SHOGUN  4.1.0
CRationalApproximation Class Referenceabstract

## Detailed Description

Abstract base class of the rational approximation of a function of a linear operator (A) times vector (v) using Cauchy's integral formula -

$f(\text{A})\text{v}=\oint_{\Gamma}f(z)(z\text{I}-\text{A})^{-1} \text{v}dz$

Computes eigenvalues of linear operator and uses Jacobi elliptic functions and conformal maps [2] for quadrature rule for discretizing the contour integral and computes complex shifts, weights and constant multiplier of the rational approximation of the above expression as

$f(\text{A})\text{v}\approx \eta\text{A}\Im-\left(\sum_{l=1}^{N}\alpha_{l} (\text{A}-\sigma_{l}\text{I})^{-1}\text{v}\right)$

where $$\alpha_{l},\sigma_{l}\in\mathbb{C}$$ are respectively the shifts and weights of the linear systems generated from the rational approximation, and $$\eta\in\mathbb{R}$$ is the constant multiplier, equals to $$\frac{-8K(\lambda_{m}\lambda_{M})^{\frac{1}{4}}}{k\pi N}$$.

The number of shifts is automatically computed based on a previously specified accuracy $$\epsilon$$ using the error bound

$-1.5\left(\log\left( \frac{\lambda_\text{max}}{\lambda_\text{min}}\right)+6.0 \right)\frac{\log(\epsilon)}{2\pi^2}.$

It can also manually be set.

Reference: [1] Aune, E., D. Simpson, and J. Eidsvik (2012). Parameter estimation in high dimensional gaussian distributions. Technical Report Statistics 5/2012, NTNU.

[2] Nicholas Hale, Nicholas J. Higham and Lloyd N. Trefethen (2008). Computing $$A^{\alpha}$$ , $$log(A)$$ and related matrix functions by contour integrals. SIAM Journal of Numerical Analysis, 46:2505-2523

Note: The implementation of compute_weights_shifts_const function has been adapted from KRYLSTAT (Copyright 2011 by Erlend Aune erlen.nosp@m.da@m.nosp@m.ath.n.nosp@m.tnu..nosp@m.no) under GPL2+. See https://github.com/Froskekongen/KRYLSTAT.

Definition at line 63 of file RationalApproximation.h.

Inheritance diagram for CRationalApproximation:
[legend]

## Public Member Functions

CRationalApproximation ()

CRationalApproximation (CLinearOperator< SGVector< float64_t >, SGVector< float64_t > > *linear_operator, CIndependentComputationEngine *computation_engine, CEigenSolver *eigen_solver, float64_t desired_accuracy, EOperatorFunction function_type)

virtual ~CRationalApproximation ()

virtual void precompute ()

int32_t compute_num_shifts_from_accuracy ()

virtual CJobResultAggregatorsubmit_jobs (SGVector< float64_t > sample)=0

SGVector< complex128_tget_shifts () const

SGVector< complex128_tget_weights () const

float64_t get_constant_multiplier () const

index_t get_num_shifts () const

void set_num_shifts (index_t num_shifts)

virtual const char * get_name () const

CLinearOperator< SGVector
< float64_t >, SGVector
< float64_t > > *
get_operator () const

virtual CSGObjectshallow_copy () const

virtual CSGObjectdeep_copy () const

virtual bool is_generic (EPrimitiveType *generic) const

template<class T >
void set_generic ()

template<>
void set_generic ()

template<>
void set_generic ()

template<>
void set_generic ()

template<>
void set_generic ()

template<>
void set_generic ()

template<>
void set_generic ()

template<>
void set_generic ()

template<>
void set_generic ()

template<>
void set_generic ()

template<>
void set_generic ()

template<>
void set_generic ()

template<>
void set_generic ()

template<>
void set_generic ()

template<>
void set_generic ()

template<>
void set_generic ()

void unset_generic ()

virtual void print_serializable (const char *prefix="")

virtual bool save_serializable (CSerializableFile *file, const char *prefix="", int32_t param_version=Version::get_version_parameter())

virtual bool load_serializable (CSerializableFile *file, const char *prefix="", int32_t param_version=Version::get_version_parameter())

