Code Overview#

Executive summary#

The fundamental building blocks required for IFT computations are best recognized from a large distance, ignoring all technical details.

From such a perspective,

  • IFT problems largely consist of the combination of several high dimensional minimization problems.

  • Within NIFTy, operators are used to define the characteristic equations and properties of the problems.

  • The equations are built mostly from the application of linear operators, but there may also be nonlinear functions involved.

  • The unknowns in the equations represent either continuous physical fields, or they are simply individual measured data points.

  • Discretized fields have geometrical information (like locations and volume elements) associated with every entry; this information is called the field’s domain.

In the following sections, the concepts briefly presented here will be discussed in more detail; this is done in reversed order of their introduction, to avoid forward references.

Domains#

Abstract base class#

One of the fundamental building blocks of the NIFTy7 framework is the domain. Its required capabilities are expressed by the abstract Domain class. A domain must be able to answer the following queries:

  • its total number of data entries (pixels), which is accessible via the size property

  • the shape of the array that is supposed to hold these data entries (obtainable by means of the shape property)

  • equality comparison to another Domain instance

Unstructured domains#

Domains can be either structured (i.e. there is geometrical information associated with them, like position in space and volume factors), or unstructured (meaning that the data points have no associated manifold).

Unstructured domains can be described by instances of NIFTy’s UnstructuredDomain class.

Structured domains#

In contrast to unstructured domains, these domains have an assigned geometry. NIFTy requires them to provide the volume elements of their grid cells. The additional methods are specified in the abstract class StructuredDomain:

NIFTy comes with several concrete subclasses of StructuredDomain:

  • RGSpace represents a regular Cartesian grid with an arbitrary number of dimensions, which is supposed to be periodic in each dimension.

  • LogRGSpace implements a Cartesian grid with logarithmically spaced bins and an arbitrary number of dimensions.

  • HPSpace and GLSpace describe pixelisations of the 2-sphere; their counterpart in harmonic space is LMSpace, which contains spherical harmonic coefficients.

  • PowerSpace is used to describe one-dimensional power spectra.

Among these, RGSpace and LogRGSpace can be harmonic or not (depending on constructor arguments), GLSpace, HPSpace, and PowerSpace are pure position domains (i.e. nonharmonic), and LMSpace is always harmonic.

Combinations of domains#

The fundamental classes described above are often sufficient to specify the domain of a field. In some cases, however, it will be necessary to define the field on a product of elementary domains instead of a single one. More sophisticated operators also require a set of several such fields. Some examples are:

  • sky emission depending on location and energy. This could be represented by a product of an HPSpace (for location) with an RGSpace (for energy).

  • a polarized field, which could be modelled as a product of any structured domain (representing location) with a four-element UnstructuredDomain holding Stokes I, Q, U and V components.

  • a model for the sky emission, which holds both the current realisation (on a harmonic domain) and a few inferred model parameters (e.g. on an unstructured grid).

Consequently, NIFTy defines a class called DomainTuple holding a sequence of Domain objects. The full domain is specified as the product of all elementary domains. Thus, an instance of DomainTuple would be suitable to describe the first two examples above. In principle, a DomainTuple can even be empty, which implies that the field living on it is a scalar.

A DomainTuple supports iteration and indexing, and also provides the properties shape and size in analogy to the elementary Domain.

An aggregation of several DomainTuple s, each member identified by a name, is described by the MultiDomain class. In contrast to a DomainTuple a MultiDomain is a collection and does not define the product space of its elements. It would be the adequate space to use in the last of the above examples.

Fields#

Fields on a single DomainTuple#

A Field object consists of the following components:

  • a domain in form of a DomainTuple object

  • a data type (e.g. numpy.float64)

  • an array containing the actual values

Usually, the array is stored in the form of a numpy.ndarray, but for very resource-intensive tasks NIFTy also provides an alternative storage method to be used with distributed memory processing.

Fields support a wide range of arithmetic operations, either involving two fields of equal domains or a field and a scalar. Arithmetic operations are performed point-wise, and the returned field has the same domain as the input field(s). Available operators are addition (“+”), subtraction (“-“), multiplication (“*”), division (“/”), floor division (“//”) and exponentiation (“**”). Inplace operators (“+=”, etc.) are not supported. Further, boolean operators, performing a point-wise comparison of a field with either another field of equal domain or a scalar, are available as well. These include equals (“==”), not equals (“!=”), less (“<”), less or equal (“<=”), greater (“>”) and greater or equal (“>=). The domain of the field returned equals that of the input field(s), while the stored data is of boolean type.

