An implementation of Sparse Coding with Dictionary Learning that achieves sparsity via an l1-norm regularizer on the codes (LASSO) or an (l1+l2)-norm regularizer on the codes (the Elastic Net). More...

Public Member Functions
template<typename DictionaryInitializer = DataDependentRandomInitializer>
	SparseCoding (const arma::mat &data, const size_t atoms, const double lambda1, const double lambda2=0, const size_t maxIterations=0, const double objTolerance=0.01, const double newtonTolerance=1e-6, const DictionaryInitializer &initializer=DictionaryInitializer())
	Set the parameters to SparseCoding. More...

	SparseCoding (const size_t atoms=0, const double lambda1=0, const double lambda2=0, const size_t maxIterations=0, const double objTolerance=0.01, const double newtonTolerance=1e-6)
	Set the parameters to SparseCoding. More...

size_t	Atoms () const
	Access the number of atoms. More...

size_t &	Atoms ()
	Modify the number of atoms. More...

const arma::mat &	Dictionary () const
	Access the dictionary. More...

arma::mat &	Dictionary ()
	Modify the dictionary. More...

void	Encode (const arma::mat &data, arma::mat &codes)
	Sparse code each point in the given dataset via LARS, using the current dictionary and store the encoded data in the codes matrix. More...

double	Lambda1 () const
	Access the L1 regularization term. More...

double &	Lambda1 ()
	Modify the L1 regularization term. More...

double	Lambda2 () const
	Access the L2 regularization term. More...

double &	Lambda2 ()
	Modify the L2 regularization term. More...

size_t	MaxIterations () const
	Get the maximum number of iterations. More...

size_t &	MaxIterations ()
	Modify the maximum number of iterations. More...

double	NewtonTolerance () const
	Get the tolerance for Newton's method (dictionary optimization step). More...

double &	NewtonTolerance ()
	Modify the tolerance for Newton's method (dictionary optimization step). More...

double	Objective (const arma::mat &data, const arma::mat &codes) const
	Compute the objective function. More...

double	ObjTolerance () const
	Get the objective tolerance. More...

double &	ObjTolerance ()
	Modify the objective tolerance. More...

double	OptimizeDictionary (const arma::mat &data, const arma::mat &codes, const arma::uvec &adjacencies)
	Learn dictionary via Newton method based on Lagrange dual. More...

void	ProjectDictionary ()
	Project each atom of the dictionary back onto the unit ball, if necessary. More...

template<typename Archive >
void	Serialize (Archive &ar, const unsigned int)
	Serialize the sparse coding model. More...

template<typename DictionaryInitializer = DataDependentRandomInitializer>
void	Train (const arma::mat &data, const DictionaryInitializer &initializer=DictionaryInitializer())
	Train the sparse coding model on the given dataset. More...

Private Attributes
size_t	atoms
	Number of atoms. More...

arma::mat	dictionary
	Dictionary (columns are atoms). More...

double	lambda1
	l1 regularization term. More...

double	lambda2
	l2 regularization term. More...

size_t	maxIterations
	Maximum number of iterations during training. More...

double	newtonTolerance
	Tolerance for Newton's method (dictionary training). More...

double	objTolerance
	Tolerance for main objective. More...

Detailed Description

An implementation of Sparse Coding with Dictionary Learning that achieves sparsity via an l1-norm regularizer on the codes (LASSO) or an (l1+l2)-norm regularizer on the codes (the Elastic Net).

Let d be the number of dimensions in the original space, m the number of training points, and k the number of atoms in the dictionary (the dimension of the learned feature space). The training data X is a d-by-m matrix where each column is a point and each row is a dimension. The dictionary D is a d-by-k matrix, and the sparse codes matrix Z is a k-by-m matrix. This program seeks to minimize the objective:

$\min_{D,Z} 0.5 ||X - D Z||_{F}^2\ + \lambda_1 \sum_{i=1}^m ||Z_i||_1 + 0.5 \lambda_2 \sum_{i=1}^m ||Z_i||_2^2$

subject to $ ||D_j||_2 <= 1 $ for $ 1 <= j <= k $ where typically $ lambda_1 > 0 $ and $ lambda_2 = 0 $ .

This problem is solved by an algorithm that alternates between a dictionary learning step and a sparse coding step. The dictionary learning step updates the dictionary D using a Newton method based on the Lagrange dual (see the paper below for details). The sparse coding step involves solving a large number of sparse linear regression problems; this can be done efficiently using LARS, an algorithm that can solve the LASSO or the Elastic Net (papers below).

