NAME¶

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

SYNOPSIS¶

Public Member Functions¶

LARS (const bool useCholesky, const double lambda1=0.0, const double lambda2=0.0, const double tolerance=1e-16) Set the parameters to LARS. 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. const std::vector< size_t > & ActiveSet () const Access the set of active dimensions. const std::vector< arma::vec > & BetaPath () const Access the set of coefficients after each iteration; the solution is the last element. const std::vector< double > & LambdaPath () const Access the set of values for lambda1 after each iteration; the solution is the last element. const arma::mat & MatUtriCholFactor () const Access the upper triangular cholesky factor. void Regress (const arma::mat &data, const arma::vec &responses, arma::vec &beta, const bool transposeData=true) Run LARS. std::string ToString () const

Private Member Functions¶

void Activate (const size_t varInd) Add dimension varInd to active set. void CholeskyDelete (const size_t colToKill) void CholeskyInsert (const arma::vec &newX, const arma::mat &X) void CholeskyInsert (double sqNormNewX, const arma::vec &newGramCol) void ComputeYHatDirection (const arma::mat &matX, const arma::vec &betaDirection, arma::vec &yHatDirection) void Deactivate (const size_t activeVarInd) Remove activeVarInd'th element from active set. void GivensRotate (const arma::vec::fixed< 2 > &x, arma::vec::fixed< 2 > &rotatedX, arma::mat &G) void Ignore (const size_t varInd) Add dimension varInd to ignores set (never removed). void InterpolateBeta ()

Private Attributes¶

std::vector< size_t > activeSet Active set of dimensions. std::vector< arma::vec > betaPath Solution path. bool elasticNet True if this is the elastic net problem. std::vector< size_t > ignoreSet Set of ignored variables (for dimensions in span{active set dimensions}). std::vector< bool > isActive Active set membership indicator (for each dimension). std::vector< bool > isIgnored Membership indicator for set of ignored variables. double lambda1 Regularization parameter for l1 penalty. double lambda2 Regularization parameter for l2 penalty. std::vector< double > lambdaPath Value of lambda_1 for each solution in solution path. bool lasso True if this is the LASSO problem. const arma::mat & matGram Reference to the Gram matrix we will use. arma::mat matGramInternal Gram matrix. arma::mat matUtriCholFactor Upper triangular cholesky factor; initially 0x0 matrix. double tolerance Tolerance for main loop. bool useCholesky Whether or not to use Cholesky decomposition when solving linear system.

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 $ X $ be a matrix where each row is a point and each column is a dimension and let $ y $ be a vector of responses. The Elastic Net problem is to solve where $ vector of regression coefficients. If $ _1 > 0 $ and $ _2 = 0 $, the problem is the LASSO. If $ _1 > 0 $ and $ _2 > 0 $, the problem is the elastic net. If $ _1 = 0 $ and $ _2 > 0 $, the problem is ridge regression. If $ _1 = 0 $ and $ _2 = 0 $, the problem is unregularized linear regression. Note: This algorithm is not recommended for use (in terms of efficiency) when $ _1 $ = 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 99 of file lars.hpp.

Constructor & Destructor Documentation¶

mlpack::regression::LARS::LARS (const booluseCholesky, const doublelambda1 = 0.0, const doublelambda2 = 0.0, const doubletolerance = 1e-16)¶

Set the parameters to LARS. Both lambda1 and lambda2 default to 0. Parameters:

mlpack::regression::LARS::LARS (const booluseCholesky, const arma::mat &gramMatrix, const doublelambda1 = 0.0, const doublelambda2 = 0.0, const doubletolerance = 1e-16)¶

Set the parameters to LARS, and pass in a precalculated Gram matrix. Both lambda1 and lambda2 default to 0. Parameters:

Member Function Documentation¶

void mlpack::regression::LARS::Activate (const size_tvarInd) [private]¶

Add dimension varInd to active set. Parameters:

const std::vector<size_t>& mlpack::regression::LARS::ActiveSet () const [inline]¶

Access the set of active dimensions. Definition at line 155 of file lars.hpp. References activeSet.

const std::vector<arma::vec>& mlpack::regression::LARS::BetaPath () const [inline]¶

