# This program is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU General Public License for more details.
#
# You should have received a copy of the GNU General Public License
# along with this program. If not, see <http://www.gnu.org/licenses/>.
#
# Copyright(C) 2013-2021 Max-Planck-Society
#
# NIFTy is being developed at the Max-Planck-Institut fuer Astrophysik.
import numpy as np
from ..domain_tuple import DomainTuple
from ..domains.unstructured_domain import UnstructuredDomain
from ..field import Field
from ..linearization import Linearization
from ..minimization.energy_adapter import StochasticEnergyAdapter
from ..multi_field import MultiField
from ..operators.einsum import MultiLinearEinsum
from ..operators.energy_operators import EnergyOperator
from ..operators.linear_operator import LinearOperator
from ..operators.multifield2vector import Multifield2Vector
from ..operators.sandwich_operator import SandwichOperator
from ..operators.simple_linear_operators import FieldAdapter
from ..sugar import from_random, full, is_fieldlike, makeDomain, makeField
from ..utilities import myassert
[docs]
class MeanFieldVI:
"""Collect the operators required for Gaussian meanfield variational
inference.
Gaussian meanfield variational inference approximates some target
distribution with a Gaussian distribution with a diagonal covariance
matrix. The parameters of the approximation, in this case the mean and
standard deviation, are obtained by minimizing a stochastic estimate of the
Kullback-Leibler divergence between the target and the approximation. In
order to obtain gradients w.r.t the parameters, the reparametrization trick
is employed, which separates the stochastic part of the approximation from
a deterministic function, the generator. Samples from the approximation are
drawn by processing samples from a standard Gaussian through this
generator.
Parameters
----------
position : :class:`nifty8.field.Field`
The initial estimate of the approximate mean parameter.
hamiltonian : Energy
Hamiltonian of the approximated probability distribution.
n_samples : int
Number of samples used to stochastically estimate the KL.
mirror_samples : bool
Whether the negative of the drawn samples are also used, as they are
equally legitimate samples. If true, the number of used samples
doubles. Mirroring samples stabilizes the KL estimate as extreme sample
variation is counterbalanced. Since it improves stability in many
cases, it is recommended to set `mirror_samples` to `True`.
initial_sig : positive :class:`nifty8.field.Field` or positive float
The initial estimate of the standard deviation.
comm : MPI communicator or None
If not None, samples will be distributed as evenly as possible across
this communicator. If `mirror_samples` is set, then a sample and its
mirror image will always reside on the same task.
nanisinf : bool
If true, nan energies which can happen due to overflows in the forward
model are interpreted as inf. Thereby, the code does not crash on these
occasions but rather the minimizer is told that the position it has
tried is not sensible.
"""
[docs]
def __init__(self, position, hamiltonian, n_samples, mirror_samples,
initial_sig=1, comm=None, nanisinf=False):
Flat = Multifield2Vector(position.domain)
self._std = FieldAdapter(Flat.target, 'std').absolute()
latent = FieldAdapter(Flat.target,'latent')
self._mean = FieldAdapter(Flat.target, 'mean')
self._generator = Flat.adjoint(self._mean + self._std * latent)
self._entropy = GaussianEntropy(self._std.target) @ self._std
self._mean = Flat.adjoint @ self._mean
self._std = Flat.adjoint @ self._std
pos = {'mean': Flat(position)}
if is_fieldlike(initial_sig):
pos['std'] = Flat(initial_sig)
else:
pos['std'] = full(Flat.target, initial_sig)
pos = MultiField.from_dict(pos)
op = hamiltonian(self._generator) + self._entropy
self._KL = StochasticEnergyAdapter.make(pos, op, ['latent',], n_samples,
mirror_samples, nanisinf=nanisinf, comm=comm)
self._samdom = latent.domain
@property
def mean(self):
return self._mean.force(self._KL.position)
@property
def std(self):
return self._std.force(self._KL.position)
@property
def entropy(self):
return self._entropy.force(self._KL.position)
@property
def KL(self):
return self._KL
[docs]
def draw_sample(self):
_, op = self._generator.simplify_for_constant_input(
from_random(self._samdom))
return op(self._KL.position)
[docs]
def minimize(self, minimizer):
self._KL, _ = minimizer(self._KL)
[docs]
class FullCovarianceVI:
"""Collect the operators required for Gaussian full-covariance variational
Gaussian meanfield variational inference approximates some target
distribution with a Gaussian distribution with a diagonal covariance
matrix. The parameters of the approximation, in this case the mean and a
lower triangular matrix corresponding to a Cholesky decomposition of the
covariance, are obtained by minimizing a stochastic estimate of the
Kullback-Leibler divergence between the target and the approximation. In
order to obtain gradients w.r.t the parameters, the reparametrization trick
is employed, which separates the stochastic part of the approximation from
a deterministic function, the generator. Samples from the approximation are
drawn by processing samples from a standard Gaussian through this
generator.
Note that the size of the covariance scales quadratically with the number
of model parameters.
