gm_to_dirac_even_closed_form< T > Class Template Reference

quadrature-free Gaussian-to-Dirac approximation for an even-dimensional standard normal target More...

#include <gm_to_dirac_even_closed_form.h>

Inheritance diagram for gm_to_dirac_even_closed_form< T >:

Public Types
using	GSLVectorType = typename gm_to_dirac_approx_standard_normal_distribution_i< T >::GSLVectorType
using	GSLMatrixType = typename gm_to_dirac_approx_standard_normal_distribution_i< T >::GSLMatrixType
Public Types inherited from gm_to_dirac_approx_standard_normal_distribution_i< T >
using	GSLVectorType = typename GSLTemplateTypeAlias< T >::VectorType
using	GSLVectorViewType = typename GSLTemplateTypeAlias< T >::VectorViewType
using	GSLMatrixType = typename GSLTemplateTypeAlias< T >::MatrixType

Public Member Functions
bool	approximate (size_t L, size_t N, double bMax, T *x, const T *wX=nullptr, GslminimizerResult *result=nullptr, const ApproximateOptions &options=ApproximateOptions{})
	approximate a standard normal by L Dirac components
void	modified_van_mises_distance_sq (T *distance, size_t L, size_t N, double bMax, T *x, const T *wX)
	true mCvM distance, D1 included
void	modified_van_mises_distance_sq_derivative (T *gradient, size_t L, size_t N, double bMax, T *x, const T *wX)
	gradient of the true mCvM distance with respect to x
bool	approximate (size_t L, size_t N, double bMax, GSLVectorType *x, const GSLVectorType *wX=nullptr, GslminimizerResult *result=nullptr, const ApproximateOptions &options=ApproximateOptions{})
void	modified_van_mises_distance_sq (T *distance, size_t L, size_t N, double bMax, GSLVectorType *x, const GSLVectorType *wX)
void	modified_van_mises_distance_sq_derivative (GSLVectorType *gradient, size_t L, size_t N, double bMax, GSLVectorType *x, const GSLVectorType *wX)
bool	approximate (size_t L, size_t N, double bMax, GSLMatrixType *x, const GSLVectorType *wX=nullptr, GslminimizerResult *result=nullptr, const ApproximateOptions &options=ApproximateOptions{})
void	modified_van_mises_distance_sq (T *distance, size_t L, size_t N, double bMax, GSLMatrixType *x, const GSLVectorType *wX)
void	modified_van_mises_distance_sq_derivative (GSLMatrixType *gradient, size_t L, size_t N, double bMax, GSLMatrixType *x, const GSLVectorType *wX)
bool	approximate_isotropic (size_t L, size_t N, double sigma, double bMax, T *x, const T *wX=nullptr, GslminimizerResult *result=nullptr, const ApproximateOptions &options=ApproximateOptions{})
	approximate an isotropic Gaussian N(0, sigma^2 * I)
void	modified_van_mises_distance_sq_isotropic (T *distance, size_t L, size_t N, double sigma, double bMax, T *x, const T *wX)
	true mCvM distance for an isotropic Gaussian N(0, sigma^2 * I)
bool	approximate (size_t L, size_t N, T *x, const T *wX, GslminimizerResult *result=nullptr, const ApproximateOptions &options=ApproximateOptions{}) override
	approximate using raw pointers
void	modified_van_mises_distance_sq (T *distance, size_t L, size_t N, size_t bMax, T *x, const T *wX) override
	calculate modified van mises distance based on standard normal deviation and x
void	modified_van_mises_distance_sq_derivative (T *gradient, size_t L, size_t N, size_t bMax, T *x, const T *wX) override
	calculate modified van mises distance based on standard normal deviation and x
bool	approximate (size_t L, size_t N, GSLVectorType *x, const GSLVectorType *wX=nullptr, GslminimizerResult *result=nullptr, const ApproximateOptions &options=ApproximateOptions{}) override
	approximate using gsl vectors
void	modified_van_mises_distance_sq (T *distance, size_t L, size_t N, size_t bMax, GSLVectorType *x, const GSLVectorType *wX) override
	calculate modified van mises distance based on standard normal deviation and x
void	modified_van_mises_distance_sq_derivative (GSLVectorType *gradient, size_t L, size_t N, size_t bMax, GSLVectorType *x, const GSLVectorType *wX) override
	calculate modified van mises distance based on standard normal deviation and x
bool	approximate (size_t L, size_t N, GSLMatrixType *x, const GSLVectorType *wX=nullptr, GslminimizerResult *result=nullptr, const ApproximateOptions &options=ApproximateOptions{}) override
	approximate using gsl vectors
void	modified_van_mises_distance_sq (T *distance, size_t L, size_t N, size_t bMax, GSLMatrixType *x, const GSLVectorType *wX) override
	calculate modified van mises distance based on standard normal deviation and x
void	modified_van_mises_distance_sq_derivative (GSLMatrixType *gradient, size_t L, size_t N, size_t bMax, GSLMatrixType *x, const GSLVectorType *wX) override
	calculate modified van mises distance based on standard normal deviation and x
bool	approximate (size_t L, size_t N, double bMax, gsl_vector_float *x, const gsl_vector_float *wX, GslminimizerResult *result, const ApproximateOptions &options)
void	modified_van_mises_distance_sq (float *distance, size_t L, size_t N, double bMax, gsl_vector_float *x, const gsl_vector_float *wX)
void	modified_van_mises_distance_sq_derivative (gsl_vector_float *gradient, size_t L, size_t N, double bMax, gsl_vector_float *x, const gsl_vector_float *wX)
bool	approximate (size_t L, size_t N, double bMax, gsl_vector *x, const gsl_vector *wX, GslminimizerResult *result, const ApproximateOptions &options)
void	modified_van_mises_distance_sq (double *distance, size_t L, size_t N, double bMax, gsl_vector *x, const gsl_vector *wX)
void	modified_van_mises_distance_sq_derivative (gsl_vector *gradient, size_t L, size_t N, double bMax, gsl_vector *x, const gsl_vector *wX)
bool	approximate (size_t L, size_t N, double bMax, gsl_vector_float *x, const gsl_vector_float *wX, GslminimizerResult *result, const ApproximateOptions &options)
bool	approximate (size_t L, size_t N, double bMax, gsl_vector *x, const gsl_vector *wX, GslminimizerResult *result, const ApproximateOptions &options)
void	modified_van_mises_distance_sq (float *distance, size_t L, size_t N, double bMax, gsl_vector_float *x, const gsl_vector_float *wX)
void	modified_van_mises_distance_sq (double *distance, size_t L, size_t N, double bMax, gsl_vector *x, const gsl_vector *wX)
void	modified_van_mises_distance_sq_derivative (gsl_vector_float *gradient, size_t L, size_t N, double bMax, gsl_vector_float *x, const gsl_vector_float *wX)
void	modified_van_mises_distance_sq_derivative (gsl_vector *gradient, size_t L, size_t N, double bMax, gsl_vector *x, const gsl_vector *wX)