Here are those papers:

@incollection{lee2007efficient,
  title = {Efficient sparse coding algorithms},
  author = {Honglak Lee and Alexis Battle and Rajat Raina and Andrew Y. Ng},
  booktitle = {Advances in Neural Information Processing Systems 19},
  editor = {B. Sch\"{o}lkopf and J. Platt and T. Hoffman},
  publisher = {MIT Press},
  address = {Cambridge, MA},
  pages = {801--808},
  year = {2007}
}

@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}
}

Note that the implementation here does not use the feature-sign search algorithm from Honglak Lee's paper, but instead the LARS algorithm suggested in that paper.

When Train() is called, the dictionary is initialized using the DictionaryInitializationPolicy class. Possible choices include the RandomInitializer, which provides an entirely random dictionary, the DataDependentRandomInitializer, which provides a random dictionary based loosely on characteristics of the dataset, and the NothingInitializer, which does not initialize the dictionary – instead, the user should set the dictionary using the Dictionary() mutator method.

Once a dictionary is trained with Train(), another matrix may be encoded with the Encode() function.

Template Parameters

DictionaryInitializationPolicy The class to use to initialize the dictionary; must have 'void Initialize(const arma::mat& data, arma::mat& dictionary)' function.

Definition at line 115 of file sparse_coding.hpp.

Constructor & Destructor Documentation

template<typename DictionaryInitializer = DataDependentRandomInitializer>

mlpack::sparse_coding::SparseCoding::SparseCoding	(	const arma::mat &	data,
		const size_t	atoms,
		const double	lambda1,
		const double	lambda2 = `0`,
		const size_t	maxIterations = `0`,
		const double	objTolerance = `0.01`,
		const double	newtonTolerance = `1e-6`,
		const DictionaryInitializer &	initializer = `DictionaryInitializer()`
	)

Set the parameters to SparseCoding.

lambda2 defaults to 0. This constructor will train the model. If that is not desired, call the other constructor that does not take a data matrix. This constructor will also initialize the dictionary using the given DictionaryInitializer before training.

If you want to initialize the dictionary to a custom matrix, consider either writing your own DictionaryInitializer class (with void Initialize(const arma::mat& data, arma::mat& dictionary) function), or call the constructor that does not take a data matrix, then call Dictionary() to set the dictionary matrix to a matrix of your choosing, and then call Train() with NothingInitializer (i.e. Train<NothingInitializer>(data)).

Parameters

data	Data matrix.
atoms	Number of atoms in dictionary.
lambda1	Regularization parameter for l1-norm penalty.
lambda2	Regularization parameter for l2-norm penalty.
maxIterations	Maximum number of iterations to run algorithm. If 0, the algorithm will run until convergence (or forever).
objTolerance	Tolerance for objective function. When an iteration of the algorithm produces an improvement smaller than this, the algorithm will terminate.
newtonTolerance	Tolerance for the Newton's method dictionary optimization step.

mlpack::sparse_coding::SparseCoding::SparseCoding	(	const size_t	atoms = `0`,
		const double	lambda1 = `0`,
		const double	lambda2 = `0`,
		const size_t	maxIterations = `0`,
		const double	objTolerance = `0.01`,
		const double	newtonTolerance = `1e-6`
	)

Set the parameters to SparseCoding.

lambda2 defaults to 0. This constructor will not train the model, and a subsequent call to Train() will be required before the model can encode points with Encode().

Parameters

atoms	Number of atoms in dictionary.
lambda1	Regularization parameter for l1-norm penalty.
lambda2	Regularization parameter for l2-norm penalty.
maxIterations	Maximum number of iterations to run algorithm. If 0, the algorithm will run until convergence (or forever).
objTolerance	Tolerance for objective function. When an iteration of the algorithm produces an improvement smaller than this, the algorithm will terminate.
newtonTolerance	Tolerance for the Newton's method dictionary optimization step.

Member Function Documentation

size_t mlpack::sparse_coding::SparseCoding::Atoms ( ) const

inline

Access the number of atoms.

Definition at line 226 of file sparse_coding.hpp.

References atoms.

size_t& mlpack::sparse_coding::SparseCoding::Atoms ( )

inline

Modify the number of atoms.

Definition at line 228 of file sparse_coding.hpp.

References atoms.

const arma::mat& mlpack::sparse_coding::SparseCoding::Dictionary ( ) const

inline

Access the dictionary.

Definition at line 221 of file sparse_coding.hpp.

References dictionary.

arma::mat& mlpack::sparse_coding::SparseCoding::Dictionary ( )

inline

Modify the dictionary.

Definition at line 223 of file sparse_coding.hpp.

References dictionary.

void mlpack::sparse_coding::SparseCoding::Encode	(	const arma::mat &	data,
		arma::mat &	codes
	)

Sparse code each point in the given dataset via LARS, using the current dictionary and store the encoded data in the codes matrix.

