Eigen::NNLS< MatrixType_ > Class Template Reference

Detailed Description

template<class MatrixType_>
class Eigen::NNLS< MatrixType_ >

Implementation of the Non-Negative Least Squares (NNLS) algorithm.

Template Parameters

MatrixType

The type of the system matrix \(A\).

This class implements the NNLS algorithm as described in "SOLVING LEAST SQUARES PROBLEMS", Charles L. Lawson and Richard J. Hanson, Prentice-Hall, 1974. This algorithm solves a least squares problem iteratively and ensures that the solution is non-negative. I.e.

\[ \min \left\Vert Ax-b\right\Vert_2^2\quad s.t.\, x\ge 0 \]

The algorithm solves the constrained least-squares problem above by iteratively improving an estimate of which constraints are active (elements of \(x\) equal to zero) and which constraints are inactive (elements of \(x\) greater than zero). Each iteration, an unconstrained linear least-squares problem solves for the components of \(x\) in the (estimated) inactive set and the sets are updated. The unconstrained problem minimizes \(\left\Vert A^Nx^N-b\right\Vert_2^2\), where \(A^N\) is a matrix formed by selecting all columns of A which are in the inactive set \(N\).

See the wikipedia page on non-negative least squares for more background information.

Note

Please note that it is possible to construct an NNLS problem for which the algorithm does not converge. In practice these cases are extremely rare.

Public Types
typedef Matrix< Scalar, RowsAtCompileTime, 1 >	RhsVectorType
typedef Matrix< Scalar, ColsAtCompileTime, 1 >	SolutionVectorType

Public Member Functions
template<typename MatrixDerived >
NNLS< MatrixType > &	compute (const EigenBase< MatrixDerived > &A)
ComputationInfo	info () const
Index	iterations () const
Index	maxIterations () const
	NNLS (const MatrixType &A, Index max_iter=-1, Scalar tol=NumTraits< Scalar >::dummy_precision())
	Constructs a NNLS sovler and initializes it with the given system matrix A. More...
NNLS< MatrixType > &	setMaxIterations (Index maxIters)
NNLS< MatrixType > &	setTolerance (const Scalar &tolerance)
const SolutionVectorType &	solve (const RhsVectorType &b)
	Solves the NNLS problem. More...
Scalar	tolerance () const
const SolutionVectorType &	x () const
	Returns the solution if a problem was solved. If not, an uninitialized vector may be returned.

Member Typedef Documentation

RhsVectorType

template<class MatrixType_ >

typedef Matrix<Scalar, RowsAtCompileTime, 1> Eigen::NNLS< MatrixType_ >::RhsVectorType

Type of a column vector of the system matrix \(A\).

SolutionVectorType

template<class MatrixType_ >

typedef Matrix<Scalar, ColsAtCompileTime, 1> Eigen::NNLS< MatrixType_ >::SolutionVectorType

Type of a row vector of the system matrix \(A\).

Constructor & Destructor Documentation

NNLS()

template<typename MatrixType >

Eigen::NNLS< MatrixType >::NNLS	(	const MatrixType &	A,
		Index	max_iter = -1,
		Scalar	tol = NumTraits<Scalar>::dummy_precision()
	)

Constructs a NNLS sovler and initializes it with the given system matrix A.

Parameters

A	Specifies the system matrix.
max_iter	Specifies the maximum number of iterations to solve the system.
tol	Specifies the precision of the optimum. This is an absolute tolerance on the gradient of the Lagrangian, \(A^T(Ax-b)-\lambda\) (with Lagrange multipliers \(\lambda\)).

Member Function Documentation

compute()

template<typename MatrixType >

template<typename MatrixDerived >

NNLS< MatrixType > & Eigen::NNLS< MatrixType >::compute

(

const EigenBase< MatrixDerived > &

A

)

Initializes the solver with the matrix A for further solving NNLS problems.

This function mostly initializes/computes the preconditioner. In the future we might, for instance, implement column reordering for faster matrix vector products.

info()

template<class MatrixType_ >

ComputationInfo Eigen::NNLS< MatrixType_ >::info ( ) const

inline

Returns

Success if the iterations converged, and an error values otherwise.

iterations()

template<class MatrixType_ >

Index Eigen::NNLS< MatrixType_ >::iterations ( ) const

inline

Returns

the number of iterations (least-squares solves) performed during the last solve

maxIterations()

template<class MatrixType_ >

Index Eigen::NNLS< MatrixType_ >::maxIterations ( ) const

inline

Returns

the max number of iterations. It is either the value set by setMaxIterations or, by default, twice the number of columns of the matrix.

setMaxIterations()

template<class MatrixType_ >

NNLS<MatrixType>& Eigen::NNLS< MatrixType_ >::setMaxIterations ( Index  maxIters )

inline

Sets the max number of iterations. Default is twice the number of columns of the matrix. The algorithm requires at least k iterations to produce a solution vector with k non-zero entries.

setTolerance()

template<class MatrixType_ >

NNLS<MatrixType>& Eigen::NNLS< MatrixType_ >::setTolerance ( const Scalar &  tolerance )

inline

Sets the tolerance threshold used by the stopping criteria.

This is an absolute tolerance on the gradient of the Lagrangian, \(A^T(Ax-b)-\lambda\) (with Lagrange multipliers \(\lambda\)).

solve()

template<typename MatrixType >

const NNLS< MatrixType >::SolutionVectorType & Eigen::NNLS< MatrixType >::solve

(

const RhsVectorType &

b

)

Solves the NNLS problem.

The dimension of b must be equal to the number of rows of A, given to the constructor.

Returns

The approximate solution vector \( x \). Use info() to determine if the solve was a success or not.

See also

info()

tolerance()

template<class MatrixType_ >

Scalar Eigen::NNLS< MatrixType_ >::tolerance ( ) const

inline

Returns

the tolerance threshold used by the stopping criteria.

See also

setTolerance()

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

• NNLS