Truncated Fourier feature map. More...

#include <TruncatedFourierFeatureMap.h>

Inheritance diagram for lucid::TruncatedFourierFeatureMap:

Public Member Functions
	TruncatedFourierFeatureMap (int num_frequencies, const Matrix &prob_per_dim, const Matrix &omega_per_dim, ConstVectorRef sigma_l, Scalar sigma_f, const RectSet &X_bounds)
	Construct a truncated Fourier feature map.

	TruncatedFourierFeatureMap (int num_frequencies, const Matrix &prob_per_dim, const Matrix &omega_per_dim, double sigma_l, Scalar sigma_f, const RectSet &X_bounds)
	Construct a truncated Fourier feature map.

Vector	map_vector (ConstVectorRef x) const
	Given a \( \texttip{d}{Dimension of the vector space} \) dimensional vector \( \texttip{x}{Element of the vector space} \), project it to the unit hypercube \( [0, 1]^d \), then compute the feature map.

Vector	invert_vector (ConstVectorRef y) const
	Given a @2M+1 dimensional vector \( \texttip{y}{Element of the vector space} \), invert the feature map to obtain the original input vector.

Matrix	map_matrix (ConstMatrixRef x) const
	Given an \( \texttip{n}{Number of samples} \times \texttip{d}{Dimension of the vector space} \) dimensional matrix \( \texttip{x}{Element of the vector space} \), project each row vector to the unit hypercube \( [0, 1]^d \), then compute the feature map.

Dimension	dimension () const
	Get read-only access to the dimension of the the feature map space.

const Matrix &	omega () const
	Get read-only access to the frequency matrix of the truncated Fourier feature map.

const Vector &	weights () const
	Get read-only access to the weights matrix of the truncated Fourier feature map.

int	num_frequencies () const
	Get read-only access to the number of frequencies per dimension of the truncated Fourier feature map.

Scalar	captured_probability () const
	Get read-only access to the probability captured by the Fourier expansion of the truncated Fourier feature map.

const RectSet &	X_bounds () const
	Get read-only access to the limits of the original input space.

double	sigma_f () const
	Get read-only access to the sigma_f value of the truncated Fourier feature map.

const Vector &	sigma_l () const
	Get read-only access to the sigma_l value of the truncated Fourier feature map.

virtual RectSet	get_periodic_set () const
	Return the periodic input domain for this linear truncated Fourier map.

std::unique_ptr< FeatureMap >	clone () const override
	Clone the feature map.

Public Member Functions inherited from lucid::FeatureMap
Matrix	operator() (ConstMatrixRef x) const
	Apply the feature map to a vector.

Matrix	invert (ConstMatrixRef y) const
	Apply the inverse feature map to a vector.

virtual std::string	to_string () const
	Obtain the string representation of this object.

Protected Member Functions
Matrix	apply_impl (ConstMatrixRef x) const override
	Given an \( \texttip{n}{Number of samples} \times \texttip{d}{Dimension of the vector space} \) dimensional matrix \( \texttip{x}{Element of the vector space} \), project each row vector to the unit hypercube \( [0, 1]^d \), then compute the feature map.

Matrix	invert_impl (ConstMatrixRef y) const override
	Concrete implementation of invert().

Protected Attributes
int	num_frequencies_
	Number of frequencies per dimension.

Matrix	omega_
	Frequencies matrix.

Vector	weights_
	Weights matrix.

Scalar	sigma_f_
	\( \sigma_f \) value

Vector	sigma_l_
	\( \sigma_l \) vector

RectSet	X_bounds_
	Limits of the input space expressed as a matrix. The set is a rectangle.

Scalar	captured_probability_
	Probability captured by the Fourier expansion. NaN if not computed.

Detailed Description

Truncated Fourier feature map.

It maps vectors from \( \texttip{\mathcal{X}}{Polish sample vector space} \subseteq \mathbb{R}^d \) to a higher-dimensional space using the truncated Fourier series. The number of frequencies, \( M > 0 \), determines the dimension of the output space. All possible combinations of frequencies are computed for each input dimension. Therefore, the output vector has a dimension of \( 2^d M + 1 \). The feature map is computed as

\[\phi_{M}(x) = \sigma_{f}\begin{bmatrix} w_{0}\\ \sqrt{2}w_{1}\cos(\omega_{1}^{\top}P(x))\\ \sqrt{2}w_{1}\sin(\omega_{1}^{\top}P(x))\\ \vdots\\ \sqrt{2}w_{M}\cos(\omega_{M}^{\top}P(x))\\ \sqrt{2}w_{M}\sin(\omega_{M}^{\top}P(x)) \end{bmatrix}, \]

The way the weights and omegas are computed depends on the subclass.

