cmp::scaler::PCA Class Reference

Principal Component Analysis (PCA) feature scaler and dimension reducer. More...

#include <scaler.h>

Inheritance diagram for cmp::scaler::PCA:

Collaboration diagram for cmp::scaler::PCA:

Public Member Functions
	PCA (size_t nComponents)
	Constructs a PCA scaler targeting a specific latent dimensionality.
	PCA (const PCA &other)=default
	PCA (PCA &&other)=default
	~PCA ()=default
PCA &	operator= (const PCA &other)=default
PCA &	operator= (PCA &&other)=default
Eigen::VectorXd	transform (const Eigen::Ref< const Eigen::VectorXd > &data) const override
	Transforms the input data from physical space to the scaled latent space.
Eigen::VectorXd	inverseTransform (const Eigen::Ref< const Eigen::VectorXd > &data) const override
	Reconstructs the data from the scaled latent space back to physical space.
Eigen::VectorXd	getIntercept () const override
	Retrieves the intercept vector \( \mu \) used in the transformation.
Eigen::MatrixXd	getScale () const override
	Retrieves the scaling matrix \( S \) used in the transformation.
void	fit (const Eigen::Ref< const Eigen::MatrixXd > &data) override
	Learns the scaling parameters (e.g., mean, variance) from the training data.
Eigen::MatrixXd	fit_transform (const Eigen::Ref< const Eigen::MatrixXd > &data) override
	Fits the scaling parameters to the data and subsequently transforms the data.
void	resize (size_t nComponents)
	Resizes the number of retained principal components \( M \) after fitting.
Eigen::VectorXd	getEigenvalues () const
	Returns the eigenvalues \( \mathbf{\Lambda} \) of the covariance matrix.
Eigen::MatrixXd	getEigenvectors () const
	Returns the eigenvectors \( \mathbf{V} \) of the covariance matrix.

Private Member Functions
void	eigenDecomposition ()
	Internal helper to execute the eigendecomposition on the current covariance matrix.

Private Attributes
Eigen::VectorXd	mean_
	The empirical mean vector \( \boldsymbol{\mu} \).
Eigen::SelfAdjointEigenSolver< Eigen::MatrixXd >	eigenSolver_
	Eigen solver storing the eigenvectors \( \mathbf{V} \) and eigenvalues \( \mathbf{\Lambda} \).
Eigen::MatrixXd	sqrtCov_
	Precomputed square root of the truncated eigenvalues matrix \( \mathbf{\Lambda}_M^{1/2} \).
Eigen::MatrixXd	sqrtCovInv_
	Precomputed inverse square root of the truncated eigenvalues matrix \( \mathbf{\Lambda}_M^{-1/2} \).
size_t	nComponents_
	The number of principal components \( M \) to retain.

Detailed Description

Principal Component Analysis (PCA) feature scaler and dimension reducer.

Mathematical Formulation

Centers data matrix \(\mathbf{X} \in \mathbb{R}^{N \times D}\) around column mean \(\boldsymbol{\mu}\) and computes the eigendecomposition of the sample covariance matrix \(\boldsymbol{\Sigma}\):

\[ \boldsymbol{\Sigma} = \mathbf{V} \mathbf{\Lambda} \mathbf{V}^T \]

where \(\mathbf{V}\) contains orthogonal eigenvectors and \(\mathbf{\Lambda}\) contains eigenvalues in descending order. The transformation projects \(\mathbf{x}\) into a lower-dimensional latent space of size \(M\):

\[ \mathbf{y} = \mathbf{\Lambda}_M^{-1/2} \mathbf{V}_M^T (\mathbf{x} - \boldsymbol{\mu}) \]

The inverse transformation reconstructs the point:

\[ \mathbf{x} = \mathbf{V}_M \mathbf{\Lambda}_M^{1/2} \mathbf{y} + \boldsymbol{\mu} \]

• ### Implementation Algorithm

1. fit() computes the mean \(\boldsymbol{\mu}\) and covariance \(\boldsymbol{\Sigma}\) of the training matrix, then runs Eigen::SelfAdjointEigenSolver.
2. transform() centers the query vector, projects it onto the principal eigenvectors, and scales it by the inverse square root of eigenvalues.
3. inverseTransform() rescales by square root eigenvalues, projects back to physical coordinates, and adds the mean.

Constructor & Destructor Documentation

PCA() [1/3]

cmp::scaler::PCA::PCA ( size_t  nComponents )

inline

Constructs a PCA scaler targeting a specific latent dimensionality.

Parameters

nComponents

Number of principal components \( M \) to retain.

