Commit 08279a38 by Martin Reinecke

Merge branch 'NIFTy_5' into privatization

parents bfdb0c7f e1e58be3
 ... ... @@ -74,7 +74,7 @@ if __name__ == '__main__': ic_sampling = ift.GradientNormController(iteration_limit=100) # Minimize the Hamiltonian H = ift.Hamiltonian(likelihood, ic_sampling) H = ift.StandardHamiltonian(likelihood, ic_sampling) H = ift.EnergyAdapter(position, H, want_metric=True) # minimizer = ift.L_BFGS(ic_newton) H, convergence = minimizer(H) ... ...
 ... ... @@ -99,7 +99,7 @@ if __name__ == '__main__': minimizer = ift.NewtonCG(ic_newton) # Compute MAP solution by minimizing the information Hamiltonian H = ift.Hamiltonian(likelihood) H = ift.StandardHamiltonian(likelihood) initial_position = ift.from_random('normal', domain) H = ift.EnergyAdapter(initial_position, H, want_metric=True) H, convergence = minimizer(H) ... ...
 ... ... @@ -100,10 +100,10 @@ if __name__ == '__main__': # Set up likelihood and information Hamiltonian likelihood = ift.GaussianEnergy(mean=data, covariance=N)(signal_response) H = ift.Hamiltonian(likelihood, ic_sampling) H = ift.StandardHamiltonian(likelihood, ic_sampling) initial_position = ift.MultiField.full(H.domain, 0.) position = initial_position initial_mean = ift.MultiField.full(H.domain, 0.) mean = initial_mean plot = ift.Plot() plot.add(signal(mock_position), title='Ground Truth') ... ... @@ -117,9 +117,9 @@ if __name__ == '__main__': # Draw new samples to approximate the KL five times for i in range(5): # Draw new samples and minimize KL KL = ift.KL_Energy(position, H, N_samples) KL = ift.MetricGaussianKL(mean, H, N_samples) KL, convergence = minimizer(KL) position = KL.position mean = KL.position # Plot current reconstruction plot = ift.Plot() ... ... @@ -128,7 +128,7 @@ if __name__ == '__main__': plot.output(ny=1, ysize=6, xsize=16, name="loop-{:02}.png".format(i)) # Draw posterior samples KL = ift.KL_Energy(position, H, N_samples) KL = ift.MetricGaussianKL(mean, H, N_samples) sc = ift.StatCalculator() for sample in KL.samples: sc.add(signal(sample + KL.position)) ... ...
 ... ... @@ -103,7 +103,7 @@ N = ift.DiagonalOperator(ift.from_global_data(d_space, var)) IC = ift.DeltaEnergyController(tol_rel_deltaE=1e-12, iteration_limit=200) likelihood = ift.GaussianEnergy(d, N)(R) Ham = ift.Hamiltonian(likelihood, IC) Ham = ift.StandardHamiltonian(likelihood, IC) H = ift.EnergyAdapter(params, Ham, want_metric=True) # Minimize ... ...