Access the set of coefficients after each iteration; the solution is the last element. Definition at line 159 of file lars.hpp. References betaPath.

void mlpack::regression::LARS::CholeskyDelete (const size_tcolToKill) [private]¶

void mlpack::regression::LARS::CholeskyInsert (const arma::vec &newX, const arma::mat &X) [private]¶

void mlpack::regression::LARS::CholeskyInsert (doublesqNormNewX, const arma::vec &newGramCol) [private]¶

void mlpack::regression::LARS::ComputeYHatDirection (const arma::mat &matX, const arma::vec &betaDirection, arma::vec &yHatDirection) [private]¶

void mlpack::regression::LARS::Deactivate (const size_tactiveVarInd) [private]¶

Remove activeVarInd'th element from active set. Parameters:

void mlpack::regression::LARS::GivensRotate (const arma::vec::fixed< 2 > &x, arma::vec::fixed< 2 > &rotatedX, arma::mat &G) [private]¶

void mlpack::regression::LARS::Ignore (const size_tvarInd) [private]¶

Add dimension varInd to ignores set (never removed). Parameters:

void mlpack::regression::LARS::InterpolateBeta () [private]¶

const std::vector<double>& mlpack::regression::LARS::LambdaPath () const [inline]¶

Access the set of values for lambda1 after each iteration; the solution is the last element. Definition at line 163 of file lars.hpp. References lambdaPath.

const arma::mat& mlpack::regression::LARS::MatUtriCholFactor () const [inline]¶

Access the upper triangular cholesky factor. Definition at line 166 of file lars.hpp. References matUtriCholFactor.

void mlpack::regression::LARS::Regress (const arma::mat &data, const arma::vec &responses, arma::vec &beta, const booltransposeData = 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:

std::string mlpack::regression::LARS::ToString () const¶

Member Data Documentation¶

std::vector<size_t> mlpack::regression::LARS::activeSet [private]¶

Active set of dimensions. Definition at line 204 of file lars.hpp. Referenced by ActiveSet().

std::vector<arma::vec> mlpack::regression::LARS::betaPath [private]¶

Solution path. Definition at line 198 of file lars.hpp. Referenced by BetaPath().

bool mlpack::regression::LARS::elasticNet [private]¶

True if this is the elastic net problem. Definition at line 190 of file lars.hpp.

std::vector<size_t> mlpack::regression::LARS::ignoreSet [private]¶

Set of ignored variables (for dimensions in span{active set dimensions}). Definition at line 212 of file lars.hpp.

std::vector<bool> mlpack::regression::LARS::isActive [private]¶

Active set membership indicator (for each dimension). Definition at line 207 of file lars.hpp.

std::vector<bool> mlpack::regression::LARS::isIgnored [private]¶

Membership indicator for set of ignored variables. Definition at line 215 of file lars.hpp.

double mlpack::regression::LARS::lambda1 [private]¶

Regularization parameter for l1 penalty. Definition at line 187 of file lars.hpp.

double mlpack::regression::LARS::lambda2 [private]¶

Regularization parameter for l2 penalty. Definition at line 192 of file lars.hpp.

std::vector<double> mlpack::regression::LARS::lambdaPath [private]¶

Value of lambda_1 for each solution in solution path. Definition at line 201 of file lars.hpp. Referenced by LambdaPath().

bool mlpack::regression::LARS::lasso [private]¶

True if this is the LASSO problem. Definition at line 185 of file lars.hpp.

const arma::mat& mlpack::regression::LARS::matGram [private]¶

Reference to the Gram matrix we will use. Definition at line 176 of file lars.hpp.

arma::mat mlpack::regression::LARS::matGramInternal [private]¶

Gram matrix. Definition at line 173 of file lars.hpp.

arma::mat mlpack::regression::LARS::matUtriCholFactor [private]¶

Upper triangular cholesky factor; initially 0x0 matrix. Definition at line 179 of file lars.hpp. Referenced by MatUtriCholFactor().

double mlpack::regression::LARS::tolerance [private]¶

Tolerance for main loop. Definition at line 195 of file lars.hpp.

bool mlpack::regression::LARS::useCholesky [private]¶

Whether or not to use Cholesky decomposition when solving linear system. Definition at line 182 of file lars.hpp.

Author¶

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