mlpack: mlpack Documentation

LinearConstrSolver for FrankWolfe algorithm. More...

Public Member Functions
	ConstrLpBallSolver (const double p)
	Construct the solver of constrained problem. More...

	ConstrLpBallSolver (const double p, const arma::vec lambda)
	Construct the solver of constrained problem, with regularization parameter lambda here. More...

arma::vec	Lambda () const
	Get the regularization parameter. More...

arma::vec &	Lambda ()
	Modify the regularization parameter. More...

void	Optimize (const arma::mat &v, arma::mat &s)
	Optimizer of Linear Constrained Problem for FrankWolfe. More...

double	P () const
	Get the p-norm. More...

double &	P ()
	Modify the p-norm. More...

bool	RegFlag () const
	Get regularization flag. More...

bool &	RegFlag ()
	Modify regularization flag. More...

Detailed Description

LinearConstrSolver for FrankWolfe algorithm.

Constraint domain given in the form of lp ball. That is, given $ v $ , solve $s:=arg\min_{s\in D} <s, v>$ when $ D $ is a regularized lp ball. That is,

$D = \{ x: (\sum_j|\lambda_j x_j|^p)^{1/p}\leq 1 \}.$

If $\lambda$ is not given in the constructor, default is using all $\lambda_j = 1$ for all $ j $ .

In applications such as Orthogonal Matching Pursuit (OMP), $\lambda$ could be ideally set to the norm of the elements in the dictionary.

For $ p=1 $ : take (one) $k = arg\max_j |v_j/\lambda_j|$ , then the solution is:

$s_k = -sign(v_k)/\lambda_k, \qquad s_j = 0, \quad j\neq k.$

For $1<p<\infty$ : the solution is

$t_j = -sign(v_j) |v_j/\lambda_j|^{q-1}, \qquad s_j = \frac{t_j}{||t||_p\cdot\lambda_j}, \quad 1/p + 1/q = 1.$

For $p=\infty$ : the solution is

$s_j = -sign(v_j)/\lambda_j$

Definition at line 55 of file constr_lpball.hpp.

Constructor & Destructor Documentation

◆ ConstrLpBallSolver() [1/2]

ConstrLpBallSolver ( const double p )

inline

Construct the solver of constrained problem.

The constrained domain should be unit lp ball for this class.

Parameters

p	The constraint is unit lp ball.

Definition at line 64 of file constr_lpball.hpp.

◆ ConstrLpBallSolver() [2/2]

ConstrLpBallSolver	(	const double	p,
		const arma::vec	lambda
	)

inline

Construct the solver of constrained problem, with regularization parameter lambda here.

Parameters

p	The constraint is unit lp ball.
lambda	Regularization parameter.

Definition at line 74 of file constr_lpball.hpp.

Member Function Documentation

◆ Lambda() [1/2]

arma::vec Lambda ( ) const

inline

Get the regularization parameter.

Definition at line 145 of file constr_lpball.hpp.

◆ Lambda() [2/2]

arma::vec& Lambda ( )

inline

Modify the regularization parameter.

Definition at line 147 of file constr_lpball.hpp.

◆ Optimize()

void Optimize	(	const arma::mat &	v,
		arma::mat &	s
	)

inline

Optimizer of Linear Constrained Problem for FrankWolfe.

Parameters

v	Input local gradient.
s	Output optimal solution in the constrained domain (lp ball).

Definition at line 85 of file constr_lpball.hpp.

References Log::Fatal, and mlpack::math::Sign().

◆ P() [1/2]

double P ( ) const

inline

Get the p-norm.

Definition at line 135 of file constr_lpball.hpp.

◆ P() [2/2]

double& P ( )

inline

Modify the p-norm.

Definition at line 137 of file constr_lpball.hpp.

◆ RegFlag() [1/2]

bool RegFlag ( ) const

inline

Get regularization flag.

Definition at line 140 of file constr_lpball.hpp.

◆ RegFlag() [2/2]

bool& RegFlag ( )

inline

Modify regularization flag.

Definition at line 142 of file constr_lpball.hpp.

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

src/mlpack/core/optimizers/fw/constr_lpball.hpp