 # rm -rf docs/build docs/source/mod sphinx-apidoc -e -o docs/source/mod nifty5 sphinx-build -b html docs/source/ docs/build/
 ... ... @@ -5,11 +5,20 @@ Theoretical Background ---------------------- Information Field Theory _ [1]_ (IFT) is information theory, the logic of reasoning under uncertainty, applied to fields. A field can be any quantity defined over some space, e.g. the air temperature over Europe, the magnetic field strength in the Milky Way, or the matter density in the Universe. IFT describes how data and knowledge can be used to infer field properties. Mathematically it is a statistical field theory and exploits many of the tools developed for such. Practically, it is a framework for signal processing and image reconstruction. Information Field Theory _ [1]_ (IFT) is information theory, the logic of reasoning under uncertainty, applied to fields. A field can be any quantity defined over some space, e.g. the air temperature over Europe, the magnetic field strength in the Milky Way, or the matter density in the Universe. IFT describes how data and knowledge can be used to infer field properties. Mathematically it is a statistical field theory and exploits many of the tools developed for such. Practically, it is a framework for signal processing and image reconstruction. IFT is fully Bayesian. How else could infinitely many field degrees of freedom be constrained by finite data? IFT is fully Bayesian. How else could infinitely many field degrees of freedom be constrained by finite data? There is a full toolbox of methods that can be used, like the classical approximation (= Maximum a posteriori = MAP), effective action (= Variational Bayes = VI), Feynman diagrams, renormalization, and more. IFT reproduces many known well working algorithms. This should be reassuring. Also, there were certainly previous works in a similar spirit. Anyhow, in many cases IFT provides novel rigorous ways to extract information from data. NIFTy comes with reimplemented MAP and VI estimators. It also provides a Hamiltonian Monte Carlo sampler for Fields (HMCF). (*FIXME* does it?) There is a full toolbox of methods that can be used, like the classical approximation (= Maximum a posteriori = MAP), effective action (= Variational Bayes = VI), Feynman diagrams, renormalization, and more. IFT reproduces many known well working algorithms, which is reassuring. Also, there were certainly previous works in a similar spirit. Anyhow, in many cases IFT provides novel rigorous ways to extract information from data. NIFTy comes with reimplemented MAP and VI estimators. .. tip:: *In-a-nutshell introductions to information field theory* can be found in [2]_, [3]_, [4]_, and [5]_, with the latter probably being the most didactical. ... ... @@ -27,7 +36,8 @@ There is a full toolbox of methods that can be used, like the classical approxim Discretized continuum --------------------- The representation of fields that are mathematically defined on a continuous space in a finite computer environment is a common necessity. The goal hereby is to preserve the continuum limit in the calculus in order to ensure a resolution independent discretization. The representation of fields that are mathematically defined on a continuous space in a finite computer environment is a common necessity. The goal hereby is to preserve the continuum limit in the calculus in order to ensure a resolution independent discretization. +-----------------------------+-----------------------------+ | .. image:: images/42vs6.png | .. image:: images/42vs9.png | ... ... @@ -43,7 +53,8 @@ Any partition of the continuous position space :math:\Omega (with volume :math V &\quad=\quad \int_\Omega \mathrm{d}x \quad=\quad \sum_{q=1}^Q \int_{\Omega_q} \mathrm{d}x \quad=\quad \sum_{q=1}^Q V_q . Here the number :math:Q characterizes the resolution of the pixelization and the continuum limit is described by :math:Q \rightarrow \infty and :math:V_q \rightarrow 0 for all :math:q \in \{1,\dots,Q\} simultaneously. Moreover, the above equation defines a discretization of continuous integrals, :math:\int_\Omega \mathrm{d}x \mapsto \sum_q V_q. Here the number :math:Q characterizes the resolution of the pixelization and the continuum limit is described by :math:Q \rightarrow \infty and :math:V_q \rightarrow 0 for all :math:q \in \{1,\dots,Q\} simultaneously. Moreover, the above equation defines a discretization of continuous integrals, :math:\int_\Omega \mathrm{d}x \mapsto \sum_q V_q. Any valid discretization scheme for a field :math:{s} can be described by a mapping, ... ... @@ -52,39 +63,48 @@ Any valid discretization scheme for a field :math:{s} can be described by a ma s(x \in \Omega_q) \quad\mapsto\quad s_q \quad=\quad \int_{\Omega_q} \mathrm{d}x \; w_q(x) \; s(x) , if the weighting function :math:w_q(x) is chosen appropriately. In order for the