Constructor & Destructor Documentation

◆ TruncatedFourierFeatureMap() [1/2]

lucid::TruncatedFourierFeatureMap::TruncatedFourierFeatureMap	(	int	num_frequencies,
		const Matrix &	prob_per_dim,
		const Matrix &	omega_per_dim,
		ConstVectorRef	sigma_l,
		Scalar	sigma_f,
		const RectSet &	X_bounds )

Construct a truncated Fourier feature map.

Precondition: num_frequencies must be greater than 0.; sigma_f must be greater than 0.

Parameters

num_frequencies	number of frequencies per dimension. Includes the zero frequency
prob_per_dim	probability distribution of frequencies per dimension, \( \mathbb{P}(\zeta_j) \)
omega_per_dim	matrix omega where each row is a dimension and each column is a frequency coefficient
sigma_l	length scales per dimension
sigma_f	scaling factor
X_bounds	domain of the input space, \( \texttip{\mathcal{X}}{Polish sample vector space} \subseteq \mathbb{R}^d \)

◆ TruncatedFourierFeatureMap() [2/2]

lucid::TruncatedFourierFeatureMap::TruncatedFourierFeatureMap	(	int	num_frequencies,
		const Matrix &	prob_per_dim,
		const Matrix &	omega_per_dim,
		double	sigma_l,
		Scalar	sigma_f,
		const RectSet &	X_bounds )

Construct a truncated Fourier feature map.

Precondition: num_frequencies must be greater than 0.; sigma_f must be greater than 0.

Parameters

num_frequencies	number of frequencies per dimension. Includes the zero frequency
prob_per_dim	probability distribution of frequencies per dimension, \( \mathbb{P}(\zeta_j) \)
omega_per_dim	matrix omega where each row is a dimension and each column is a frequency coefficient
sigma_l	length scales per dimension
sigma_f	scaling factor
X_bounds	domain of the input space, \( \texttip{\mathcal{X}}{Polish sample vector space} \subseteq \mathbb{R}^d \)

Member Function Documentation

◆ apply_impl()

Matrix lucid::TruncatedFourierFeatureMap::apply_impl ( ConstMatrixRef x ) const

nodiscardoverrideprotectedvirtual

Given an \( \texttip{n}{Number of samples} \times \texttip{d}{Dimension of the vector space} \) dimensional matrix \( \texttip{x}{Element of the vector space} \), project each row vector to the unit hypercube \( [0, 1]^d \), then compute the feature map.

Parameters

x	input vector

Returns: \( n \times 2 M + 1 \) dimensional feature map

Implements lucid::FeatureMap.

◆ captured_probability()

Scalar lucid::TruncatedFourierFeatureMap::captured_probability ( ) const

inlinenodiscard

Get read-only access to the probability captured by the Fourier expansion of the truncated Fourier feature map.

NaN if not computed

Returns: probability captured by the Fourier expansion of the truncated Fourier feature map

◆ clone()

std::unique_ptr< FeatureMap > lucid::TruncatedFourierFeatureMap::clone ( ) const

nodiscardoverridevirtual

Clone the feature map.

Create a new instance of the feature map with the same parameters.

Returns: new instance of the feature map

Implements lucid::FeatureMap.

◆ dimension()

Dimension lucid::TruncatedFourierFeatureMap::dimension ( ) const

inlinenodiscard

Get read-only access to the dimension of the the feature map space.

Returns: dimension of the the feature map space

◆ get_periodic_set()

RectSet lucid::TruncatedFourierFeatureMap::get_periodic_set ( ) const

nodiscardvirtual

Return the periodic input domain for this linear truncated Fourier map.

We want to find a space such that the smallest frequency is able to complete a full period. Given that frequencies are defined as \( \omega P(x) \), where \( P(x) \) is the projection to the unit hypercube of the original domain, we want to find the upper bound of the periodic domain \( \bar{x} \) such that

\[\omega P(\bar{x}) = 2\pi \]

for the smallest \( \omega \). Graphically,

                             New max (x̄)
┌───────────────────────────────●
│                               │
│                               │
│                     Old max   │
├───────────────────────●       │
│                       │       │
│                       │       │
│                       │       │
│                       │       │
│ Min                   │       │
●───────────────────────┴───────┘

Notice how the lower bound remains fixed, while the upper bound is shifted to create the periodic domain.