Contractions (like summation, integration, minimum/maximum, computation of statistical moments) can be carried out either over an entire field (producing a scalar result) or over sub-domains (resulting in a field defined on a smaller domain). Scalar products of two fields can also be computed easily as well. See the documentation of Field for details.

There is also a set of convenience functions to generate fields with constant values or fields filled with random numbers according to a user-specified distribution: full, from_random.

Like almost all NIFTy objects, fields are immutable: their value or any other attribute cannot be modified after construction. To manipulate a field in ways that are not covered by the provided standard operations, its data content must be extracted first, then changed, and a new field has to be created from the result.

Fields defined on a MultiDomain#

The MultiField class can be seen as a dictionary of individual Field s, each identified by a name, which is defined on a MultiDomain.

Operators#

All transformations between different NIFTy fields are expressed in the form of Operator objects. The interface of this class is rather minimalistic: it has a property called domain which returns a DomainTuple or MultiDomain object specifying the structure of the Field or MultiField it expects as input, another property target describing its output, and finally an overloaded apply method, which can take:

This is the interface that all objects derived from Operator must implement. In addition, Operator objects can be added/subtracted, multiplied, chained (via the __call__ method or the @ operator) and support point-wise application of functions like exp(), log(), sqrt(), conjugate().

Advanced operators#

NIFTy provides a library of commonly employed operators which can be used for specific inference problems. Currently these are:

  • SLAmplitude, which returns a smooth power spectrum.

  • InverseGammaOperator, which models point sources which are distributed according to a inverse-gamma distribution.

  • CorrelatedField, which models a diffuse field whose correlation structure is described by an amplitude operator.

Linear Operators#

A linear operator (represented by NIFTy7’s abstract LinearOperator class) is derived from Operator and can be interpreted as an (implicitly defined) matrix. Since its operation is linear, it can provide some additional functionality which is not available for the more generic Operator class.

Linear Operator basics#

There are four basic ways of applying an operator A to a field s:

  • direct application: A(s)

  • adjoint application: A^\dagger (s)

  • inverse application: A^{-1} (s)

  • adjoint inverse application: (A^\dagger)^{-1} (s)

Note: The inverse of the adjoint of a linear map and the adjoint of the inverse of a linear map (if all those exist) are the same.

These different actions of a linear operator Op on a field f can be invoked in various ways:

  • direct multiplication: Op(f) or Op.times(f) or Op.apply(f, Op.TIMES)

  • adjoint multiplication: Op.adjoint_times(f) or Op.apply(f, Op.ADJOINT_TIMES)

  • inverse multiplication: Op.inverse_times(f) or Op.apply(f, Op.INVERSE_TIMES)

  • adjoint inverse multiplication: Op.adjoint_inverse_times(f) or Op.apply(f, Op.ADJOINT_INVERSE_TIMES)

Operator classes defined in NIFTy may implement an arbitrary subset of these four operations. This subset can be queried using the capability property.

If needed, the set of supported operations can be enhanced by iterative inversion methods; for example, an operator defining direct and adjoint multiplication could be enhanced by this approach to support the complete set. This functionality is provided by NIFTy’s InversionEnabler class, which is itself a linear operator.

Direct multiplication and adjoint inverse multiplication transform a field defined on the operator’s domain to one defined on the operator’s target, whereas adjoint multiplication and inverse multiplication transform from target to domain.

Operators with identical domain and target can be derived from EndomorphicOperator. Typical examples for this category are the ScalingOperator, which simply multiplies its input by a scalar value, and DiagonalOperator, which multiplies every value of its input field with potentially different values.

Further operator classes provided by NIFTy are

  • HarmonicTransformOperator for transforms from a harmonic domain to its counterpart in position space, and their adjoint

  • PowerDistributor for transforms from a PowerSpace to an associated harmonic domain, and their adjoint.

  • GeometryRemover, which transforms from structured domains to unstructured ones. This is typically needed when building instrument response operators.

Syntactic sugar#

NIFTy allows simple and intuitive construction of altered and combined operators. As an example, if A, B and C are of type LinearOperator and f1 and f2 are of type Field, writing:

X = A(B.inverse(A.adjoint)) + C
f2 = X(f1)

will perform the operation suggested intuitively by the notation, checking domain compatibility while building the composed operator. The properties adjoint and inverse return a new operator which behaves as if it were the original operator’s adjoint or inverse, respectively. The combined operator infers its domain and target from its constituents, as well as the set of operations it can support. Instantiating operator adjoints or inverses by adjoint and similar methods is to be distinguished from the instant application of operators performed by adjoint_times and similar methods.

