RKHS Gaussian kernel. More...

#include <GaussianKernel.h>

Inheritance diagram for lucid::GaussianKernel:

Public Member Functions
	GaussianKernel (Vector sigma_l, double sigma_f=1.0)
	Construct a new anisotropic GaussianKernel object with the given parameters.

	GaussianKernel (double sigma_l=1.0, double sigma_f=1.0)
	Construct a new isotropic GaussianKernel object with the given parameters.

Scalar	sigma_f () const
	Get read-only access to the \( \sigma_f \) value of the kernel.

const Vector &	sigma_l () const
	Get read-only access to the \( \sigma_l \) of the kernel.

bool	is_stationary () const override
	Check whether the kernel is is stationary.

bool	is_isotropic () const
	Check whether the kernel is isotropic.

std::unique_ptr< Kernel >	clone () const override
	Clone the kernel.

void	set (Parameter parameter, double value) override
	Set the `parameter` to the indicated `value`.

void	set (Parameter parameter, const Vector &value) override
	Set the `parameter` to the indicated `value`.

std::string	to_string () const override
	Get string representation of the kernel.

template<class DerivedX, class DerivedY>
Matrix	operator() (const MatrixBase< DerivedX > &x1, const MatrixBase< DerivedY > &x2) const
	Compute the kernel function on \( x_1 \) and \( x_2 \), both being matrices of row vectors in \( \texttip{\mathcal{X}}{Polish sample vector space} \subseteq \mathbb{R}^d \),.

template<class Derived>
Matrix	operator() (const MatrixBase< Derived > &x) const
	Compute the kernel function on \( \texttip{x}{Element of the vector space} \), which is a matrix of row vectors in \( \texttip{\mathcal{X}}{Polish sample vector space} \subseteq \mathbb{R}^d \),.

template<class Derived>
Matrix	operator() (const MatrixBase< Derived > &x, std::vector< Matrix > &gradient) const
	Compute the kernel function on \( \texttip{x}{Element of the vector space} \), which is a matrix of row vectors in \( \texttip{\mathcal{X}}{Polish sample vector space} \subseteq \mathbb{R}^d \),.

Matrix	operator() (ConstMatrixRef x1, ConstMatrixRef x2, std::vector< Matrix > *gradient) const
	Compute the kernel function on \( x_1 \) and \( x_2 \), both being matrices of row vectors in \( \texttip{\mathcal{X}}{Polish sample vector space} \subseteq \mathbb{R}^d \),.

Public Member Functions inherited from lucid::Kernel
template<class DerivedX, class DerivedY>
Matrix	operator() (const MatrixBase< DerivedX > &x1, const MatrixBase< DerivedY > &x2) const
	Compute the kernel function on \( x_1 \) and \( x_2 \), both being matrices of row vectors in \( \texttip{\mathcal{X}}{Polish sample vector space} \subseteq \mathbb{R}^d \),.

template<class Derived>
Matrix	operator() (const MatrixBase< Derived > &x) const
	Compute the kernel function on \( \texttip{x}{Element of the vector space} \), which is a matrix of row vectors in \( \texttip{\mathcal{X}}{Polish sample vector space} \subseteq \mathbb{R}^d \),.

template<class Derived>
Matrix	operator() (const MatrixBase< Derived > &x, std::vector< Matrix > &gradient) const
	Compute the kernel function on \( \texttip{x}{Element of the vector space} \), which is a matrix of row vectors in \( \texttip{\mathcal{X}}{Polish sample vector space} \subseteq \mathbb{R}^d \),.