PCA() [2/3]

cmp::scaler::PCA::PCA ( const PCA &  other )

default

PCA() [3/3]

cmp::scaler::PCA::PCA ( PCA &&  other )

default

~PCA()

cmp::scaler::PCA::~PCA ( )

default

Member Function Documentation

eigenDecomposition()

void cmp::scaler::PCA::eigenDecomposition ( )

private

Internal helper to execute the eigendecomposition on the current covariance matrix.

fit()

void cmp::scaler::PCA::fit ( const Eigen::Ref< const Eigen::MatrixXd > &  data )

overridevirtual

Learns the scaling parameters (e.g., mean, variance) from the training data.

Parameters

data	A matrix \( X \in \mathbb{R}^{n \times d} \) of training samples.

Implements cmp::scaler::Scaler.

fit_transform()

Eigen::MatrixXd cmp::scaler::PCA::fit_transform ( const Eigen::Ref< const Eigen::MatrixXd > &  data )

overridevirtual

Fits the scaling parameters to the data and subsequently transforms the data.

Parameters

data	A matrix \( X \in \mathbb{R}^{n \times d} \) of training samples.

Returns

The transformed dataset matrix \( Y \).

Implements cmp::scaler::Scaler.

getEigenvalues()

Eigen::VectorXd cmp::scaler::PCA::getEigenvalues ( ) const

inline

Returns the eigenvalues \( \mathbf{\Lambda} \) of the covariance matrix.

Returns

A vector of eigenvalues.

getEigenvectors()

Eigen::MatrixXd cmp::scaler::PCA::getEigenvectors ( ) const

inline

Returns the eigenvectors \( \mathbf{V} \) of the covariance matrix.

Returns

A matrix whose columns are the eigenvectors.

getIntercept()

Eigen::VectorXd cmp::scaler::PCA::getIntercept ( ) const

inlineoverridevirtual

Retrieves the intercept vector \( \mu \) used in the transformation.

Returns

The intercept vector.

Implements cmp::scaler::Scaler.

getScale()

Eigen::MatrixXd cmp::scaler::PCA::getScale ( ) const

inlineoverridevirtual

Retrieves the scaling matrix \( S \) used in the transformation.

Returns

The scaling matrix.

Implements cmp::scaler::Scaler.

inverseTransform()

Eigen::VectorXd cmp::scaler::PCA::inverseTransform ( const Eigen::Ref< const Eigen::VectorXd > &  data ) const

overridevirtual

Reconstructs the data from the scaled latent space back to physical space.

Parameters

data	A column vector \( y \) in the scaled space.

Returns

The unscaled physical data vector \( x \).

Implements cmp::scaler::Scaler.

operator=() [1/2]

PCA & cmp::scaler::PCA::operator= ( const PCA &  other )

default

operator=() [2/2]

PCA & cmp::scaler::PCA::operator= ( PCA &&  other )

default

resize()

void cmp::scaler::PCA::resize

(

size_t

nComponents

)

Resizes the number of retained principal components \( M \) after fitting.

Parameters

nComponents

The new number of components to retain.

transform()

Eigen::VectorXd cmp::scaler::PCA::transform ( const Eigen::Ref< const Eigen::VectorXd > &  data ) const

overridevirtual

Transforms the input data from physical space to the scaled latent space.

Parameters

data	A column vector \( x \) representing a single data point in physical space.

Returns

The scaled data vector \( y \).

Implements cmp::scaler::Scaler.

Member Data Documentation

eigenSolver_

Eigen::SelfAdjointEigenSolver<Eigen::MatrixXd> cmp::scaler::PCA::eigenSolver_

private

Eigen solver storing the eigenvectors \( \mathbf{V} \) and eigenvalues \( \mathbf{\Lambda} \).

mean_

Eigen::VectorXd cmp::scaler::PCA::mean_

private

The empirical mean vector \( \boldsymbol{\mu} \).

nComponents_

size_t cmp::scaler::PCA::nComponents_

private

The number of principal components \( M \) to retain.

sqrtCov_

Eigen::MatrixXd cmp::scaler::PCA::sqrtCov_

private

Precomputed square root of the truncated eigenvalues matrix \( \mathbf{\Lambda}_M^{1/2} \).

sqrtCovInv_

Eigen::MatrixXd cmp::scaler::PCA::sqrtCovInv_

private

Precomputed inverse square root of the truncated eigenvalues matrix \( \mathbf{\Lambda}_M^{-1/2} \).

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

• include/scaler.h
• src/scaler.cpp