This class implements the popular nuclear norm minimization heuristic for matrix completion problems. More...

Public Member Functions
	MatrixCompletion (const size_t m, const size_t n, const arma::umat &indices, const arma::vec &values, const size_t r)
	Construct a matrix completion problem, specifying the maximum rank of the solution. More...

	MatrixCompletion (const size_t m, const size_t n, const arma::umat &indices, const arma::vec &values, const arma::mat &initialPoint)
	Construct a matrix completion problem, specifying the initial point of the optimization. More...

	MatrixCompletion (const size_t m, const size_t n, const arma::umat &indices, const arma::vec &values)
	Construct a matrix completion problem. More...

void	Recover (arma::mat &recovered)
	Solve the underlying SDP to fill in the remaining values. More...

const optimization::LRSDP< optimization::SDP< arma::sp_mat > > &	Sdp () const
	Return the underlying SDP. More...

optimization::LRSDP< optimization::SDP< arma::sp_mat > > &	Sdp ()
	Modify the underlying SDP. More...

Private Member Functions
void	CheckValues ()
	Validate the input matrices. More...

void	InitSDP ()
	Initialize the SDP. More...

Static Private Member Functions
static size_t	DefaultRank (const size_t m, const size_t n, const size_t p)
	Select a rank of the matrix given that is of size m x n and has p known elements. More...

Private Attributes
arma::umat	indices
	Matrix containing the indices of the known entries (has two rows). More...

size_t	m
	Number of rows in original matrix. More...

size_t	n
	Number of columns in original matrix. More...

optimization::LRSDP< optimization::SDP< arma::sp_mat > >	sdp
	The underlying SDP to be solved. More...

arma::mat	values
	Vector containing the values of the known entries. More...

Detailed Description

This class implements the popular nuclear norm minimization heuristic for matrix completion problems.

That is, given known values M_ij's, the following optimization problem (semi-definite program) is solved to fill in the remaining unknown values of X

min ||X||_* subj to X_ij = M_ij

where ||X||_* denotes the nuclear norm (sum of singular values of X).

For a theoretical treatment of the conditions necessary for exact recovery, see the following paper:

A Simpler Appoarch to Matrix Completion. Benjamin Recht. JMLR 11. http://arxiv.org/pdf/0910.0651v2.pdf

An example of how to use this class is shown below:

size_t m, n;         // size of unknown matrix
arma::umat indices;  // contains the known indices [2 x n_entries]
arma::vec values;    // contains the known values [n_entries]
arma::mat recovered; // will contain the completed matrix
MatrixCompletion mc(m, n, indices, values);
mc.Recover(recovered);

See also: LRSDP

Definition at line 53 of file matrix_completion.hpp.

Constructor & Destructor Documentation

mlpack::matrix_completion::MatrixCompletion::MatrixCompletion	(	const size_t	m,
		const size_t	n,
		const arma::umat &	indices,
		const arma::vec &	values,
		const size_t	r
	)

Construct a matrix completion problem, specifying the maximum rank of the solution.

Parameters

m	Number of rows of original matrix.
n	Number of columns of original matrix.
indices	Matrix containing the indices of the known entries (must be [2 x p]).
values	Vector containing the values of the known entries (must be length p).
r	Maximum rank of solution.

mlpack::matrix_completion::MatrixCompletion::MatrixCompletion	(	const size_t	m,
		const size_t	n,
		const arma::umat &	indices,
		const arma::vec &	values,
		const arma::mat &	initialPoint
	)

Construct a matrix completion problem, specifying the initial point of the optimization.

Parameters

m	Number of rows of original matrix.
n	Number of columns of original matrix.
indices	Matrix containing the indices of the known entries (must be [2 x p]).
values	Vector containing the values of the known entries (must be length p).
initialPoint	Starting point for the SDP optimization.

mlpack::matrix_completion::MatrixCompletion::MatrixCompletion	(	const size_t	m,
		const size_t	n,
		const arma::umat &	indices,
		const arma::vec &	values
	)

Construct a matrix completion problem.

Parameters

m	Number of rows of original matrix.
n	Number of columns of original matrix.
indices	Matrix containing the indices of the known entries (must be [2 x p]).
values	Vector containing the values of the known entries (must be length p).

Member Function Documentation

void mlpack::matrix_completion::MatrixCompletion::CheckValues ( )

private

Validate the input matrices.

static size_t mlpack::matrix_completion::MatrixCompletion::DefaultRank	(	const size_t	m,
		const size_t	n,
		const size_t	p
	)

staticprivate

Select a rank of the matrix given that is of size m x n and has p known elements.

void mlpack::matrix_completion::MatrixCompletion::InitSDP ( )

private

Initialize the SDP.

void mlpack::matrix_completion::MatrixCompletion::Recover ( arma::mat & recovered )

Solve the underlying SDP to fill in the remaining values.

Parameters

recovered Will contain the completed matrix.

const optimization::LRSDP<optimization::SDP<arma::sp_mat> >& mlpack::matrix_completion::MatrixCompletion::Sdp ( ) const

inline

Return the underlying SDP.

Definition at line 115 of file matrix_completion.hpp.

References sdp.

optimization::LRSDP<optimization::SDP<arma::sp_mat> >& mlpack::matrix_completion::MatrixCompletion::Sdp ( )

inline

Modify the underlying SDP.

Definition at line 117 of file matrix_completion.hpp.

References sdp.

Member Data Documentation

arma::umat mlpack::matrix_completion::MatrixCompletion::indices

private

Matrix containing the indices of the known entries (has two rows).

Definition at line 125 of file matrix_completion.hpp.

size_t mlpack::matrix_completion::MatrixCompletion::m

private

Number of rows in original matrix.

Definition at line 121 of file matrix_completion.hpp.

size_t mlpack::matrix_completion::MatrixCompletion::n

private

Number of columns in original matrix.

Definition at line 123 of file matrix_completion.hpp.

optimization::LRSDP<optimization::SDP<arma::sp_mat> > mlpack::matrix_completion::MatrixCompletion::sdp

private

The underlying SDP to be solved.

Definition at line 130 of file matrix_completion.hpp.

Referenced by Sdp().

arma::mat mlpack::matrix_completion::MatrixCompletion::values

private

Vector containing the values of the known entries.

Definition at line 127 of file matrix_completion.hpp.

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

src/mlpack/methods/matrix_completion/matrix_completion.hpp

Public Member Functions

Private Member Functions

Static Private Member Functions

Private Attributes

Detailed Description

Constructor & Destructor Documentation

Member Function Documentation

Member Data Documentation