DynArray< TParameter * > * load_file_parameters (const SGParamInfo *param_info, int32_t file_version, CSerializableFile *file, const char *prefix="")

DynArray< TParameter * > * load_all_file_parameters (int32_t file_version, int32_t current_version, CSerializableFile *file, const char *prefix="")

void map_parameters (DynArray< TParameter * > *param_base, int32_t &base_version, DynArray< const SGParamInfo * > *target_param_infos)

void set_global_io (SGIO *io)

SGIOget_global_io ()

void set_global_parallel (Parallel *parallel)

Parallelget_global_parallel ()

void set_global_version (Version *version)

Versionget_global_version ()

SGStringList< char > get_modelsel_names ()

void print_modsel_params ()

char * get_modsel_param_descr (const char *param_name)

index_t get_modsel_param_index (const char *param_name)

void build_gradient_parameter_dictionary (CMap< TParameter *, CSGObject * > *dict)

virtual void update_parameter_hash ()

virtual bool parameter_hash_changed ()

virtual bool equals (CSGObject *other, float64_t accuracy=0.0, bool tolerant=false)

virtual CSGObjectclone ()

## Public Attributes

SGIOio

Parallelparallel

Versionversion

Parameterm_parameters

Parameterm_model_selection_parameters

ParameterMapm_parameter_map

uint32_t m_hash

## Protected Member Functions

virtual TParametermigrate (DynArray< TParameter * > *param_base, const SGParamInfo *target)

virtual void one_to_one_migration_prepare (DynArray< TParameter * > *param_base, const SGParamInfo *target, TParameter *&replacement, TParameter *&to_migrate, char *old_name=NULL)

virtual void load_serializable_pre () throw (ShogunException)

virtual void load_serializable_post () throw (ShogunException)

virtual void save_serializable_pre () throw (ShogunException)

virtual void save_serializable_post () throw (ShogunException)

## Protected Attributes

CEigenSolverm_eigen_solver

SGVector< complex128_tm_shifts

SGVector< complex128_tm_weights

float64_t m_constant_multiplier

int32_t m_num_shifts

float64_t m_desired_accuracy

CLinearOperator< SGVector
< float64_t >, SGVector
< float64_t > > *
m_linear_operator

CIndependentComputationEnginem_computation_engine

const EOperatorFunction m_function_type

## Constructor & Destructor Documentation

 CRationalApproximation ( )

default constructor

Definition at line 25 of file RationalApproximation.cpp.

 CRationalApproximation ( CLinearOperator< SGVector< float64_t >, SGVector< float64_t > > * linear_operator, CIndependentComputationEngine * computation_engine, CEigenSolver * eigen_solver, float64_t desired_accuracy, EOperatorFunction function_type )

Constructor. Number of shifts will be computed using a specified accuracy.

Parameters
 linear_operator real valued linear operator for this operator function computation_engine engine that computes the independent jobs eigen_solver eigen solver for computing min and max eigenvalues needed for computing shifts, weights and constant multiplier desired_accuracy desired error bound on approximation. Computes the number of shifts automatically function_type operator function type

Definition at line 33 of file RationalApproximation.cpp.

 ~CRationalApproximation ( )
virtual

destructor

Definition at line 52 of file RationalApproximation.cpp.

## Member Function Documentation

 void build_gradient_parameter_dictionary ( CMap< TParameter *, CSGObject * > * dict )
inherited

Builds a dictionary of all parameters in SGObject as well of those of SGObjects that are parameters of this object. Dictionary maps parameters to the objects that own them.

Parameters
 dict dictionary of parameters to be built.

Definition at line 1244 of file SGObject.cpp.

 CSGObject * clone ( )
virtualinherited

Creates a clone of the current object. This is done via recursively traversing all parameters, which corresponds to a deep copy. Calling equals on the cloned object always returns true although none of the memory of both objects overlaps.

