mlpack::regression::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...

List of all members.

Public Member Functions

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

Private Member Functions

void Activate (const size_t varInd)
 Add dimension varInd to active set.
void CholeskyDelete (const size_t colToKill)
void CholeskyInsert (double sqNormNewX, const arma::vec &newGramCol)
void CholeskyInsert (const arma::vec &newX, const arma::mat &X)
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 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< bool > isActive
 Active set membership indicator (for each dimension).
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

\[ \min_{\beta} 0.5 || X \beta - y ||_2^2 + \lambda_1 || \beta ||_1 + 0.5 \lambda_2 || \beta ||_2^2 \]

where $ \beta $ is the vector of regression coefficients.

If $ \lambda_1 > 0 $ and $ \lambda_2 = 0 $, the problem is the LASSO. If $ \lambda_1 > 0 $ and $ \lambda_2 > 0 $, the problem is the elastic net. If $ \lambda_1 = 0 $ and $ \lambda_2 > 0 $, the problem is ridge regression. If $ \lambda_1 = 0 $ and $ \lambda_2 = 0 $, the problem is unregularized linear regression.

Note: This algorithm is not recommended for use (in terms of efficiency) when $ \lambda_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 100 of file lars.hpp.


Constructor & Destructor Documentation

mlpack::regression::LARS::LARS ( const bool  useCholesky,
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.
mlpack::regression::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.

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.

Member Function Documentation

void mlpack::regression::LARS::Activate ( const size_t  varInd  )  [private]

Add dimension varInd to active set.

Parameters:
varInd Dimension to add to active set.
const std::vector<size_t>& mlpack::regression::LARS::ActiveSet (  )  const [inline]

Access the set of active dimensions.

Definition at line 156 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 160 of file lars.hpp.

References betaPath.

void mlpack::regression::LARS::CholeskyDelete ( const size_t  colToKill  )  [private]
void mlpack::regression::LARS::CholeskyInsert ( double  sqNormNewX,
const arma::vec &  newGramCol 
) [private]
void mlpack::regression::LARS::CholeskyInsert ( const arma::vec &  newX,
const arma::mat &  X 
) [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_t  activeVarInd  )  [private]

Remove activeVarInd'th element from active set.

Parameters:
activeVarInd Index of element to remove from active set.
void mlpack::regression::LARS::GivensRotate ( const arma::vec::fixed< 2 > &  x,
arma::vec::fixed< 2 > &  rotatedX,
arma::mat &  G 
) [private]
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 164 of file lars.hpp.

References lambdaPath.

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

Access the upper triangular cholesky factor.

Definition at line 167 of file lars.hpp.

References matUtriCholFactor.

void mlpack::regression::LARS::Regress ( const arma::mat &  data,
const arma::vec &  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.
rowMajor Set to false if the data is row-major.

Member Data Documentation

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

Active set of dimensions.

Definition at line 202 of file lars.hpp.

Referenced by ActiveSet().

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

Solution path.

Definition at line 196 of file lars.hpp.

Referenced by BetaPath().

True if this is the elastic net problem.

Definition at line 188 of file lars.hpp.

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

Active set membership indicator (for each dimension).

Definition at line 205 of file lars.hpp.

Regularization parameter for l1 penalty.

Definition at line 185 of file lars.hpp.

Regularization parameter for l2 penalty.

Definition at line 190 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 199 of file lars.hpp.

Referenced by LambdaPath().

True if this is the LASSO problem.

Definition at line 183 of file lars.hpp.

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

Reference to the Gram matrix we will use.

Definition at line 174 of file lars.hpp.

Gram matrix.

Definition at line 171 of file lars.hpp.

Upper triangular cholesky factor; initially 0x0 matrix.

Definition at line 177 of file lars.hpp.

Referenced by MatUtriCholFactor().

Tolerance for main loop.

Definition at line 193 of file lars.hpp.

Whether or not to use Cholesky decomposition when solving linear system.

Definition at line 180 of file lars.hpp.


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

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