Note: The periodic domain could be smaller than the original domain, depending on the values of \( \sigma_l \).

Precondition: \( \sigma_l \) must have the same dimension as the input space.; All values in \( \sigma_l \) must be greater than 0.

Returns: new RectSet representing the periodic input domain

Reimplemented in lucid::ConstantTruncatedFourierFeatureMap, and lucid::LinearTruncatedFourierFeatureMap.

◆ invert_impl()

Matrix lucid::TruncatedFourierFeatureMap::invert_impl ( ConstMatrixRef y ) const

nodiscardoverrideprotectedvirtual

Concrete implementation of invert().

Parameters

y	\( n \times M \) input vector in the feature space

Returns: \( \texttip{n}{Number of samples} \times \texttip{d}{Dimension of the vector space} \) output in the original space

Implements lucid::FeatureMap.

◆ invert_vector()

Vector lucid::TruncatedFourierFeatureMap::invert_vector ( ConstVectorRef y ) const

nodiscard

Given a @2M+1 dimensional vector \( \texttip{y}{Element of the vector space} \), invert the feature map to obtain the original input vector.

Precondition: \( \texttip{y}{Element of the vector space} \) must have the same dimension as the feature map output.

Parameters

y	input vector in the feature space

Returns: \( \texttip{d}{Dimension of the vector space} \) dimensional output in the original space

◆ map_matrix()

Matrix lucid::TruncatedFourierFeatureMap::map_matrix ( ConstMatrixRef x ) const

nodiscard

Given an \( \texttip{n}{Number of samples} \times \texttip{d}{Dimension of the vector space} \) dimensional matrix \( \texttip{x}{Element of the vector space} \), project each row vector to the unit hypercube \( [0, 1]^d \), then compute the feature map.

Parameters

x	input vector

Returns: \( n \times 2 M + 1 \) dimensional feature map

◆ map_vector()

Vector lucid::TruncatedFourierFeatureMap::map_vector ( ConstVectorRef x ) const

nodiscard

Given a \( \texttip{d}{Dimension of the vector space} \) dimensional vector \( \texttip{x}{Element of the vector space} \), project it to the unit hypercube \( [0, 1]^d \), then compute the feature map.

Parameters

x	input vector

Returns: \( 2 M + 1 \) dimensional feature map

◆ num_frequencies()

int lucid::TruncatedFourierFeatureMap::num_frequencies ( ) const

inlinenodiscard

Get read-only access to the number of frequencies per dimension of the truncated Fourier feature map.

Returns: number of frequencies per dimension of the truncated Fourier feature map

◆ omega()

const Matrix & lucid::TruncatedFourierFeatureMap::omega ( ) const

inlinenodiscard

Get read-only access to the frequency matrix of the truncated Fourier feature map.

Returns: frequency matrix of the truncated Fourier feature map

◆ sigma_f()

double lucid::TruncatedFourierFeatureMap::sigma_f ( ) const

inlinenodiscard

Get read-only access to the sigma_f value of the truncated Fourier feature map.

Returns: sigma_f value of the truncated Fourier feature map

◆ sigma_l()

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

inlinenodiscard

Get read-only access to the sigma_l value of the truncated Fourier feature map.

Returns: sigma_l value of the truncated Fourier feature map

◆ weights()

const Vector & lucid::TruncatedFourierFeatureMap::weights ( ) const

inlinenodiscard

Get read-only access to the weights matrix of the truncated Fourier feature map.

Returns: weights matrix of the truncated Fourier feature map

◆ X_bounds()

const RectSet & lucid::TruncatedFourierFeatureMap::X_bounds ( ) const

inlinenodiscard

Get read-only access to the limits of the original input space.

Returns: limits of the original input space

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

lucid/model/TruncatedFourierFeatureMap.h
lucid/model/TruncatedFourierFeatureMap.cpp

Public Member Functions

Protected Member Functions

Protected Attributes

Detailed Description

Constructor & Destructor Documentation

◆ TruncatedFourierFeatureMap() [1/2]

◆ TruncatedFourierFeatureMap() [2/2]

Member Function Documentation

◆ apply_impl()

◆ captured_probability()

◆ clone()

◆ dimension()

◆ get_periodic_set()

◆ invert_impl()

◆ invert_vector()

◆ map_matrix()

◆ map_vector()

◆ num_frequencies()

◆ omega()

◆ sigma_f()

◆ sigma_l()

◆ weights()

◆ X_bounds()