discretized version of the field to converge to the actual field in the continuum limit, the weighting functions need to be normalized in each subset; i.e., :math:\forall q: \int_{\Omega_q} \mathrm{d}x \; w_q(x) = 1. Choosing such a weighting function that is constant with respect to :math:x yields if the weighting function :math:w_q(x) is chosen appropriately. In order for the discretized version of the field to converge to the actual field in the continuum limit, the weighting functions need to be normalized in each subset; i.e., :math:\forall q: \int_{\Omega_q} \mathrm{d}x \; w_q(x) = 1. Choosing such a weighting function that is constant with respect to :math:x yields .. math:: s_q = \frac{\int_{\Omega_q} \mathrm{d}x \; s(x)}{\int_{\Omega_q} \mathrm{d}x} = \left< s(x) \right>_{\Omega_q} , which corresponds to a discretization of the field by spatial averaging. Another common and equally valid choice is :math:w_q(x) = \delta(x-x_q), which distinguishes some position :math:x_q \in \Omega_q, and evaluates the continuous field at this position, which corresponds to a discretization of the field by spatial averaging. Another common and equally valid choice is :math:w_q(x) = \delta(x-x_q), which distinguishes some position :math:x_q \in \Omega_q, and evaluates the continuous field at this position, .. math:: s_q \quad=\quad \int_{\Omega_q} \mathrm{d}x \; \delta(x-x_q) \; s(x) \quad=\quad s(x_q) . In practice, one often makes use of the spatially averaged pixel position, :math:x_q = \left< x \right>_{\Omega_q}. If the resolution is high enough to resolve all features of the signal field :math:{s}, both of these discretization schemes approximate each other, :math:\left< s(x) \right>_{\Omega_q} \approx s(\left< x \right>_{\Omega_q}), since they approximate the continuum limit by construction. (The approximation of :math:\left< s(x) \right>_{\Omega_q} \approx s(x_q \in \Omega_q) marks a resolution threshold beyond which further refinement of the discretization reveals no new features; i.e., no new information content of the field :math:{s}.) In practice, one often makes use of the spatially averaged pixel position, :math:x_q = \left< x \right>_{\Omega_q}. If the resolution is high enough to resolve all features of the signal field :math:{s}, both of these discretization schemes approximate each other, :math:\left< s(x) \right>_{\Omega_q} \approx s(\left< x \right>_{\Omega_q}), since they approximate the continuum limit by construction. (The approximation of :math:\left< s(x) \right>_{\Omega_q} \approx s(x_q \in \Omega_q) marks a resolution threshold beyond which further refinement of the discretization reveals no new features; i.e., no new information content of the field :math:{s}.) All operations involving position integrals can be normalized in accordance with the above definitions. For example, the scalar product between two fields :math:{s} and :math:{u} is defined as All operations involving position integrals can be normalized in accordance with the above definitions. For example, the scalar product between two fields :math:{s} and :math:{u} is defined as .. math:: {s}^\dagger {u} \quad=\quad \int_\Omega \mathrm{d}x \; s^*(x) \; u(x) \quad\approx\quad \sum_{q=1}^Q V_q^{\phantom{*}} \; s_q^* \; u_q^{\phantom{*}} , where :math:\dagger denotes adjunction and :math:* complex conjugation. Since the above approximation becomes an equality in the continuum limit, the scalar product is independent of the pixelization scheme and resolution, if the latter is sufficiently high. where :math:\dagger denotes adjunction and :math:* complex conjugation. Since the above approximation becomes an equality in the continuum limit, the scalar product is independent of the pixelization scheme and resolution, if the latter is sufficiently high. The above line of argumentation analogously applies to the discretization of operators. For a linear operator :math:{A} acting on some field :math:{s} as :math:{A} {s} = \int_\Omega \mathrm{d}y \; A(x,y) \; s(y), a matrix representation discretized with constant weighting functions is given by The above line of argumentation analogously applies to the discretization of operators. For a linear operator :math:{A} acting on some field :math:{s} as :math:{A} {s} = \int_\Omega \mathrm{d}y \; A(x,y) \; s(y), a matrix representation discretized with constant weighting functions is given by .. math:: A(x \in \Omega_p, y \in \Omega_q) \quad\mapsto\quad A_{pq} \quad=\quad \frac{\iint_{\Omega_p \Omega_q} \mathrm{d}x \, \mathrm{d}y \; A(x,y)}{\iint_{\Omega_p \Omega_q} \mathrm{d}x \, \mathrm{d}y} \quad=\quad \big< \big< A(x,y) \big>_{\Omega_p} \big>_{\Omega_q} . 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. 