regarded as part of the response, :math:`{R'=R\,A}`. In general, more sophisticated responses can be constructed out of the composition of simpler operators.
......
......@@ -46,9 +46,10 @@ from .operators.outer_product_operator import OuterProduct
from .operators.simple_linear_operators import (
VdotOperator, ConjugationOperator, Realizer,
FieldAdapter, ducktape, GeometryRemover, NullOperator)
from .operators.value_inserter import ValueInserter
from .operators.energy_operators import (
EnergyOperator, GaussianEnergy, PoissonianEnergy, InverseGammaLikelihood,
BernoulliEnergy, Hamiltonian, SampledKullbachLeiblerDivergence)
BernoulliEnergy, Hamiltonian, AveragedEnergy)
from .probing import probe_with_posterior_samples, probe_diagonal, \
StatCalculator
......
......@@ -442,7 +442,7 @@ class Field(object):
----------
spaces : None, int or tuple of int
The summation is only carried out over the sub-domains in this
tuple. If None, it is carried out over all sub-domains. Default: None.
tuple. If None, it is carried out over all sub-domains.
Returns
-------
......@@ -463,7 +463,6 @@ class Field(object):
spaces : None, int or tuple of int
The summation is only carried out over the sub-domains in this
tuple. If None, it is carried out over all sub-domains.
Default: None.
Returns
-------
......@@ -547,7 +546,7 @@ class Field(object):
----------
spaces : None, int or tuple of int
The operation is only carried out over the sub-domains in this
tuple. If None, it is carried out over all sub-domains. Default: None.
tuple. If None, it is carried out over all sub-domains.
Returns
-------
......
......@@ -34,7 +34,7 @@ def _float_or_listoffloat(inp):
return [float(x) for x in inp] if isinstance(inp, list) else float(inp)
def _make_dynamic_operator(domain,
def _make_dynamic_operator(target,
harmonic_padding,
sm_s0,
sm_x0,
......@@ -44,8 +44,10 @@ def _make_dynamic_operator(domain,
minimum_phase,
sigc=None,
quant=None):
if not isinstance(domain, RGSpace):
if not isinstance(target, RGSpace):
raise TypeError("RGSpace required")
if not target.harmonic:
raise TypeError("Target space must be harmonic")
if not (isinstance(harmonic_padding, int) or harmonic_padding is None
or all(isinstance(ii, int) for ii in harmonic_padding)):
raise TypeError
......@@ -62,7 +64,7 @@ def _make_dynamic_operator(domain,
if cone and (sigc is None or quant is None):
raise RuntimeError
dom = DomainTuple.make(domain)
dom = DomainTuple.make(target.get_default_codomain())
ops = {}
FFT = FFTOperator(dom)
Real = Realizer(dom)
......@@ -134,7 +136,8 @@ def _make_dynamic_operator(domain,
return m, ops
def dynamic_operator(domain,
def dynamic_operator(*,
target,
harmonic_padding,
sm_s0,
sm_x0,
......@@ -143,11 +146,22 @@ def dynamic_operator(domain,
minimum_phase=False):
'''Constructs an operator encoding the Green's function of a linear
homogeneous dynamic system.
When evaluated, this operator returns the Green's function representation
in harmonic space. This result can be used as a convolution kernel to
construct solutions of the homogeneous stochastic differential equation
encoded in this operator. Note that if causal is True, the Green's function
is convolved with a step function in time, where the temporal axis is the
first axis of the space. In this case the resulting function only extends
up to half the length of the first axis of the space to avoid boundary
effects during convolution. If minimum_phase is true then the spectrum of
the Green's function is used to construct a corresponding minimum phase
filter.
Parameters
----------
domain : RGSpace
The position space in which the Green's function shall be constructed.
target : RGSpace
The harmonic space in which the Green's function shall be constructed.
harmonic_padding : None, int, list of int
Amount of central padding in harmonic space in pixels. If None the
field is not padded at all.
......@@ -159,13 +173,15 @@ def dynamic_operator(domain,
key for dynamics encoding parameter.
causal : boolean
Whether or not the Green's function shall be causal in time.
Default is True.
minimum_phase: boolean
Whether or not the Green's function shall be a minimum phase filter.