Returns
an identical copy of the given object, which is disjoint in memory. NULL if the clone fails. Note that the returned object is SG_REF'ed

Definition at line 1361 of file SGObject.cpp.

 int32_t compute_num_shifts_from_accuracy ( )

Computes the number of shifts from the current set accuracy $$\epsilon$$ using

$-1.5\left(\log\left( \frac{\lambda_\text{max}}{\lambda_\text{min}}\right)+6.0 \right)\frac{\log(\epsilon)}{2\pi^2},$

Returns
number of shift to reach the above error bound

Definition at line 129 of file RationalApproximation.cpp.

 CSGObject * deep_copy ( ) const
virtualinherited

A deep copy. All the instance variables will also be copied.

Definition at line 200 of file SGObject.cpp.

 bool equals ( CSGObject * other, float64_t accuracy = 0.0, bool tolerant = false )
virtualinherited

Recursively compares the current SGObject to another one. Compares all registered numerical parameters, recursion upon complex (SGObject) parameters. Does not compare pointers!

May be overwritten but please do with care! Should not be necessary in most cases.

Parameters
 other object to compare with accuracy accuracy to use for comparison (optional) tolerant allows linient check on float equality (within accuracy)
Returns
true if all parameters were equal, false if not

Definition at line 1265 of file SGObject.cpp.

 float64_t get_constant_multiplier ( ) const
Returns
constant multiplier

Definition at line 96 of file RationalApproximation.cpp.

 SGIO * get_global_io ( )
inherited

get the io object

Returns
io object

Definition at line 237 of file SGObject.cpp.

 Parallel * get_global_parallel ( )
inherited

get the parallel object

Returns
parallel object

Definition at line 279 of file SGObject.cpp.

 Version * get_global_version ( )
inherited

get the version object

Returns
version object

Definition at line 292 of file SGObject.cpp.

 SGStringList< char > get_modelsel_names ( )
inherited
Returns
vector of names of all parameters which are registered for model selection

Definition at line 1136 of file SGObject.cpp.

 char * get_modsel_param_descr ( const char * param_name )
inherited

Returns description of a given parameter string, if it exists. SG_ERROR otherwise

Parameters
 param_name name of the parameter
Returns
description of the parameter

Definition at line 1160 of file SGObject.cpp.

 index_t get_modsel_param_index ( const char * param_name )
inherited

Returns index of model selection parameter with provided index

Parameters
 param_name name of model selection parameter
Returns
index of model selection parameter with provided name, -1 if there is no such

Definition at line 1173 of file SGObject.cpp.

 virtual const char* get_name ( ) const
virtual
Returns
object name

Reimplemented from COperatorFunction< float64_t >.

Reimplemented in CLogRationalApproximationIndividual, and CLogRationalApproximationCGM.

Definition at line 142 of file RationalApproximation.h.

 index_t get_num_shifts ( ) const
Returns
number of shifts

Definition at line 101 of file RationalApproximation.cpp.

 CLinearOperator, SGVector >* get_operator ( ) const
inherited
Returns
the operator

Definition at line 81 of file OperatorFunction.h.

 SGVector< complex128_t > get_shifts ( ) const
Returns
shifts

Definition at line 86 of file RationalApproximation.cpp.

 SGVector< complex128_t > get_weights ( ) const
Returns
weights

Definition at line 91 of file RationalApproximation.cpp.

 bool is_generic ( EPrimitiveType * generic ) const
virtualinherited

If the SGSerializable is a class template then TRUE will be returned and GENERIC is set to the type of the generic.

Parameters
 generic set to the type of the generic if returning TRUE
Returns
TRUE if a class template.

Definition at line 298 of file SGObject.cpp.

 DynArray< TParameter * > * load_all_file_parameters ( int32_t file_version, int32_t current_version, CSerializableFile * file, const char * prefix = "" )
inherited

maps all parameters of this instance to the provided file version and loads all parameter data from the file into an array, which is sorted (basically calls load_file_parameter(...) for all parameters and puts all results into a sorted array)

Parameters
 file_version parameter version of the file current_version version from which mapping begins (you want to use Version::get_version_parameter() for this in most cases) file file to load from prefix prefix for members
Returns
(sorted) array of created TParameter instances with file data

Definition at line 705 of file SGObject.cpp.

