LARS Class Reference

An implementation of LARS, a stage-wise homotopy-based algorithm for l1-regularized linear regression (LASSO) and l1+l2 regularized linear regression (Elastic Net). More...

## Public Member Functions

LARS (const bool useCholesky=false, const double lambda1=0.0, const double lambda2=0.0, const double tolerance=1e-16)
Set the parameters to LARS. More...

LARS (const bool useCholesky, const arma::mat &gramMatrix, const double lambda1=0.0, const double lambda2=0.0, const double tolerance=1e-16)
Set the parameters to LARS, and pass in a precalculated Gram matrix. More...

LARS (const arma::mat &data, const arma::rowvec &responses, const bool transposeData=true, const bool useCholesky=false, const double lambda1=0.0, const double lambda2=0.0, const double tolerance=1e-16)
Set the parameters to LARS and run training. More...

LARS (const arma::mat &data, const arma::rowvec &responses, const bool transposeData, const bool useCholesky, const arma::mat &gramMatrix, const double lambda1=0.0, const double lambda2=0.0, const double tolerance=1e-16)
Set the parameters to LARS, pass in a precalculated Gram matrix, and run training. More...

LARS (const LARS &other)
Construct the LARS object by copying the given LARS object. More...

LARS (LARS &&other)
Construct the LARS object by taking ownership of the given LARS object. More...

const std::vector< size_t > & ActiveSet () const
Access the set of active dimensions. More...

const arma::vec & Beta () const
Access the solution coefficients. More...

const std::vector< arma::vec > & BetaPath () const
Access the set of coefficients after each iteration; the solution is the last element. More...

double ComputeError (const arma::mat &matX, const arma::rowvec &y, const bool rowMajor=false)
Compute cost error of the given data matrix using the currently-trained LARS model. More...

const std::vector< double > & LambdaPath () const
Access the set of values for lambda1 after each iteration; the solution is the last element. More...

const arma::mat & MatUtriCholFactor () const
Access the upper triangular cholesky factor. More...

LARSoperator= (const LARS &other)
Copy the given LARS object. More...

LARSoperator= (LARS &&other)
Take ownership of the given LARS object. More...

void Predict (const arma::mat &points, arma::rowvec &predictions, const bool rowMajor=false) const
Predict y_i for each data point in the given data matrix using the currently-trained LARS model. More...

template
<
typename
Archive
>
void serialize (Archive &ar, const unsigned int)
Serialize the LARS model. More...

double Train (const arma::mat &data, const arma::rowvec &responses, arma::vec &beta, const bool transposeData=true)
Run LARS. More...

double Train (const arma::mat &data, const arma::rowvec &responses, const bool transposeData=true)
Run LARS. More...

## Detailed Description

An implementation of LARS, a stage-wise homotopy-based algorithm for l1-regularized linear regression (LASSO) and l1+l2 regularized linear regression (Elastic Net).

Let be a matrix where each row is a point and each column is a dimension and let be a vector of responses.

The Elastic Net problem is to solve where is the vector of regression coefficients.

If and , the problem is the LASSO. If and , the problem is the elastic net. If and , the problem is ridge regression. If and , the problem is unregularized linear regression.

Note: This algorithm is not recommended for use (in terms of efficiency) when = 0.

For more details, see the following papers:

@article{efron2004least,
title={Least angle regression},
author={Efron, B. and Hastie, T. and Johnstone, I. and Tibshirani, R.},
journal={The Annals of statistics},
volume={32},
number={2},
pages={407--499},
year={2004},
publisher={Institute of Mathematical Statistics}
}
@article{zou2005regularization,
title={Regularization and variable selection via the elastic net},
author={Zou, H. and Hastie, T.},
journal={Journal of the Royal Statistical Society Series B},
volume={67},
number={2},
pages={301--320},
year={2005},
publisher={Royal Statistical Society}
}

Definition at line 89 of file lars.hpp.

## ◆ LARS() [1/6]

 LARS ( const bool useCholesky = false, const double lambda1 = 0.0, const double lambda2 = 0.0, const double tolerance = 1e-16 )

Set the parameters to LARS.

Both lambda1 and lambda2 default to 0.

Parameters
 useCholesky Whether or not to use Cholesky decomposition when solving linear system (as opposed to using the full Gram matrix). lambda1 Regularization parameter for l1-norm penalty. lambda2 Regularization parameter for l2-norm penalty. tolerance Run until the maximum correlation of elements in (X^T y) is less than this.

## ◆ LARS() [2/6]

 LARS ( const bool useCholesky, const arma::mat & gramMatrix, const double lambda1 = 0.0, const double lambda2 = 0.0, const double tolerance = 1e-16 )

Set the parameters to LARS, and pass in a precalculated Gram matrix.

Both lambda1 and lambda2 default to 0.

Parameters
 useCholesky Whether or not to use Cholesky decomposition when solving linear system (as opposed to using the full Gram matrix). gramMatrix Gram matrix. lambda1 Regularization parameter for l1-norm penalty. lambda2 Regularization parameter for l2-norm penalty. tolerance Run until the maximum correlation of elements in (X^T y) is less than this.

## ◆ LARS() [3/6]

 LARS ( const arma::mat & data, const arma::rowvec & responses, const bool transposeData = true, const bool useCholesky = false, const double lambda1 = 0.0, const double lambda2 = 0.0, const double tolerance = 1e-16 )

Set the parameters to LARS and run training.

