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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< 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< 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="")
 
virtual bool load_serializable (CSerializableFile *file, const char *prefix="")
 
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
 
Parameterm_gradient_parameters
 
uint32_t m_hash
 

Protected Member Functions

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< float64_t > * m_linear_operator
 
CIndependentComputationEnginem_computation_engine
 
const EOperatorFunction m_function_type
 

Constructor & Destructor Documentation

default constructor

Definition at line 25 of file RationalApproximation.cpp.

CRationalApproximation ( CLinearOperator< 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_operatorreal valued linear operator for this operator function
computation_engineengine that computes the independent jobs
eigen_solvereigen solver for computing min and max eigenvalues needed for computing shifts, weights and constant multiplier
desired_accuracydesired error bound on approximation. Computes the number of shifts automatically
function_typeoperator 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
dictdictionary of parameters to be built.

Definition at line 597 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 714 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 198 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
otherobject to compare with
accuracyaccuracy to use for comparison (optional)
tolerantallows linient check on float equality (within accuracy)
Returns
true if all parameters were equal, false if not

Definition at line 618 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 235 of file SGObject.cpp.

Parallel * get_global_parallel ( )
inherited

get the parallel object

Returns
parallel object

Definition at line 277 of file SGObject.cpp.

Version * get_global_version ( )
inherited

get the version object

Returns
version object

Definition at line 290 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 498 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_namename of the parameter
Returns
description of the parameter

Definition at line 522 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_namename of model selection parameter
Returns
index of model selection parameter with provided name, -1 if there is no such

Definition at line 535 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<float64_t >* 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
genericset to the type of the generic if returning TRUE
Returns
TRUE if a class template.

Definition at line 296 of file SGObject.cpp.

bool load_serializable ( CSerializableFile file,
const char *  prefix = "" 
)
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
filewhere to load from
prefixprefix for members
Returns
TRUE if done, otherwise FALSE

Definition at line 369 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
ShogunExceptionwill be thrown if an error occurs.

Reimplemented in CKernel, CWeightedDegreePositionStringKernel, CList, CAlphabet, CLinearHMM, CGaussianKernel, CInverseMultiQuadricKernel, CCircularKernel, and CExponentialKernel.

Definition at line 426 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
ShogunExceptionwill be thrown if an error occurs.

Reimplemented in CDynamicArray< T >, CDynamicArray< float64_t >, CDynamicArray< float32_t >, CDynamicArray< int32_t >, CDynamicArray< char >, CDynamicArray< bool >, and CDynamicObjectArray.

Definition at line 421 of file SGObject.cpp.

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

Definition at line 262 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 474 of file SGObject.cpp.

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

prints registered parameters out

Parameters
prefixprefix for members

Definition at line 308 of file SGObject.cpp.

bool save_serializable ( CSerializableFile file,
const char *  prefix = "" 
)
virtualinherited

Save this object to file.

Parameters
filewhere to save the object; will be closed during returning if PREFIX is an empty string.
prefixprefix for members
Returns
TRUE if done, otherwise FALSE

Definition at line 314 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
ShogunExceptionwill be thrown if an error occurs.

Reimplemented in CKernel.

Definition at line 436 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
ShogunExceptionwill be thrown if an error occurs.

Reimplemented in CKernel, CDynamicArray< T >, CDynamicArray< float64_t >, CDynamicArray< float32_t >, CDynamicArray< int32_t >, CDynamicArray< char >, CDynamicArray< bool >, and CDynamicObjectArray.

Definition at line 431 of file SGObject.cpp.

void set_generic ( )
inherited

Definition at line 41 of file SGObject.cpp.

void set_generic ( )
inherited

Definition at line 46 of file SGObject.cpp.

void set_generic ( )
inherited

Definition at line 51 of file SGObject.cpp.

void set_generic ( )
inherited

Definition at line 56 of file SGObject.cpp.

void set_generic ( )
inherited

Definition at line 61 of file SGObject.cpp.

void set_generic ( )
inherited

Definition at line 66 of file SGObject.cpp.

void set_generic ( )
inherited

Definition at line 71 of file SGObject.cpp.

void set_generic ( )
inherited

Definition at line 76 of file SGObject.cpp.

void set_generic ( )
inherited

Definition at line 81 of file SGObject.cpp.

void set_generic ( )
inherited

Definition at line 86 of file SGObject.cpp.

void set_generic ( )
inherited

Definition at line 91 of file SGObject.cpp.

void set_generic ( )
inherited

Definition at line 96 of file SGObject.cpp.

void set_generic ( )
inherited

Definition at line 101 of file SGObject.cpp.

void set_generic ( )
inherited

Definition at line 106 of file SGObject.cpp.

void set_generic ( )
inherited

Definition at line 111 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
ioio object to use

Definition at line 228 of file SGObject.cpp.

void set_global_parallel ( Parallel parallel)
inherited

set the parallel object

Parameters
parallelparallel object to use

Definition at line 241 of file SGObject.cpp.

void set_global_version ( Version version)
inherited

set the version object

Parameters
versionversion object to use

Definition at line 283 of file SGObject.cpp.

void set_num_shifts ( index_t  num_shifts)
Parameters
num_shiftsnumber 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 192 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
samplethe 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 303 of file SGObject.cpp.

void update_parameter_hash ( )
virtualinherited

Updates the hash of current parameter combination

Definition at line 248 of file SGObject.cpp.

Member Data Documentation

SGIO* io
inherited

io

Definition at line 369 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.

Parameter* m_gradient_parameters
inherited

parameters wrt which we can compute gradients

Definition at line 384 of file SGObject.h.

uint32_t m_hash
inherited

Hash of parameter values

Definition at line 387 of file SGObject.h.

CLinearOperator<float64_t >* 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 381 of file SGObject.h.

int32_t m_num_shifts
protected

number of shifts

Definition at line 161 of file RationalApproximation.h.

Parameter* m_parameters
inherited

parameters

Definition at line 378 of file SGObject.h.

SGVector<complex128_t> m_shifts
protected

complex shifts in the systems coming from rational approximation

Definition at line 152 of file RationalApproximation.h.

SGVector<complex128_t> 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 372 of file SGObject.h.

Version* version
inherited

version

Definition at line 375 of file SGObject.h.


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

SHOGUN Machine Learning Toolbox - Documentation