Parameters
----------
position : :class:`nifty8.field.Field`
The initial estimate of the approximate mean parameter.
hamiltonian : Energy
Hamiltonian of the approximated probability distribution.
n_samples : int
Number of samples used to stochastically estimate the KL.
mirror_samples : bool
Whether the negative of the drawn samples are also used, as they are
equally legitimate samples. If true, the number of used samples
doubles. Mirroring samples stabilizes the KL estimate as extreme sample
variation is counterbalanced. Since it improves stability in many
cases, it is recommended to set `mirror_samples` to `True`.
initial_sig : positive float
The initial estimate for the standard deviation. Initially no
correlation between the parameters is assumed.
comm : MPI communicator or None
If not None, samples will be distributed as evenly as possible across
this communicator. If `mirror_samples` is set, then a sample and its
mirror image will always reside on the same task.
nanisinf : bool
If true, nan energies which can happen due to overflows in the forward
model are interpreted as inf. Thereby, the code does not crash on these
occasions but rather the minimizer is told that the position it has
tried is not sensible.
"""
[docs]
def __init__(self, position, hamiltonian, n_samples, mirror_samples,
initial_sig=1, comm=None, nanisinf=False):
Flat = Multifield2Vector(position.domain)
flat_domain = Flat.target[0]
mat_space = DomainTuple.make((flat_domain,flat_domain))
lat = FieldAdapter(Flat.target,'latent')
LT = LowerTriangularInserter(mat_space)
tri = FieldAdapter(LT.domain, 'cov')
mean = FieldAdapter(flat_domain,'mean')
cov = LT @ tri
matmul_setup = lat.adjoint @ lat + cov.ducktape_left('co')
MatMult = MultiLinearEinsum(matmul_setup.target,'ij,j->i',
key_order=('co','latent'))
self._generator = Flat.adjoint @ (mean + MatMult @ matmul_setup)
diag_cov = (DiagonalSelector(cov.target) @ cov).absolute()
self._entropy = GaussianEntropy(diag_cov.target) @ diag_cov
diag_tri = np.diag(np.full(flat_domain.shape[0], initial_sig))
pos = MultiField.from_dict(
{'mean': Flat(position),
'cov': LT.adjoint(makeField(mat_space, diag_tri))})
op = hamiltonian(self._generator) + self._entropy
self._KL = StochasticEnergyAdapter.make(pos, op, ['latent',], n_samples,
mirror_samples, nanisinf=nanisinf, comm=comm)
self._mean = Flat.adjoint @ mean
self._samdom = lat.domain
@property
def mean(self):
return self._mean.force(self._KL.position)
@property
def entropy(self):
return self._entropy.force(self._KL.position)
@property
def KL(self):
return self._KL
[docs]
def draw_sample(self):
_, op = self._generator.simplify_for_constant_input(
from_random(self._samdom))
return op(self._KL.position)
[docs]
def minimize(self, minimizer):
self._KL, _ = minimizer(self._KL)
[docs]
class GaussianEntropy(EnergyOperator):
"""Entropy of a Gaussian distribution given the diagonal of a triangular
decomposition of the covariance.
As metric a `SandwichOperator` of the Jacobian is used. This is not a
proper Fisher metric but may be useful for second order minimization.
Parameters
----------
domain: Domain, DomainTuple, list of Domain
The domain of the diagonal.
"""
[docs]
def __init__(self, domain):
self._domain = DomainTuple.make(domain)
[docs]
def apply(self, x):
self._check_input(x)
if isinstance(x, Field):
if not np.issubdtype(x.dtype, np.floating):
raise NotImplementedError("only real fields are allowed")
if isinstance(x, MultiField):
for key in x.keys():
if not np.issubdtype(x[key].dtype, np.floating):
raise NotImplementedError("only real fields are allowed")
res = (x*x).scale(2*np.pi*np.e).log().sum().scale(-0.5)
if not isinstance(x, Linearization):
return res
if not x.want_metric:
return res
return res.add_metric(SandwichOperator.make(res.jac))
[docs]
class LowerTriangularInserter(LinearOperator):
"""Insert the entries of a lower triangular matrix into a matrix.
Parameters
----------
target: Domain, DomainTuple, list of Domain
A two-dimensional domain with NxN entries.
"""
[docs]
def __init__(self, target):
myassert(len(target.shape) == 2)
myassert(target.shape[0] == target.shape[1])
self._target = makeDomain(target)
ndof = (target.shape[0]*(target.shape[0]+1))//2
self._domain = makeDomain(UnstructuredDomain(ndof))
self._indices = np.tril_indices(target.shape[0])
self._capability = self.TIMES | self.ADJOINT_TIMES
[docs]
def apply(self, x, mode):
self._check_input(x, mode)
x = x.val
if mode == self.TIMES:
res = np.zeros(self._target.shape)
res[self._indices] = x
else:
res = x[self._indices].reshape(self._domain.shape)
return makeField(self._tgt(mode), res)
[docs]
class DiagonalSelector(LinearOperator):
"""Extract the diagonal of a two-dimensional field.
Parameters
----------
domain: Domain, DomainTuple, list of Domain
The two-dimensional domain of the input field. Must be of shape NxN.
"""
[docs]
def __init__(self, domain):
myassert(len(domain.shape) == 2)
myassert(domain.shape[0] == domain.shape[1])
self._domain = makeDomain(domain)
self._target = makeDomain(UnstructuredDomain(domain.shape[0]))
self._capability = self.TIMES | self.ADJOINT_TIMES
[docs]
def apply(self, x, mode):
self._check_input(x, mode)
return makeField(self._tgt(mode), np.diag(x.val))