Data Fields
bool	includeD1InObjective = true
	carry the sample-independent constant inside the minimizer's objective

Detailed Description

template<typename T>
class gm_to_dirac_even_closed_form< T >

quadrature-free Gaussian-to-Dirac approximation for an even-dimensional standard normal target

Same objective as gm_to_dirac_short_standard_normal_deviation, but every integral over the kernel-width variable b is evaluated in closed form (see lcd_even_closed_form.h).

One further differences from the quadrature path:

• the reported distance is the TRUE mCvM distance: it includes the D1 term, which gm_to_dirac_short omits.

Note

PRECONDITIONS: the target must be the STANDARD normal N(0, I) (or an isotropic sigma^2 * I via approximate_isotropic()), and N must be EVEN and >= 2. Odd N fails loudly rather than falling back.

Member Function Documentation

approximate() [1/4]

template<typename T >

bool gm_to_dirac_even_closed_form< T >::approximate	(	size_t	L,
		size_t	N,
		double	bMax,
		T *	x,
		const T *	wX = nullptr,
		GslminimizerResult *	result = nullptr,
		const ApproximateOptions &	options = ApproximateOptions{}
	)

approximate a standard normal by L Dirac components

Parameters

L	number of Dirac components
N	dimension, must be even and >= 2
bMax	upper bound of the b-integral, must be > 0
x	initial guess and output locations (L * N)
wX	weights, nullptr for uniform
result	minimizer result
options	minimizer options; options.bMax is ignored here

Returns

true on success, false otherwise (including odd N)

approximate() [2/4]

template<typename T >

bool gm_to_dirac_even_closed_form< T >::approximate ( size_t  L, size_t  N, GSLMatrixType *  x, const GSLVectorType *  wX = nullptr, GslminimizerResult *  result = nullptr, const ApproximateOptions &  options = ApproximateOptions{}  )

overridevirtual

approximate using gsl vectors

Parameters

L	number of data points for apprioximation
N	dimension of the data
bMax	bMax
x	first guess for the approximation and return value
wX	weights for the x data points
result	minimizer result
options	options for minimizer