Default is False.
Returns
-------
Operator
The Operator encoding the dynamic Green's function in harmonic space.
The Operator encoding the dynamic Green's function in target space.
Dictionary of Operator
A collection of sub-chains of Operator which can be used for plotting
and evaluation.
......@@ -175,7 +191,7 @@ def dynamic_operator(domain,
The first axis of the domain is interpreted the time axis.
'''
dct = {
'domain': domain,
'target': target,
'harmonic_padding': harmonic_padding,
'sm_s0': sm_s0,
'sm_x0': sm_x0,
......@@ -187,7 +203,8 @@ def dynamic_operator(domain,
return _make_dynamic_operator(**dct)
def dynamic_lightcone_operator(domain,
def dynamic_lightcone_operator(*,
target,
harmonic_padding,
sm_s0,
sm_x0,
......@@ -199,11 +216,16 @@ def dynamic_lightcone_operator(domain,
minimum_phase=False):
'''Extends the functionality of :function: dynamic_operator to a Green's
function which is constrained to be within a light cone.
The resulting Green's function is constrained to be within a light cone.
This is achieved via convolution of the function with a light cone in
space-time. Thereby the first axis of the space is set to be the teporal
axis.
Parameters
----------
domain : RGSpace
The position space in which the Green's function shall be constructed.
target : RGSpace
The harmonic space in which the Green's function shall be constructed.
It needs to have at least two dimensions.
harmonic_padding : None, int, list of int
Amount of central padding in harmonic space in pixels. If None the
......@@ -222,8 +244,10 @@ def dynamic_lightcone_operator(domain,
Quantization of the light cone in pixels.
causal : boolean
Whether or not the Green's function shall be causal in time.
Default is True.
minimum_phase: boolean
Whether or not the Green's function shall be a minimum phase filter.
Default is False.
Returns
-------
......@@ -238,10 +262,10 @@ def dynamic_lightcone_operator(domain,
The first axis of the domain is interpreted the time axis.
'''
if len(domain.shape) < 2:
if len(target.shape) < 2:
raise ValueError("Space must be at least 2 dimensional!")
dct = {
'domain': domain,
'target': target,
'harmonic_padding': harmonic_padding,
'sm_s0': sm_s0,
'sm_x0': sm_x0,
......
......@@ -187,6 +187,18 @@ class Linearization(object):
return self.__mul__(other)
def outer(self, other):
"""Computes the outer product of this Linearization with a Field or
another Linearization
Parameters
----------
other : Field or MultiField or Linearization
Returns
-------
Linearization
the outer product of self and other
"""
from .operators.outer_product_operator import OuterProduct
if isinstance(other, Linearization):
return self.new(
......@@ -200,6 +212,18 @@ class Linearization(object):
OuterProduct(self._jac(self._val), other.domain))
def vdot(self, other):
"""Computes the inner product of this Linearization with a Field or
another Linearization
Parameters
----------
other : Field or MultiField or Linearization
Returns
-------
Linearization
the inner product of self and other
"""
from .operators.simple_linear_operators import VdotOperator
if isinstance(other, (Field, MultiField)):
return self.new(
......@@ -211,6 +235,19 @@ class Linearization(object):
VdotOperator(other._val)(self._jac))
def sum(self, spaces=None):
"""Computes the (partial) sum over self
Parameters
----------
spaces : None, int or list of int
- if None, sum over the entire domain
- else sum over the specified subspaces
Returns
-------
Linearization
the (partial) sum
"""
from .operators.contraction_operator import ContractionOperator
if spaces is None:
return self.new(
......@@ -222,6 +259,19 @@ class Linearization(object):
ContractionOperator(self._jac.target, spaces)(self._jac))
def integrate(self, spaces=None):
"""Computes the (partial) integral over self
Parameters
----------
spaces : None, int or list of int
- if None, integrate over the entire domain
- else integrate over the specified subspaces
Returns
-------
Linearization
the (partial) integral
"""
from .operators.contraction_operator import ContractionOperator
if spaces is None:
return self.new(
......@@ -309,18 +359,72 @@ class Linearization(object):
@staticmethod
def make_var(field, want_metric=False):
"""Converts a Field to a Linearization, with a unity Jacobian
Parameters
----------
field : Field or Multifield
the field to be converted
want_metric : bool
If True, the metric will be computed for other Linearizations
derived from this one. Default: False.