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 -------------------------------- ... ... @@ -135,18 +155,24 @@ the posterior covariance operator, and j = R^\dagger N^{-1} d the information source. The operation in :math:{m = D\,R^\dagger N^{-1} d} is also called the generalized Wiener filter. the information source. The operation in :math:{m = 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} implicitly, 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}}. 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 coordinate :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}}. These implicit operators can be combined into new operators, e.g. to :math:{D^{-1} = S^{-1} + R^\dagger N^{-1} R}, as well as their inverses, e.g. :math:{D = \left( D^{-1} \right)^{-1}}. The invocation of an inverse operator applied to a vector might trigger the execution of a numerical linear algebra solver. Thus, when NIFTy calculates :math:{m = D\, j} it actually solves :math:{D^{-1} m = j} for :math:{m} behind the scenes. The advantage of implicit operators to explicit matrices is the reduced memory requirements. The reconstruction of only a Megapixel image would otherwithe require the storage and processing of matrices with sizes of several Terrabytes. Larger images could not be dealt with due to the quadratic memory requirements of explicit operator representations. Thus, when NIFTy calculates :math:{m = D\, j}, it actually solves :math:{D^{-1} m = j} for :math:{m} behind the scenes. The advantage of implicit operators to explicit matrices is the reduced memory requirements. The reconstruction of only a Megapixel image would otherwithe require the storage and processing of matrices with sizes of several Terabytes. Larger images could not be dealt with due to the quadratic memory requirements of explicit operator representations. The demo codes demos/getting_started_1.py and demos/Wiener_Filter.ipynb illustrate this. The demo codes demos/getting_started_1.py and demos/Wiener_Filter.ipynb illustrate this. Generative Models ... ... @@ -164,7 +190,7 @@ Let us rewrite the above free theory as a generative model: with :math:{A} the amplitude operator such that it generates signal field realizations with the correct covariance :math:{S=A\,A^\dagger} when being applied to a white Gaussian field :math:{\xi} with :math:{\mathcal{P}(\xi)= \mathcal{G}(\xi, 1)}. The joint information Hamiltonian for the whitened signal field :math:{\xi} reads: The joint information Hamiltonian for the standardized signal field :math:{\xi} reads: .. math:: ... ... @@ -172,26 +198,35 @@ The joint information Hamiltonian for the whitened signal field :math:{\xi} re 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. 3) The response can be non-linear, e.g. :math:{R'(s)=R \exp(A\,\xi)}, see demos/getting_started_2.py. 4) The amplitude operator can be made dependent on unknowns as well, e.g. :math:A=A(\tau)= F\, \widehat{e^\tau} represents an amplitude operator with a positive definite, unknown spectrum defined in the Fourier domain. The amplitude field :math:{\tau} would get its own amplitude operator, with a cepstrum (spectrum of a log spectrum) defined in quefrency space (harmonic space of a logarithmically binned harmonic space) to regularize its degrees of freedom by imposing some (user-defined degree of) spectral smoothness. 5) NIFTy can calculate the gradient of the information Hamiltonian and the Fisher information metric with respect to all unknown parameters, here :math:{\xi} and :math:{\tau}, by automatic differentiation. The gradients are used for MAP and HMCF estimates, and the Fisher matrix is required in addition to the gradient by Metric Gaussian Variational Inference (MGVI), which is available in NIFTy as well. MGVI is an implicit operator extension of Automatic Differentiation Variational Inference (ADVI). 1) All prior degrees of freedom have unit covariance, which improves the condition number of operators that need to be inverted. The reconstruction of a non-Gaussian signal with unknown covariance from a non-trivial (tomographic) response is demonstrated in demos/getting_started_3.py. Here, the uncertainty of the field and the power spectrum of its generating process are probed via posterior samples provided by the MGVI algorithm. 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. 3) The response can be non-linear, e.g. :math:{R'(s)=R \exp(A\,\xi)}, see demos/getting_started_2.py. 4) The amplitude operator may dependent on further parameters, e.g. :math:A=A(\tau)= F\, \widehat{e^\tau} represents an amplitude operator with a positive definite, unknown spectrum defined in the Fourier domain. The amplitude field :math:{\tau} would get its own amplitude operator, with a cepstrum (spectrum of a log spectrum) defined in quefrency space (harmonic space of a logarithmically binned harmonic space) to regularize its degrees of freedom by imposing some (user-defined degree of) spectral smoothness. 