Public Member Functions inherited from lucid::Parametrizable
template<IsAnyOf< int, double, const Vector & > T>
T	get (Parameter parameter) const
	Get the value of the specified `parameter`.

template<Parameter P>
internal::ParameterType< P >::ref_type	get () const
	Get the value of the specified `parameter`.

std::variant< int, double, Vector >	get (Parameter parameter) const
	Get the value of the specified `parameter`.

void	set (Parameter parameter, const std::variant< int, double, Vector > &value)
	Set the `parameter` to the indicated `value`.

template<Parameter P>
void	set (const std::variant< int, double, Vector > &value)
	Set the `parameter` to the indicated `value`.

void	set (Parameter parameter, std::size_t idx, const std::variant< std::vector< int >, std::vector< double >, std::vector< Vector > > &values)
	Set the `parameter` to the `index`-th value among the indicated `values`.

template<Parameter P>
void	set (std::size_t idx, const std::variant< std::vector< int >, std::vector< double >, std::vector< Vector > > &values)
	Set the `parameter` to the `index`-th value among the indicated `values`.

virtual void	set (Parameter parameter, int value)
	Set the `parameter` to the indicated `value`.

template<Parameter P>
void	set (typename internal::ParameterType< P >::ref_type value)
	Set the `parameter` to the indicated `value`.

bool	has (const Parameter parameter) const
	Check whether the `parameter` is present in this object.

Parameters	parameters () const
	Get read-only access to the parameters of the parametrizable object.

std::vector< Parameter >	parameters_list () const
	Get read-only access to the list of parameters of the parametrizable object.

Parametrizable &	load (const Parametrizable &o)
	Load parameters from another `Parametrizable` object.

Private Member Functions
Matrix	apply_impl (ConstMatrixRef x1, ConstMatrixRef x2, std::vector< Matrix > *gradient) const override
	Concrete implementation of operator()().

double	get_d (Parameter parameter) const override
	Get the value of the specified `parameter`.

const Vector &	get_v (Parameter parameter) const override
	Get the value of the specified `parameter`.

Private Attributes
Vector	sigma_l_
	\( \sigma_l \) value

Vector	log_parameters_
	[ \( \sigma_f \) \( \sigma_l \) ] value in log space. Used for optimization

double	sigma_f_
	\( \sigma_f \) value

bool	is_isotropic_
	True if the kernel is isotropic (i.e., \( \sigma_l \) is the same for all dimensions)

Additional Inherited Members
Protected Member Functions inherited from lucid::Kernel
Matrix	operator() (ConstMatrixRef x1, ConstMatrixRef x2, std::vector< Matrix > *gradient) const
	Compute the kernel function on \( x_1 \) and \( x_2 \), both being matrices of row vectors in \( \texttip{\mathcal{X}}{Polish sample vector space} \subseteq \mathbb{R}^d \),.

Protected Member Functions inherited from lucid::Parametrizable
virtual int	get_i (Parameter parameter) const
	Get the value of the specified `parameter`.

Protected Attributes inherited from lucid::Parametrizable
Parameters	parameters_
	Parameters supported by this object.

Detailed Description

RKHS Gaussian kernel.

Given a vector space \( \texttip{\mathcal{X}}{Polish sample vector space} \) and two vectors \( x_1, x_2 \in \mathcal{X} \), the Gaussian kernel is defined as

\[k(x_1, x_2) = \sigma_f^2 \exp\left(-\frac{1}{2} (x_1 - x_2)^T\Sigma(x_1 - x_2)\right) \]

where \( \Sigma = \text{diag}( \sigma _l )^{-2} \). The type of \( \sigma_l \) on kernel construction determines whether the kernel is isotropic or anisotropic. The kernel can be isotropic or anisotropic. In the anisotropic case, each dimension has its own length scale (i.e., \( \sigma_l \) value). In the isotropic case, the same length scale is used for all dimensions. If requested, the kernel can also compute the gradient of the kernel matrix with respect to \( \sigma_l \) and \( \sigma_f \).

Constructor & Destructor Documentation

◆ GaussianKernel() [1/2]

lucid::GaussianKernel::GaussianKernel	(	Vector	sigma_l,
		double	sigma_f = 1.0 )

explicit

Construct a new anisotropic GaussianKernel object with the given parameters.

Note: Even if all the elements of sigma_l are equal, the kernel will be anisotropic. This may play a role during hyperparameter optimization.

Precondition: sigma_l must contain at least one element all elements must be greater than 0

Parameters

sigma_l	\( \sigma_l \) value
sigma_f	\( \sigma_f \) value

◆ GaussianKernel() [2/2]

lucid::GaussianKernel::GaussianKernel	(	double	sigma_l = 1.0,
		double	sigma_f = 1.0 )

explicit

Construct a new isotropic GaussianKernel object with the given parameters.