Returns

true, on success, false otherwise

Implements gm_to_dirac_approx_standard_normal_distribution_i< T >.

approximate() [3/4]

template<typename T >

bool gm_to_dirac_even_closed_form< T >::approximate ( size_t  L, size_t  N, GSLVectorType *  x, const GSLVectorType *  wX = nullptr, GslminimizerResult *  result = nullptr, const ApproximateOptions &  options = ApproximateOptions{}  )

overridevirtual

approximate using gsl vectors

Parameters

L	number of data points for apprioximation
N	dimension of the data
bMax	bMax
x	first guess for the approximation and return value
wX	weights for the x data points
result	minimizer result
options	options for minimizer

Returns

true, on success, false otherwise

Implements gm_to_dirac_approx_standard_normal_distribution_i< T >.

approximate() [4/4]

template<typename T >

bool gm_to_dirac_even_closed_form< T >::approximate ( size_t  L, size_t  N, T *  x, const T *  wX, GslminimizerResult *  result = nullptr, const ApproximateOptions &  options = ApproximateOptions{}  )

overridevirtual

approximate using raw pointers

Parameters

L	number of data points for approximation
N	dimension of the data
bMax	bMax
x	first guess for the approximation and return value
result	minimizer result
options	options for minimizer

Returns

true, on success, false otherwise

Implements gm_to_dirac_approx_standard_normal_distribution_i< T >.

approximate_isotropic()

template<typename T >

bool gm_to_dirac_even_closed_form< T >::approximate_isotropic	(	size_t	L,
		size_t	N,
		double	sigma,
		double	bMax,
		T *	x,
		const T *	wX = nullptr,
		GslminimizerResult *	result = nullptr,
		const ApproximateOptions &	options = ApproximateOptions{}
	)

approximate an isotropic Gaussian N(0, sigma^2 * I)

Solves the standard-normal problem at bMax / sigma and scales the result by sigma. This is why the new API takes a double bMax: bMax / sigma is generally not an integer.

Parameters

L	number of Dirac components
N	dimension, must be even and >= 2
sigma	standard deviation, must be > 0
bMax	upper bound of the b-integral, must be > 0
x	initial guess and output locations (L * N)
wX	weights, nullptr for uniform
result	minimizer result
options	minimizer options; options.bMax is ignored here

Returns

true on success, false otherwise

Note

only isotropic covariance is supported; general diagonal covariance admits no such reduction

modified_van_mises_distance_sq() [1/4]

template<typename T >

void gm_to_dirac_even_closed_form< T >::modified_van_mises_distance_sq	(	T *	distance,
		size_t	L,
		size_t	N,
		double	bMax,
		T *	x,
		const T *	wX
	)

true mCvM distance, D1 included

Parameters

distance	output distance
L	number of Dirac components
N	dimension, must be even and >= 2
bMax	upper bound of the b-integral, must be > 0
x	sample locations (L * N)
wX	weights, nullptr for uniform

modified_van_mises_distance_sq() [2/4]

template<typename T >

void gm_to_dirac_even_closed_form< T >::modified_van_mises_distance_sq ( T *  distance, size_t  L, size_t  N, size_t  bMax, GSLMatrixType *  x, const GSLVectorType *  wX  )

overridevirtual

calculate modified van mises distance based on standard normal deviation and x

Parameters

distance	pointer to distance value to be calculated
L	number of data points for approximation
N	dimension of the data
bMax	bMax
x	first guess for the approximation and return value
result	minimizer result
options	options for minimizer

Returns

true, on success, false otherwise

Implements gm_to_dirac_approx_standard_normal_distribution_i< T >.

modified_van_mises_distance_sq() [3/4]

template<typename T >

void gm_to_dirac_even_closed_form< T >::modified_van_mises_distance_sq ( T *  distance, size_t  L, size_t  N, size_t  bMax, GSLVectorType *  x, const GSLVectorType *  wX  )

overridevirtual

calculate modified van mises distance based on standard normal deviation and x

Parameters

distance	pointer to distance value to be calculated
L	number of data points for approximation
N	dimension of the data
bMax	bMax
x	first guess for the approximation and return value
result	minimizer result
options	options for minimizer

Returns

true, on success, false otherwise

Implements gm_to_dirac_approx_standard_normal_distribution_i< T >.