5) NIFTy calculates the gradient of the information Hamiltonian and the Fisher information metric with respect to all unknown parameters, here :math:{\xi} and :math:{\tau}, by automatic differentiation. The gradients are used for MAP and HMCF estimates, and the Fisher matrix is required in addition to the gradient by Metric Gaussian Variational Inference (MGVI), which is available in NIFTy as well. MGVI is an implicit operator extension of Automatic Differentiation Variational Inference (ADVI). The reconstruction of a non-Gaussian signal with unknown covariance from a non-trivial (tomographic) response is demonstrated in demos/getting_started_3.py. Here, the uncertainty of the field and the power spectrum of its generating process are probed via posterior samples provided by the MGVI algorithm. +----------------------------------------------------+ | **Output of tomography demo getting_started_3.py** | +----------------------------------------------------+ | .. image:: images/getting_started_3_setup.png | | | | :width: 50 % | +----------------------------------------------------+ | Non-Gaussian signal field, | | data backprojected into the image domain, power | | spectrum of underlying Gausssian process. | +----------------------------------------------------+ | .. image:: images/getting_started_3_results.png | | | | :width: 50 % | +----------------------------------------------------+ | Posterior mean field signal | | reconstruction, its uncertainty, and the power | ... ... @@ -199,3 +234,73 @@ The reconstruction of a non-Gaussian signal with unknown covariance from a non-t | samples in comparison to the correct one (thick | | orange line). | +----------------------------------------------------+ Maximim a Posteriori -------------------- One popular field estimation method is Maximim a Posteriori (MAP). It only requires to minimize the information Hamiltonian, e.g by a gradient descent method that stops when .. math:: \frac{\partial \mathcal{H}(d,\xi)}{\partial \xi} = 0. NIFTy5 automatically calculates the necessary gradient from a generative model of the signal and the data and to minimize the Hamiltonian. However, MAP often provides unsatisfactory results in cases of deep hirachical Bayesian networks. The reason for this is that MAP ignores the volume factors in parameter space, which are not to be neglected in deciding whether a solution is reasonable or not. In the high dimensional setting of field inference these volume factors can differ by large ratios. A MAP estimate, which is only representative for a tiny fraction of the parameter space, might be a poorer choice (with respect to an error norm) compared to a slightly worse location with slightly lower posterior probability, which, however, is associated with a much larger volume (of nearby locations with similar probability). This causes MAP signal estimates to be more prone to overfitting the noise as well as to perception thresholds than methods that take volume effects into account. Variational Inference --------------------- One method that takes volume effects into account is Variational Inference (VI). In VI, the posterior :math:\mathcal{P}(\xi|d) is approximated by a simpler, parametrized distribution, often a Gaussian :math:\mathcal{Q}(\xi)=\mathcal{G}(\xi-m,D). The parameters of :math:\mathcal{Q}, the mean :math:m and its covariance :math:D are obtained by minimization of an appropriate information distance measure between :math:\mathcal{Q} and :math:\mathcal{P}. As a compromise between being optimal and being computationally affordable, the variational Kullback-Leibler (KL) divergence is used: .. math:: \mathrm{KL}(m,D|d)= \mathcal{D}_\mathrm{KL}(\mathcal{Q}||\mathcal{P})= \int \mathcal{D}\xi \,\mathcal{Q}(\xi) \log \left( \frac{\mathcal{Q}(\xi)}{\mathcal{P}(\xi)} \right) Minimizing this with respect to all entries of the covariance :math:D is unfeasible for fields. Therefore, Metric Gaussian Variational Inference (MGVI) approximates the precision matrix at the location of the current mean :math:M=D^{-1} by the Bayesian Fisher information metric, .. math:: M \approx \left\langle \frac{\partial \mathcal{H}(d,\xi)}{\partial \xi} \, \frac{\partial \mathcal{H}(d,\xi)}{\partial \xi}^\dagger \right\rangle_{(d,\xi)}. In practice the average is performed over :math:\mathcal{P}(d,\xi)\approx \mathcal{P}(d|\xi)\,\delta(\xi-m) by evaluating the expression at the current mean :math:m. This results in a Fisher information metric of the likelihood evaluated at the mean plus the prior information metric. Therefore we will only have to infer the mean of the approximate distribution. The only term within the KL-divergence that explicitly depends on it is the Hamiltonian of the true problem averaged over the approximation: .. math:: \mathrm{KL}(m|d) \;\widehat{=}\; \left\langle \mathcal{H}(\xi,d) \right\rangle_{\mathcal{Q}(\xi)}, where :math:\widehat{=} expresses