Returns
-------
Linearization
the requested Linearization
"""
from .operators.scaling_operator import ScalingOperator
return Linearization(field, ScalingOperator(1., field.domain),
want_metric=want_metric)
@staticmethod
def make_const(field, want_metric=False):
"""Converts a Field to a Linearization, with a zero Jacobian
Parameters
----------
field : Field or Multifield
the field to be converted
want_metric : bool
If True, the metric will be computed for other Linearizations
derived from this one. Default: False.
Returns
-------
Linearization
the requested Linearization
Notes
-----
The Jacobian is square and contains only zeroes.
"""
from .operators.simple_linear_operators import NullOperator
return Linearization(field, NullOperator(field.domain, field.domain),
want_metric=want_metric)
@staticmethod
def make_const_empty_input(field, want_metric=False):
"""Converts a Field to a Linearization, with a zero Jacobian
Parameters
----------
field : Field or Multifield
the field to be converted
want_metric : bool
If True, the metric will be computed for other Linearizations
derived from this one. Default: False.
Returns
-------
Linearization
the requested Linearization
Notes
-----
The Jacobian has an empty input domain, i.e. its matrix representation
has 0 columns.
"""
from .operators.simple_linear_operators import NullOperator
from .multi_domain import MultiDomain
return Linearization(
......@@ -329,6 +433,29 @@ class Linearization(object):
@staticmethod
def make_partial_var(field, constants, want_metric=False):
"""Converts a MultiField to a Linearization, with a Jacobian that is
unity for some MultiField components and a zero matrix for others.
Parameters
----------
field : Multifield
the field to be converted
constants : list of string
the MultiField components for which the Jacobian should be
a zero matrix.
want_metric : bool
If True, the metric will be computed for other Linearizations
derived from this one. Default: False.
Returns
-------
Linearization
the requested Linearization
Notes
-----
The Jacobian is square.
"""
from .operators.scaling_operator import ScalingOperator
from .operators.block_diagonal_operator import BlockDiagonalOperator
if len(constants) == 0:
......
......@@ -15,11 +15,14 @@
#
# NIFTy is being developed at the Max-Planck-Institut fuer Astrophysik.
import numpy as np
from .. import utilities
from ..domain_tuple import DomainTuple
from ..field import Field
from ..linearization import Linearization
from ..sugar import makeOp, makeDomain
from ..sugar import makeDomain, makeOp
from .linear_operator import LinearOperator
from .operator import Operator
from .sampling_enabler import SamplingEnabler
from .sandwich_operator import SandwichOperator
......@@ -27,11 +30,28 @@ from .simple_linear_operators import VdotOperator
class EnergyOperator(Operator):
"""Operator which has a scalar domain as target domain."""
"""Operator which has a scalar domain as target domain.
It is intended as an objective function for field inference.
Examples
--------
- Information Hamiltonian, i.e. negative-log-probabilities.
- Gibbs free energy, i.e. an averaged Hamiltonian, aka Kullbach-Leibler
divergence.
"""
_target = DomainTuple.scalar_domain()
class SquaredNormOperator(EnergyOperator):
"""Computes the L2-norm of the output of an operator.
Parameters
----------
domain : Domain, DomainTuple or tuple of Domain
Target domain of the operator in which the L2-norm shall be computed.
"""
def __init__(self, domain):
self._domain = domain
......@@ -45,12 +65,24 @@ class SquaredNormOperator(EnergyOperator):
class QuadraticFormOperator(EnergyOperator):
def __init__(self, op):
"""Computes the L2-norm of a Field or MultiField with respect to a
specific metric `endo`.
.. math ::
E(f) = \\frac12 f^\\dagger \\text{endo}(f)
Parameters
----------
endo : EndomorphicOperator
Kernel of quadratic form.