Minimization#

Most problems in IFT are solved by (possibly nested) minimizations of high-dimensional functions, which are often nonlinear.

Energy functionals#

In NIFTy7 such functions are represented by objects of type Energy. These hold the prescription how to calculate the function’s value, gradient and (optionally) metric at any given position in parameter space. Function values are floating-point scalars, gradients have the form of fields defined on the energy’s position domain, and metrics are represented by linear operator objects.

Energies are classes that typically have to be provided by the user when tackling new IFT problems. An example of concrete energy classes delivered with NIFTy7 is QuadraticEnergy (with position-independent metric, mainly used with conjugate gradient minimization).

For MGVI and GeoVI, NIFTy provides MetricGaussianKL() and GeoMetricKL() that instantiate objects containing the sampled estimate of the KL divergence, its gradient and the Fisher metric. These constructors require an instance of StandardHamiltonian, an operator to compute the negative log-likelihood of the problem in standardized coordinates at a given position in parameter space. Finally, the StandardHamiltonian can be constructed from the likelihood, represented by a LikelihoodEnergyOperator instance. Several commonly used forms of the likelihoods are already provided in NIFTy, such as GaussianEnergy, PoissonianEnergy, InverseGammaEnergy or BernoulliEnergy, but the user is free to implement any likelihood customized to the problem at hand. The demo code demos/getting_started_3.py illustrates how to set up an energy functional for MGVI and minimize it.

Iteration control#

Iterative minimization of an energy requires some means of checking the quality of the current solution estimate and stopping once it is sufficiently accurate. In case of numerical problems, the iteration needs to be terminated as well, returning a suitable error description.

In NIFTy7, this functionality is encapsulated in the abstract IterationController class, which is provided with the initial energy object before starting the minimization, and is updated with the improved energy after every iteration. Based on this information, it can either continue the minimization or return the current estimate indicating convergence or failure.

Sensible stopping criteria can vary significantly with the problem being solved; NIFTy provides a concrete sub-class of IterationController called GradientNormController, which should be appropriate in many circumstances. A full list of the available IterationController s in NIFTy can be found below, but users have complete freedom to implement custom IterationController sub-classes for their specific applications.

Minimization algorithms#

All minimization algorithms in NIFTy inherit from the abstract Minimizer class, which presents a minimalistic interface consisting only of a __call__() method taking an Energy object and optionally a preconditioning operator, and returning the energy at the discovered minimum and a status code.

For energies with a quadratic form (i.e. which can be expressed by means of a QuadraticEnergy object), an obvious choice of algorithm is the ConjugateGradient minimizer.

A similar algorithm suited for nonlinear problems is provided by NonlinearCG.

Many minimizers for nonlinear problems can be characterized as

  • First deciding on a direction for the next step.

  • Then finding a suitable step length along this direction, resulting in the next energy estimate.

This family of algorithms is encapsulated in NIFTy’s DescentMinimizer class, which currently has three generally usable concrete implementations: NewtonCG, L_BFGS and VL_BFGS. Of these algorithms, only NewtonCG requires the energy object to provide a metric property, the others only need energy values and gradients. Further available descent minimizers are RelaxedNewton and SteepestDescent.

The flexibility of NIFTy’s design allows using externally provided minimizers. With only small effort, adaptors for two SciPy minimizers were written; they are available under the names ScipyCG and L_BFGS_B.

Application to operator inversion#

The machinery presented here cannot only be used for minimizing functionals derived from IFT, but also for the numerical inversion of linear operators, if the desired application mode is not directly available. A classical example is the information propagator whose inverse is defined as:

D^{-1} = \left(R^\dagger N^{-1} R + S^{-1}\right).

It needs to be applied in forward direction in order to calculate the Wiener filter solution, but only its inverse application is straightforward. To use it in forward direction, we make use of NIFTy’s InversionEnabler class, which internally applies the (approximate) inverse of the given operator x = Op^{-1} (y) by solving the equation y = Op (x) for x. This is accomplished by minimizing a suitable QuadraticEnergy with the ConjugateGradient algorithm. An example is provided in WienerFilterCurvature().

Posterior analysis and visualization#

After the minimization of an energy functional has converged, samples can be drawn from the posterior distribution at the current position to investigate the result. The probing module offers class called StatCalculator which allows to evaluate the mean and the unbiased variance var of these samples.

Fields can be visualized using the Plot class, which invokes matplotlib for plotting.