 DynArray< TParameter * > * load_file_parameters ( const SGParamInfo * param_info, int32_t file_version, CSerializableFile * file, const char * prefix = "" )
inherited

loads some specified parameters from a file with a specified version The provided parameter info has a version which is recursively mapped until the file parameter version is reached. Note that there may be possibly multiple parameters in the mapping, therefore, a set of TParameter instances is returned

Parameters
 param_info information of parameter file_version parameter version of the file, must be <= provided parameter version file file to load from prefix prefix for members
Returns
new array with TParameter instances with the attached data

Definition at line 546 of file SGObject.cpp.

 bool load_serializable ( CSerializableFile * file, const char * prefix = "", int32_t param_version = Version::get_version_parameter() )
virtualinherited

Load this object from file. If it will fail (returning FALSE) then this object will contain inconsistent data and should not be used!

Parameters
 file where to load from prefix prefix for members param_version (optional) a parameter version different to (this is mainly for testing, better do not use)
Returns
TRUE if done, otherwise FALSE

Definition at line 375 of file SGObject.cpp.

 void load_serializable_post ( ) throw ( ShogunException )
protectedvirtualinherited

Can (optionally) be overridden to post-initialize some member variables which are not PARAMETER::ADD'ed. Make sure that at first the overridden method BASE_CLASS::LOAD_SERIALIZABLE_POST is called.

Exceptions
 ShogunException will be thrown if an error occurs.

Definition at line 1063 of file SGObject.cpp.

 void load_serializable_pre ( ) throw ( ShogunException )
protectedvirtualinherited

Can (optionally) be overridden to pre-initialize some member variables which are not PARAMETER::ADD'ed. Make sure that at first the overridden method BASE_CLASS::LOAD_SERIALIZABLE_PRE is called.

Exceptions
 ShogunException will be thrown if an error occurs.

Definition at line 1058 of file SGObject.cpp.

 void map_parameters ( DynArray< TParameter * > * param_base, int32_t & base_version, DynArray< const SGParamInfo * > * target_param_infos )
inherited

Takes a set of TParameter instances (base) with a certain version and a set of target parameter infos and recursively maps the base level wise to the current version using CSGObject::migrate(...). The base is replaced. After this call, the base version containing parameters should be of same version/type as the initial target parameter infos. Note for this to work, the migrate methods and all the internal parameter mappings have to match

Parameters
 param_base set of TParameter instances that are mapped to the provided target parameter infos base_version version of the parameter base target_param_infos set of SGParamInfo instances that specify the target parameter base

Definition at line 743 of file SGObject.cpp.

 TParameter * migrate ( DynArray< TParameter * > * param_base, const SGParamInfo * target )
protectedvirtualinherited

creates a new TParameter instance, which contains migrated data from the version that is provided. The provided parameter data base is used for migration, this base is a collection of all parameter data of the previous version. Migration is done FROM the data in param_base TO the provided param info Migration is always one version step. Method has to be implemented in subclasses, if no match is found, base method has to be called.

If there is an element in the param_base which equals the target, a copy of the element is returned. This represents the case when nothing has changed and therefore, the migrate method is not overloaded in a subclass

Parameters
 param_base set of TParameter instances to use for migration target parameter info for the resulting TParameter
Returns
a new TParameter instance with migrated data from the base of the type which is specified by the target parameter

Definition at line 950 of file SGObject.cpp.

 void one_to_one_migration_prepare ( DynArray< TParameter * > * param_base, const SGParamInfo * target, TParameter *& replacement, TParameter *& to_migrate, char * old_name = NULL )
protectedvirtualinherited