"""
def __init__(self, endo):
from .endomorphic_operator import EndomorphicOperator
if not isinstance(op, EndomorphicOperator):
if not isinstance(endo, EndomorphicOperator):
raise TypeError("op must be an EndomorphicOperator")
self._op = op
self._domain = op.domain
self._op = endo
self._domain = endo.domain
def apply(self, x):
self._check_input(x)
......@@ -63,7 +95,38 @@ class QuadraticFormOperator(EnergyOperator):
class GaussianEnergy(EnergyOperator):
"""Class for energies of fields with Gaussian probability distribution.
Represents up to constants in :math:`m`:
.. math ::
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.
Parameters
----------
mean : Field
Mean of the Gaussian. Default is 0.
covariance : LinearOperator
Covariance of the Gaussian. Default is the identity operator.
domain : Domain, DomainTuple, tuple of Domain or MultiDomain
Operator domain. By default it is inferred from `mean` or
`covariance` if specified
Note
----
At least one of the arguments has to be provided.
"""
def __init__(self, mean=None, covariance=None, domain=None):
if mean is not None and not isinstance(mean, Field):
raise TypeError
if covariance is not None and not isinstance(covariance,
LinearOperator):
raise TypeError
self._domain = None
if mean is not None:
self._checkEquivalence(mean.domain)
......@@ -90,7 +153,7 @@ class GaussianEnergy(EnergyOperator):
def apply(self, x):
self._check_input(x)
residual = x if self._mean is None else x-self._mean
residual = x if self._mean is None else x - self._mean
res = self._op(residual).real
if not isinstance(x, Linearization) or not x.want_metric:
return res
......@@ -99,7 +162,29 @@ class GaussianEnergy(EnergyOperator):
class PoissonianEnergy(EnergyOperator):
"""Class for likelihood Hamiltonians of expected count field constrained
by Poissonian count data.
Represents up to an f-independent term :math:`log(d!)`:
.. math ::
E(f) = -\\log \\text{Poisson}(d|f) = \\sum f - d^\\dagger \\log(f),
where f is a :class:`Field` in data space with the expectation values for
the counts.
Parameters
----------
d : Field
Data field with counts. Needs to have integer dtype and all field
values need to be non-negative.
"""
def __init__(self, d):
if not isinstance(d, Field) or not np.issubdtype(d.dtype, np.integer):
raise TypeError
if np.any(d.local_data < 0):
raise ValueError
self._d = d
self._domain = DomainTuple.make(d.domain)
......@@ -115,7 +200,13 @@ class PoissonianEnergy(EnergyOperator):
class InverseGammaLikelihood(EnergyOperator):
"""
FIXME
"""
def __init__(self, d):
if not isinstance(d, Field):
raise TypeError
self._d = d
self._domain = DomainTuple.make(d.domain)
......@@ -131,23 +222,78 @@ class InverseGammaLikelihood(EnergyOperator):
class BernoulliEnergy(EnergyOperator):
"""Computes likelihood energy of expected event frequency constrained by
event data.
.. math ::
E(f) = -\\log \\text{Bernoulli}(d|f)
= -d^\\dagger \\log f - (1-d)^\\dagger \\log(1-f),
where f is a field defined on `d.domain` with the expected
frequencies of events.
Parameters
----------
d : Field
Data field with events (1) or non-events (0).
"""
def __init__(self, d):
print(d.dtype)
if not isinstance(d, Field) or not np.issubdtype(d.dtype, np.integer):
raise TypeError
if not np.all(np.logical_or(d.local_data == 0, d.local_data == 1)):
raise ValueError
self._d = d
self._domain = DomainTuple.make(d.domain)
def apply(self, x):
self._check_input(x)
v = x.log().vdot(-self._d) - (1.-x).log().vdot(1.-self._d)
v = -(x.log().vdot(self._d) + (1. - x).log().vdot(1. - self._d))
if not isinstance(x, Linearization):
return Field.scalar(v)
if not x.want_metric:
return v
met = makeOp(1./(x.val*(1.-x.val)))
met = makeOp(1./(x.val*(1. - x.val)))
met = SandwichOperator.make(x.jac, met)
return v.add_metric(met)
class Hamiltonian(EnergyOperator):
"""Computes an information Hamiltonian in its standard form, i.e. with the
prior being a Gaussian with unit covariance.