equality up to irrelvant (here not :math:m-dependent) terms. Thus, only the gradient of the KL is needed with respect to this, which can be expressed as .. math:: \frac{\partial \mathrm{KL}(m|d)}{\partial m} = \left\langle \frac{\partial \mathcal{H}(d,\xi)}{\partial \xi} \right\rangle_{\mathcal{G}(\xi-m,D)}. We stochastically estimate the KL-divergence and gradients with a set of samples drawn from the approximate posterior distribution. The particular structure of the covariance allows us to draw independent samples solving a certain system of equations. This KL-divergence for MGVI is implemented in the class MetricGaussianKL within NIFTy5. The demo getting_started_3.py for example not only infers a field this way, but also the power spectrum of the process that has generated the field. The cross-correlation of field and power spectrum is taken care of in this process. Posterior samples can be obtained to study this cross-correlation. It should be noted that MGVI, as any VI method, can typically only provide a lower bound on the variance.
 ... ... @@ -19,6 +19,7 @@ from .field import Field from .multi_field import MultiField from .operators.operator import Operator from .operators.adder import Adder from .operators.diagonal_operator import DiagonalOperator from .operators.distributors import DOFDistributor, PowerDistributor from .operators.domain_tuple_field_inserter import DomainTupleFieldInserter ... ... @@ -33,7 +34,6 @@ from .operators.field_zero_padder import FieldZeroPadder from .operators.inversion_enabler import InversionEnabler from .operators.linear_operator import LinearOperator from .operators.mask_operator import MaskOperator from .operators.offset_operator import OffsetOperator from .operators.qht_operator import QHTOperator from .operators.regridding_operator import RegriddingOperator from .operators.sampling_enabler import SamplingEnabler ... ... @@ -49,7 +49,7 @@ from .operators.simple_linear_operators import ( from .operators.value_inserter import ValueInserter from .operators.energy_operators import ( EnergyOperator, GaussianEnergy, PoissonianEnergy, InverseGammaLikelihood, BernoulliEnergy, Hamiltonian, AveragedEnergy) BernoulliEnergy, StandardHamiltonian, AveragedEnergy) from .probing import probe_with_posterior_samples, probe_diagonal, \ StatCalculator ... ... @@ -68,7 +68,7 @@ from .minimization.scipy_minimizer import L_BFGS_B from .minimization.energy import Energy from .minimization.quadratic_energy import QuadraticEnergy from .minimization.energy_adapter import EnergyAdapter from .minimization.kl_energy import KL_Energy from .minimization.metric_gaussian_kl import MetricGaussianKL from .sugar import * from .plot import Plot ... ...
 ... ... @@ -16,10 +16,9 @@ # NIFTy is being developed at the Max-Planck-Institut fuer Astrophysik. from ..minimization.energy_adapter import EnergyAdapter from ..multi_domain import MultiDomain from ..multi_field import MultiField from ..operators.distributors import PowerDistributor from ..operators.energy_operators import Hamiltonian, InverseGammaLikelihood from ..operators.energy_operators import StandardHamiltonian, InverseGammaLikelihood from ..operators.scaling_operator import ScalingOperator from ..operators.simple_linear_operators import ducktape ... ... @@ -35,25 +34,27 @@ def make_adjust_variances(a, Constructs a Hamiltonian to solve constant likelihood optimizations of the form phi = a * xi under the constraint that phi remains constant. FIXME xi is white. Parameters ---------- a : Operator Operator which gives the amplitude when evaluated at a position Gives the amplitude when evaluated at a position. xi : Operator Operator which gives the excitation when evaluated at a position Gives the excitation when evaluated at a position. position : Field, MultiField Position of the whole problem Position of the entire problem. samples : Field, MultiField Residual samples of the whole problem Residual samples of the whole problem. scaling : Float Optional rescaling of the Likelihood Optional rescaling of the Likelihood. ic_samp : Controller Iteration Controller for Hamiltonian Iteration Controller for Hamiltonian. Returns ------- Hamiltonian A Hamiltonian that can be used for further minimization StandardHamiltonian A Hamiltonian that can be used for further minimization. """ d = a*xi ... ... @@ -71,7 +72,7 @@ def make_adjust_variances(a, if scaling is not None: x = ScalingOperator(scaling, x.target)(x) return Hamiltonian(InverseGammaLikelihood(d_eval)(x), ic_samp=ic_samp) return StandardHamiltonian(InverseGammaLikelihood(d_eval)(x), ic_samp=ic_samp) def do_adjust_variances(position, ... ... @@ -79,6 +80,9 @@ def do_adjust_variances(position, minimizer, xi_key='xi', samples=[]): ''' FIXME ''' h_space = position[xi_key].domain[0] pd = PowerDistributor(h_space, amplitude_operator.target[0]) ... ...