Precondition: sigma_l must be greater than 0

Parameters

sigma_l	\( \sigma_l \) value. It is equal for all dimensions.
sigma_f	\( \sigma_f \) value

Member Function Documentation

◆ apply_impl()

Matrix lucid::GaussianKernel::apply_impl	(	ConstMatrixRef	x1,
		ConstMatrixRef	x2,
		std::vector< Matrix > *	gradient ) const

overrideprivatevirtual

Concrete implementation of operator()().

Parameters

	x1	\( \texttip{n_1}{Number of samples in the first matrix} \times \texttip{d}{Dimension of the vector space} \) first input matrix
	x2	\( \texttip{n_2}{Number of samples in the second matrix} \times \texttip{d}{Dimension of the vector space} \) second input matrix
[out]	gradient	pointer to store the gradient of the kernel function with respect to the kernel parameters

Returns: kernel value

Implements lucid::Kernel.

◆ clone()

std::unique_ptr< Kernel > lucid::GaussianKernel::clone ( ) const

nodiscardoverridevirtual

Clone the kernel.

Create a new instance of the kernel with the same parameters.

Returns: new instance of the kernel

Implements lucid::Kernel.

◆ get_d()

double lucid::GaussianKernel::get_d ( Parameter parameter ) const

nodiscardoverrideprivatevirtual

Get the value of the specified parameter.

Parameters

parameter parameter to retrieve

Returns: value of the parameter

Precondition: The parameter must be present and be associated with a value of type double.

Reimplemented from lucid::Parametrizable.

◆ get_v()

const Vector & lucid::GaussianKernel::get_v ( Parameter parameter ) const

nodiscardoverrideprivatevirtual

Get the value of the specified parameter.

Parameters

parameter parameter to retrieve

Returns: value of the parameter

Precondition: The parameter must be present and be associated with a value of type Vector.

Reimplemented from lucid::Parametrizable.

◆ is_isotropic()

bool lucid::GaussianKernel::is_isotropic ( ) const

inlinenodiscard

Check whether the kernel is isotropic.

Returns: true if the kernel is isotropic; false if the kernel is not isotropic

◆ is_stationary()

bool lucid::GaussianKernel::is_stationary ( ) const

inlinenodiscardoverridevirtual

Check whether the kernel is is stationary.

Returns: true if the kernel is is stationary; false if the kernel is not is stationary

Implements lucid::Kernel.

◆ operator()() [1/4]

template<class Derived>

Matrix lucid::Kernel::operator() ( const MatrixBase< Derived > & x ) const

inline

Compute the kernel function on \( \texttip{x}{Element of the vector space} \), which is a matrix of row vectors in \( \texttip{\mathcal{X}}{Polish sample vector space} \subseteq \mathbb{R}^d \),.

\[k(x, x) . \]

Template Parameters

Derived type of the input matrix

Parameters

x	\( \texttip{n}{Number of samples} \times \texttip{d}{Dimension of the vector space} \) input matrix

Returns: kernel value

◆ operator()() [2/4]

template<class Derived>

Matrix lucid::Kernel::operator()	(	const MatrixBase< Derived > &	x,
		std::vector< Matrix > &	gradient ) const

inline

Compute the kernel function on \( \texttip{x}{Element of the vector space} \), which is a matrix of row vectors in \( \texttip{\mathcal{X}}{Polish sample vector space} \subseteq \mathbb{R}^d \),.

\[k(x, x) . \]

Moreover, compute the gradient of the kernel function and store it in gradient.