This method prepares everything for a one-to-one parameter migration. One to one here means that only ONE element of the parameter base is needed for the migration (the one with the same name as the target). Data is allocated for the target (in the type as provided in the target SGParamInfo), and a corresponding new TParameter instance is written to replacement. The to_migrate pointer points to the single needed TParameter instance needed for migration. If a name change happened, the old name may be specified by old_name. In addition, the m_delete_data flag of to_migrate is set to true. So if you want to migrate data, the only thing to do after this call is converting the data in the m_parameter fields. If unsure how to use - have a look into an example for this. (base_migration_type_conversion.cpp for example)

Parameters
 param_base set of TParameter instances to use for migration target parameter info for the resulting TParameter replacement (used as output) here the TParameter instance which is returned by migration is created into to_migrate the only source that is used for migration old_name with this parameter, a name change may be specified

Definition at line 890 of file SGObject.cpp.

 bool parameter_hash_changed ( )
virtualinherited
Returns
whether parameter combination has changed since last update

Definition at line 264 of file SGObject.cpp.

 void precompute ( )
virtual

precompute method that computes extremal eigenvalues using the eigensolver and then computes complex shifts, weights and constant multiplier coming from rational approximation of operator function times vector

Automatically computes the number of shifts if they have not been specified or are zero using set_shifts_from_accuracy().

Implements COperatorFunction< float64_t >.

Definition at line 111 of file RationalApproximation.cpp.

 void print_modsel_params ( )
inherited

prints all parameter registered for model selection and their type

Definition at line 1112 of file SGObject.cpp.

 void print_serializable ( const char * prefix = "" )
virtualinherited

prints registered parameters out

Parameters
 prefix prefix for members

Definition at line 310 of file SGObject.cpp.

 bool save_serializable ( CSerializableFile * file, const char * prefix = "", int32_t param_version = Version::get_version_parameter() )
virtualinherited

Save this object to file.

Parameters
 file where to save the object; will be closed during returning if PREFIX is an empty string. prefix prefix for members param_version (optional) a parameter version different to (this is mainly for testing, better do not use)
Returns
TRUE if done, otherwise FALSE

Definition at line 316 of file SGObject.cpp.

 void save_serializable_post ( ) throw ( ShogunException )
protectedvirtualinherited

Can (optionally) be overridden to post-initialize some member variables which are not PARAMETER::ADD'ed. Make sure that at first the overridden method BASE_CLASS::SAVE_SERIALIZABLE_POST is called.

Exceptions
 ShogunException will be thrown if an error occurs.

Reimplemented in CKernel.

Definition at line 1073 of file SGObject.cpp.

 void save_serializable_pre ( ) throw ( ShogunException )
protectedvirtualinherited

Can (optionally) be overridden to pre-initialize some member variables which are not PARAMETER::ADD'ed. Make sure that at first the overridden method BASE_CLASS::SAVE_SERIALIZABLE_PRE is called.

Exceptions
 ShogunException will be thrown if an error occurs.

Definition at line 1068 of file SGObject.cpp.

 void set_generic ( )
inherited

Definition at line 42 of file SGObject.cpp.

 void set_generic ( )
inherited

Definition at line 47 of file SGObject.cpp.

 void set_generic ( )
inherited

Definition at line 52 of file SGObject.cpp.

 void set_generic ( )
inherited

Definition at line 57 of file SGObject.cpp.

 void set_generic ( )
inherited

Definition at line 62 of file SGObject.cpp.

 void set_generic ( )
inherited

Definition at line 67 of file SGObject.cpp.

 void set_generic ( )
inherited

Definition at line 72 of file SGObject.cpp.

 void set_generic ( )
inherited

Definition at line 77 of file SGObject.cpp.

 void set_generic ( )
inherited

Definition at line 82 of file SGObject.cpp.

 void set_generic ( )
inherited

Definition at line 87 of file SGObject.cpp.

 void set_generic ( )
inherited

Definition at line 92 of file SGObject.cpp.

 void set_generic ( )
inherited

Definition at line 97 of file SGObject.cpp.

 void set_generic ( )
inherited

Definition at line 102 of file SGObject.cpp.