 ... ... @@ -24,7 +24,7 @@ from ..operators.harmonic_operators import HarmonicTransformOperator from ..operators.simple_linear_operators import ducktape def CorrelatedField(target, amplitude_operator, name='xi'): def CorrelatedField(target, amplitude_operator, name='xi', codomain=None): """Constructs an operator which turns a white Gaussian excitation field into a correlated field. ... ... @@ -42,16 +42,21 @@ def CorrelatedField(target, amplitude_operator, name='xi'): amplitude_operator: Operator name : string :class:MultiField key for the xi-field. codomain : Domain The codomain for target[0]. If not supplied, it is inferred. Returns ------- Correlated field : Operator Operator Correlated field """ tgt = DomainTuple.make(target) if len(tgt) > 1: raise ValueError h_space = tgt[0].get_default_codomain() ht = HarmonicTransformOperator(h_space, tgt[0]) if codomain is None: codomain = tgt[0].get_default_codomain() h_space = codomain ht = HarmonicTransformOperator(h_space, target=tgt[0]) p_space = amplitude_operator.target[0] power_distributor = PowerDistributor(h_space, p_space) A = power_distributor(amplitude_operator) ... ... @@ -70,7 +75,7 @@ def MfCorrelatedField(target, amplitudes, name='xi'): Parameters ---------- target : Domain, DomainTuple or tuple of Domain Target of the operator. Must contain exactly one space. Target of the operator. Must contain exactly two spaces. amplitudes: iterable of Operator List of two amplitude operators. name : string ... ... @@ -78,7 +83,8 @@ def MfCorrelatedField(target, amplitudes, name='xi'): Returns ------- Correlated field : Operator Operator Correlated field """ tgt = DomainTuple.make(target) if len(tgt) != 2: ... ... @@ -88,7 +94,7 @@ def MfCorrelatedField(target, amplitudes, name='xi'): hsp = DomainTuple.make([tt.get_default_codomain() for tt in tgt]) ht1 = HarmonicTransformOperator(hsp, target=tgt[0], space=0) ht2 = HarmonicTransformOperator(ht1.target, space=1) ht2 = HarmonicTransformOperator(ht1.target, target=tgt[1], space=1) ht = ht2 @ ht1 psp = [aa.target[0] for aa in amplitudes] ... ...
 ... ... @@ -43,7 +43,8 @@ def _make_dynamic_operator(target, causal, minimum_phase, sigc=None, quant=None): quant=None, codomain=None): if not isinstance(target, RGSpace): raise TypeError("RGSpace required") if not target.harmonic: ... ... @@ -64,7 +65,9 @@ def _make_dynamic_operator(target, if cone and (sigc is None or quant is None): raise RuntimeError dom = DomainTuple.make(target.get_default_codomain()) if codomain is None: codomain = target.get_default_codomain() dom = DomainTuple.make(codomain) ops = {} FFT = FFTOperator(dom) Real = Realizer(dom) ... ...