modified_van_mises_distance_sq() [4/4]

template<typename T >

void gm_to_dirac_even_closed_form< T >::modified_van_mises_distance_sq ( T *  distance, size_t  L, size_t  N, size_t  bMax, T *  x, const T *  wX  )

overridevirtual

calculate modified van mises distance based on standard normal deviation and x

Parameters

distance	pointer to distance value to be calculated
L	number of data points for approximation
N	dimension of the data
bMax	bMax
x	first guess for the approximation and return value
result	minimizer result
options	options for minimizer

Returns

true, on success, false otherwise

Implements gm_to_dirac_approx_standard_normal_distribution_i< T >.

modified_van_mises_distance_sq_derivative() [1/4]

template<typename T >

void gm_to_dirac_even_closed_form< T >::modified_van_mises_distance_sq_derivative ( GSLMatrixType *  gradient, size_t  L, size_t  N, size_t  bMax, GSLMatrixType *  x, const GSLVectorType *  wX  )

overridevirtual

calculate modified van mises distance based on standard normal deviation and x

Parameters

gradient	pointer to gradient to be calculated
L	number of data points for approximation
N	dimension of the data
bMax	bMax
x	first guess for the approximation and return value
result	minimizer result
options	options for minimizer

Returns

true, on success, false otherwise

Implements gm_to_dirac_approx_standard_normal_distribution_i< T >.

modified_van_mises_distance_sq_derivative() [2/4]

template<typename T >

void gm_to_dirac_even_closed_form< T >::modified_van_mises_distance_sq_derivative ( GSLVectorType *  gradient, size_t  L, size_t  N, size_t  bMax, GSLVectorType *  x, const GSLVectorType *  wX  )

overridevirtual

calculate modified van mises distance based on standard normal deviation and x

Parameters

gradient	pointer to gradient to be calculated
L	number of data points for approximation
N	dimension of the data
bMax	bMax
x	first guess for the approximation and return value
result	minimizer result
options	options for minimizer

Returns

true, on success, false otherwise

Implements gm_to_dirac_approx_standard_normal_distribution_i< T >.

modified_van_mises_distance_sq_derivative() [3/4]

template<typename T >

void gm_to_dirac_even_closed_form< T >::modified_van_mises_distance_sq_derivative	(	T *	gradient,
		size_t	L,
		size_t	N,
		double	bMax,
		T *	x,
		const T *	wX
	)

gradient of the true mCvM distance with respect to x

Parameters

gradient	output gradient (L * N)
L	number of Dirac components
N	dimension, must be even and >= 2
bMax	upper bound of the b-integral, must be > 0
x	sample locations (L * N)
wX	weights, nullptr for uniform

modified_van_mises_distance_sq_derivative() [4/4]

template<typename T >

void gm_to_dirac_even_closed_form< T >::modified_van_mises_distance_sq_derivative ( T *  gradient, size_t  L, size_t  N, size_t  bMax, T *  x, const T *  wX  )

overridevirtual

calculate modified van mises distance based on standard normal deviation and x

Parameters

gradient	pointer to gradient to be calculated
L	number of data points for approximation
N	dimension of the data
bMax	bMax
x	first guess for the approximation and return value
result	minimizer result
options	options for minimizer

Returns

true, on success, false otherwise

Implements gm_to_dirac_approx_standard_normal_distribution_i< T >.

modified_van_mises_distance_sq_isotropic()

template<typename T >

void gm_to_dirac_even_closed_form< T >::modified_van_mises_distance_sq_isotropic	(	T *	distance,
		size_t	L,
		size_t	N,
		double	sigma,
		double	bMax,
		T *	x,
		const T *	wX
	)

true mCvM distance for an isotropic Gaussian N(0, sigma^2 * I)

D(sigma, bMax) = sigma^2 * D(1, bMax / sigma).

Parameters

distance	output distance
L	number of Dirac components
N	dimension, must be even and >= 2
sigma	standard deviation, must be > 0
bMax	upper bound of the b-integral, must be > 0
x	sample locations (L * N)
wX	weights, nullptr for uniform

Field Documentation

includeD1InObjective

template<typename T >

bool gm_to_dirac_even_closed_form< T >::includeD1InObjective = true

carry the sample-independent constant inside the minimizer's objective

The constant does not depend on x, so omitting it leaves the gradient and the minimum unchanged and saves one scalar evaluation per objective call. With it, the objective IS the distance.

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

• lib/gm_to_dirac_even/gm_to_dirac_even_closed_form.h
• lib/gm_to_dirac_even/gm_to_dirac_even_closed_form.cpp