 void set_generic ( )
inherited

Definition at line 107 of file SGObject.cpp.

 void set_generic ( )
inherited

Definition at line 112 of file SGObject.cpp.

 void set_generic ( )
inherited

set generic type to T

 void set_global_io ( SGIO * io )
inherited

set the io object

Parameters
 io io object to use

Definition at line 230 of file SGObject.cpp.

 void set_global_parallel ( Parallel * parallel )
inherited

set the parallel object

Parameters
 parallel parallel object to use

Definition at line 243 of file SGObject.cpp.

 void set_global_version ( Version * version )
inherited

set the version object

Parameters
 version version object to use

Definition at line 285 of file SGObject.cpp.

 void set_num_shifts ( index_t num_shifts )
Parameters
 num_shifts number of shifts

Definition at line 106 of file RationalApproximation.cpp.

 CSGObject * shallow_copy ( ) const
virtualinherited

A shallow copy. All the SGObject instance variables will be simply assigned and SG_REF-ed.

Reimplemented in CGaussianKernel.

Definition at line 194 of file SGObject.cpp.

 virtual CJobResultAggregator* submit_jobs ( SGVector< float64_t > sample )
pure virtual

abstract method that creates a job result aggregator, then creates a number of jobs based on its implementation, attaches the aggregator with all those jobs, hands over the responsility of those to the computation engine and then returns the aggregator for collecting the job results

Parameters
 sample the vector for which new computation job(s) are to be created
Returns
the array of generated independent jobs

Implements COperatorFunction< float64_t >.

Implemented in CLogRationalApproximationIndividual, and CLogRationalApproximationCGM.

 void unset_generic ( )
inherited

unset generic type

this has to be called in classes specializing a template class

Definition at line 305 of file SGObject.cpp.

 void update_parameter_hash ( )
virtualinherited

Updates the hash of current parameter combination

Definition at line 250 of file SGObject.cpp.

## Member Data Documentation

 SGIO* io
inherited

io

Definition at line 496 of file SGObject.h.

 CIndependentComputationEngine* m_computation_engine
protectedinherited

the computation engine

Definition at line 115 of file OperatorFunction.h.

 float64_t m_constant_multiplier
protected

constant multiplier

Definition at line 158 of file RationalApproximation.h.

 float64_t m_desired_accuracy
protected

desired accuracy from which number of shifts might be computed

Definition at line 164 of file RationalApproximation.h.

 CEigenSolver* m_eigen_solver
protected

the eigen solver for computing extremal eigenvalues

Definition at line 149 of file RationalApproximation.h.

 const EOperatorFunction m_function_type
protectedinherited

the linear operator function type

Definition at line 118 of file OperatorFunction.h.

inherited

parameters wrt which we can compute gradients

Definition at line 511 of file SGObject.h.

 uint32_t m_hash
inherited

Hash of parameter values

Definition at line 517 of file SGObject.h.

 CLinearOperator, SGVector >* m_linear_operator
protectedinherited

the linear operator

Definition at line 112 of file OperatorFunction.h.

 Parameter* m_model_selection_parameters
inherited

model selection parameters

Definition at line 508 of file SGObject.h.

 int32_t m_num_shifts
protected

number of shifts

Definition at line 161 of file RationalApproximation.h.

 ParameterMap* m_parameter_map
inherited

map for different parameter versions

Definition at line 514 of file SGObject.h.

 Parameter* m_parameters
inherited

parameters

Definition at line 505 of file SGObject.h.

 SGVector m_shifts
protected

complex shifts in the systems coming from rational approximation

Definition at line 152 of file RationalApproximation.h.

 SGVector m_weights
protected

complex weights in the systems coming from rational approximation

Definition at line 155 of file RationalApproximation.h.

 Parallel* parallel
inherited

parallel

Definition at line 499 of file SGObject.h.

 Version* version
inherited

version

Definition at line 502 of file SGObject.h.

The documentation for this class was generated from the following files:

SHOGUN Machine Learning Toolbox - Documentation