Both lambda1 and lambda2 are set by default to 0.

Parameters
 data Input data. responses A vector of targets. transposeData Should be true if the input data is column-major and false otherwise. useCholesky Whether or not to use Cholesky decomposition when solving linear system (as opposed to using the full Gram matrix). lambda1 Regularization parameter for l1-norm penalty. lambda2 Regularization parameter for l2-norm penalty. tolerance Run until the maximum correlation of elements in (X^T y) is less than this.

## ◆ LARS() [4/6]

 LARS ( const arma::mat & data, const arma::rowvec & responses, const bool transposeData, const bool useCholesky, const arma::mat & gramMatrix, const double lambda1 = 0.0, const double lambda2 = 0.0, const double tolerance = 1e-16 )

Set the parameters to LARS, pass in a precalculated Gram matrix, and run training.

Both lambda1 and lambda2 are set by default to 0.

Parameters
 data Input data. responses A vector of targets. transposeData Should be true if the input data is column-major and false otherwise. useCholesky Whether or not to use Cholesky decomposition when solving linear system (as opposed to using the full Gram matrix). gramMatrix Gram matrix. lambda1 Regularization parameter for l1-norm penalty. lambda2 Regularization parameter for l2-norm penalty. tolerance Run until the maximum correlation of elements in (X^T y) is less than this.

## ◆ LARS() [5/6]

 LARS ( const LARS & other )

Construct the LARS object by copying the given LARS object.

Parameters
 other LARS object to copy.

## ◆ LARS() [6/6]

 LARS ( LARS && other )

Construct the LARS object by taking ownership of the given LARS object.

Parameters
 other LARS object to take ownership of.

## ◆ ActiveSet()

 const std::vector& ActiveSet ( ) const
inline

Access the set of active dimensions.

Definition at line 253 of file lars.hpp.

## ◆ Beta()

 const arma::vec& Beta ( ) const
inline

Access the solution coefficients.

Definition at line 260 of file lars.hpp.

## ◆ BetaPath()

 const std::vector& BetaPath ( ) const
inline

Access the set of coefficients after each iteration; the solution is the last element.

Definition at line 257 of file lars.hpp.

## ◆ ComputeError()

 double ComputeError ( const arma::mat & matX, const arma::rowvec & y, const bool rowMajor = false )

Compute cost error of the given data matrix using the currently-trained LARS model.

Only ||y-beta*X||2 is used to calculate cost error.

Parameters
 data Column-major input data (or row-major input data if rowMajor = true). responses A vector of targets. rowMajor Should be true if the data points matrix is row-major and false otherwise.
Returns
The minimum cost error.

Referenced by LARS::MatUtriCholFactor().

## ◆ LambdaPath()

 const std::vector& LambdaPath ( ) const
inline

Access the set of values for lambda1 after each iteration; the solution is the last element.

Definition at line 264 of file lars.hpp.

## ◆ MatUtriCholFactor()

 const arma::mat& MatUtriCholFactor ( ) const
inline

Access the upper triangular cholesky factor.

Definition at line 267 of file lars.hpp.

References LARS::ComputeError(), and LARS::serialize().

## ◆ operator=() [1/2]

 LARS& operator= ( const LARS & other )

Copy the given LARS object.

Parameters
 other LARS object to copy.

## ◆ operator=() [2/2]

 LARS& operator= ( LARS && other )

Take ownership of the given LARS object.

Parameters
 other LARS object to take ownership of.

## ◆ Predict()

 void Predict ( const arma::mat & points, arma::rowvec & predictions, const bool rowMajor = false ) const

Predict y_i for each data point in the given data matrix using the currently-trained LARS model.

Parameters
 points The data points to regress on. predictions y, which will contained calculated values on completion. rowMajor Should be true if the data points matrix is row-major and false otherwise.

## ◆ serialize()

 void serialize ( Archive & ar, const unsigned int )

Serialize the LARS model.

Referenced by LARS::MatUtriCholFactor().

## ◆ Train() [1/2]

 double Train ( const arma::mat & data, const arma::rowvec & responses, arma::vec & beta, const bool transposeData = true )

Run LARS.

The input matrix (like all mlpack matrices) should be column-major – each column is an observation and each row is a dimension. However, because LARS is more efficient on a row-major matrix, this method will (internally) transpose the matrix. If this transposition is not necessary (i.e., you want to pass in a row-major matrix), pass 'false' for the transposeData parameter.

Parameters
 data Column-major input data (or row-major input data if rowMajor = true). responses A vector of targets. beta Vector to store the solution (the coefficients) in. transposeData Set to false if the data is row-major.
Returns
minimum cost error(||y-beta*X||2 is used to calculate error).

## ◆ Train() [2/2]

 double Train ( const arma::mat & data, const arma::rowvec & responses, const bool transposeData = true )

Run LARS.

The input matrix (like all mlpack matrices) should be column-major – each column is an observation and each row is a dimension. However, because LARS is more efficient on a row-major matrix, this method will (internally) transpose the matrix. If this transposition is not necessary (i.e., you want to pass in a row-major matrix), pass 'false' for the transposeData parameter.

Parameters
 data Input data. responses A vector of targets. transposeData Should be true if the input data is column-major and false otherwise.
Returns
minimum cost error(||y-beta*X||2 is used to calculate error).

The documentation for this class was generated from the following file:
• /home/ryan/src/mlpack.org/_src/mlpack-3.3.1/src/mlpack/methods/lars/lars.hpp