 ... ... @@ -20,8 +20,8 @@ import numpy as np from ..domain_tuple import DomainTuple from ..domains.power_space import PowerSpace from ..field import Field from ..operators.adder import Adder from ..operators.exp_transform import ExpTransform from ..operators.offset_operator import OffsetOperator from ..operators.qht_operator import QHTOperator from ..operators.slope_operator import SlopeOperator from ..operators.symmetrizing_operator import SymmetrizingOperator ... ... @@ -29,7 +29,7 @@ from ..sugar import makeOp def _ceps_kernel(k, a, k0): return (a/(1+np.sum((k.T/k0)**2, axis=-1).T))**2 return (a/(1 + np.sum((k.T/k0)**2, axis=-1).T))**2 def CepstrumOperator(target, a, k0): ... ... @@ -189,7 +189,7 @@ def SLAmplitude(*, target, n_pix, a, k0, sm, sv, im, iv, keys=['tau', 'phi']): sig = np.array([sv, iv]) mean = Field.from_global_data(sl.domain, mean) sig = Field.from_global_data(sl.domain, sig) linear = (sl @ OffsetOperator(mean) @ makeOp(sig)).ducktape(keys[1]) linear = sl @ Adder(mean) @ makeOp(sig).ducktape(keys[1]) # Combine linear and smooth component loglog_ampl = 0.5*(smooth + linear) ... ...
 ... ... @@ -20,31 +20,70 @@ from ..linearization import Linearization from .. import utilities class KL_Energy(Energy): def __init__(self, position, h, nsamp, constants=[], constants_samples=None, gen_mirrored_samples=False, class MetricGaussianKL(Energy): """Provides the sampled Kullback-Leibler divergence between a distribution and a Metric Gaussian. A Metric Gaussian is used to approximate some other distribution. It is a Gaussian distribution that uses the Fisher Information Metric of the other distribution at the location of its mean to approximate the variance. In order to infer the mean, the a stochastic estimate of the Kullback-Leibler divergence is minimized. This estimate is obtained by drawing samples from the Metric Gaussian at the current mean. During minimization these samples are kept constant, updating only the mean. Due to the typically nonlinear structure of the true distribution these samples have to be updated by re-initializing this class at some point. Here standard parametrization of the true distribution is assumed. Parameters ---------- mean : Field The current mean of the Gaussian. hamiltonian : StandardHamiltonian The StandardHamiltonian of the approximated probability distribution. n_samples : integer The number of samples used to stochastically estimate the KL. constants : list A list of parameter keys that are kept constant during optimization. point_estimates : list A list of parameter keys for which no samples are drawn, but that are optimized for, corresponding to point estimates of these. mirror_samples : boolean Whether the negative of the drawn samples are also used, as they are equaly legitimate samples. If true, the number of used samples doubles. Mirroring samples stabilizes the KL estimate as extreme sample variation is counterbalanced. (default : False) Notes ----- For further details see: Metric Gaussian Variational Inference (in preparation) """ def __init__(self, mean, hamiltonian, n_sampels, constants=[], point_estimates=None, mirror_samples=False, _samples=None): super(KL_Energy, self).__init__(position) if h.domain is not position.domain: super(MetricGaussianKL, self).__init__(mean) if hamiltonian.domain is not mean.domain: raise TypeError self._h = h self._hamiltonian = hamiltonian self._constants = constants if constants_samples is None: constants_samples = constants self._constants_samples = constants_samples if point_estimates is None: point_estimates = constants self._constants_samples = point_estimates if _samples is None: met = h(Linearization.make_partial_var( position, constants_samples, True)).metric met = hamiltonian(Linearization.make_partial_var( mean, point_estimates, True)).metric _samples = tuple(met.draw_sample(from_inverse=True) for _ in range(nsamp)) if gen_mirrored_samples: for _ in range(n_sampels)) if mirror_samples: _samples += tuple(-s for s in _samples) self._samples = _samples self._lin = Linearization.make_partial_var(position, constants) self._lin = Linearization.make_partial_var(mean, constants) v, g = None, None for s in self._samples: tmp = self._h(self._lin+s) tmp = self._hamiltonian(self._lin+s) if v is None: v = tmp.val.local_data[()] g = tmp.gradient ... ... @@ -56,7 +95,7 @@ class KL_Energy(Energy): self._metric = None def at(self, position):