Template Parameters

Derived type of the input matrix

Parameters

	x	\( \texttip{n}{Number of samples} \times \texttip{d}{Dimension of the vector space} \) input matrix
[out]	gradient	gradient of the kernel function with respect to the parameters dimensions

Returns: kernel value

◆ operator()() [3/4]

template<class DerivedX, class DerivedY>

Matrix lucid::Kernel::operator()	(	const MatrixBase< DerivedX > &	x1,
		const MatrixBase< DerivedY > &	x2 ) const

inline

Compute the kernel function on \( x_1 \) and \( x_2 \), both being matrices of row vectors in \( \texttip{\mathcal{X}}{Polish sample vector space} \subseteq \mathbb{R}^d \),.

\[k(x_1, x_2) . \]

Precondition: x1 and x2 must have the same number of columns.

Template Parameters

DerivedX	type of the first input matrix
DerivedY	type of the second input matrix

Parameters

x1	\( \texttip{n_1}{Number of samples in the first matrix} \times \texttip{d}{Dimension of the vector space} \) first input row matrix
x2	\( \texttip{n_2}{Number of samples in the second matrix} \times \texttip{d}{Dimension of the vector space} \) second input row matrix

Returns: kernel value

◆ operator()() [4/4]

Matrix lucid::Kernel::operator()	(	ConstMatrixRef	x1,
		ConstMatrixRef	x2,
		std::vector< Matrix > *	gradient ) const

Compute the kernel function on \( x_1 \) and \( x_2 \), both being matrices of row vectors in \( \texttip{\mathcal{X}}{Polish sample vector space} \subseteq \mathbb{R}^d \),.

\[k(x_1, x_2) . \]

If gradient is not nullptr, the gradient of the kernel function with respect to the parameters is computed and stored in *gradient.

Precondition: x1 and x2 must have the same number of columns.

Parameters

	x1	\( \texttip{n_1}{Number of samples in the first matrix} \times \texttip{d}{Dimension of the vector space} \) first input matrix
	x2	\( \texttip{n_2}{Number of samples in the second matrix} \times \texttip{d}{Dimension of the vector space} \) second input matrix
[out]	gradient	pointer to store the gradient of the kernel function with respect to the kernel parameters

Returns: kernel value

◆ set() [1/2]

void lucid::GaussianKernel::set	(	Parameter	parameter,
		const Vector &	value )

overridevirtual

Set the parameter to the indicated value.

Parameters

parameter	parameter to set
value	value to assign to the specified parameter

Precondition: The parameter must be present and be associated with a value of type Vector.

Reimplemented from lucid::Parametrizable.

◆ set() [2/2]

void lucid::GaussianKernel::set	(	Parameter	parameter,
		double	value )

overridevirtual

Set the parameter to the indicated value.

Parameters

parameter	parameter to set
value	value to assign to the specified parameter

Precondition: The parameter must be present and be associated with a value of type double.

Reimplemented from lucid::Parametrizable.

◆ sigma_f()

Scalar lucid::GaussianKernel::sigma_f ( ) const

inlinenodiscard

Get read-only access to the \( \sigma_f \) value of the kernel.

Returns: \( \sigma_f \) value of the kernel

◆ sigma_l()

const Vector & lucid::GaussianKernel::sigma_l ( ) const

inlinenodiscard

Get read-only access to the \( \sigma_l \) of the kernel.

If the kernel is isotropic, the size of the vector will be 1, regardless of the input dimension.

Returns: dimension of the kernel

◆ to_string()

std::string lucid::GaussianKernel::to_string ( ) const

nodiscardoverridevirtual

Get string representation of the kernel.

Returns: string representation Obtain the string representation of this object.; string representation of this object

Reimplemented from lucid::Kernel.

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

lucid/model/GaussianKernel.h
lucid/model/GaussianKernel.cpp

Public Member Functions

Private Member Functions

Private Attributes

Additional Inherited Members

Detailed Description

Constructor & Destructor Documentation

◆ GaussianKernel() [1/2]

◆ GaussianKernel() [2/2]

Member Function Documentation

◆ apply_impl()

◆ clone()

◆ get_d()

◆ get_v()

◆ is_isotropic()

◆ is_stationary()

◆ operator()() [1/4]

◆ operator()() [2/4]

◆ operator()() [3/4]

◆ operator()() [4/4]

◆ set() [1/2]

◆ set() [2/2]

◆ sigma_f()

◆ sigma_l()

◆ to_string()