Parameters

data	Input data matrix to be encoded.
codes	Output codes matrix.

double mlpack::sparse_coding::SparseCoding::Lambda1 ( ) const

inline

Access the L1 regularization term.

Definition at line 231 of file sparse_coding.hpp.

References lambda1.

double& mlpack::sparse_coding::SparseCoding::Lambda1 ( )

inline

Modify the L1 regularization term.

Definition at line 233 of file sparse_coding.hpp.

References lambda1.

double mlpack::sparse_coding::SparseCoding::Lambda2 ( ) const

inline

Access the L2 regularization term.

Definition at line 236 of file sparse_coding.hpp.

References lambda2.

double& mlpack::sparse_coding::SparseCoding::Lambda2 ( )

inline

Modify the L2 regularization term.

Definition at line 238 of file sparse_coding.hpp.

References lambda2.

size_t mlpack::sparse_coding::SparseCoding::MaxIterations ( ) const

inline

Get the maximum number of iterations.

Definition at line 241 of file sparse_coding.hpp.

References maxIterations.

size_t& mlpack::sparse_coding::SparseCoding::MaxIterations ( )

inline

Modify the maximum number of iterations.

Definition at line 243 of file sparse_coding.hpp.

References maxIterations.

double mlpack::sparse_coding::SparseCoding::NewtonTolerance ( ) const

inline

Get the tolerance for Newton's method (dictionary optimization step).

Definition at line 251 of file sparse_coding.hpp.

References newtonTolerance.

double& mlpack::sparse_coding::SparseCoding::NewtonTolerance ( )

inline

Modify the tolerance for Newton's method (dictionary optimization step).

Definition at line 253 of file sparse_coding.hpp.

References newtonTolerance, and Serialize().

double mlpack::sparse_coding::SparseCoding::Objective	(	const arma::mat &	data,
		const arma::mat &	codes
	)		const

Compute the objective function.

double mlpack::sparse_coding::SparseCoding::ObjTolerance ( ) const

inline

Get the objective tolerance.

Definition at line 246 of file sparse_coding.hpp.

References objTolerance.

double& mlpack::sparse_coding::SparseCoding::ObjTolerance ( )

inline

Modify the objective tolerance.

Definition at line 248 of file sparse_coding.hpp.

References objTolerance.

double mlpack::sparse_coding::SparseCoding::OptimizeDictionary	(	const arma::mat &	data,
		const arma::mat &	codes,
		const arma::uvec &	adjacencies
	)

Learn dictionary via Newton method based on Lagrange dual.

Parameters

data	Data matrix.
codes	Matrix of codes.
adjacencies	Indices of entries (unrolled column by column) of the coding matrix Z that are non-zero (the adjacency matrix for the bipartite graph of points and atoms).

Returns: the norm of the gradient of the Lagrange dual with respect to the dual variables

void mlpack::sparse_coding::SparseCoding::ProjectDictionary ( )

Project each atom of the dictionary back onto the unit ball, if necessary.

template<typename Archive >

void mlpack::sparse_coding::SparseCoding::Serialize	(	Archive &	ar,
		const unsigned	int
	)

Serialize the sparse coding model.

Referenced by NewtonTolerance().

template<typename DictionaryInitializer = DataDependentRandomInitializer>

void mlpack::sparse_coding::SparseCoding::Train	(	const arma::mat &	data,
		const DictionaryInitializer &	initializer = `DictionaryInitializer()`
	)

Train the sparse coding model on the given dataset.

Member Data Documentation

size_t mlpack::sparse_coding::SparseCoding::atoms

private

Number of atoms.

Definition at line 261 of file sparse_coding.hpp.

Referenced by Atoms().

arma::mat mlpack::sparse_coding::SparseCoding::dictionary

private

Dictionary (columns are atoms).

Definition at line 264 of file sparse_coding.hpp.

Referenced by Dictionary().

double mlpack::sparse_coding::SparseCoding::lambda1

private

l1 regularization term.

Definition at line 267 of file sparse_coding.hpp.

Referenced by Lambda1().

double mlpack::sparse_coding::SparseCoding::lambda2

private

l2 regularization term.

Definition at line 269 of file sparse_coding.hpp.

Referenced by Lambda2().

size_t mlpack::sparse_coding::SparseCoding::maxIterations

private

Maximum number of iterations during training.

Definition at line 272 of file sparse_coding.hpp.

Referenced by MaxIterations().

double mlpack::sparse_coding::SparseCoding::newtonTolerance

private

Tolerance for Newton's method (dictionary training).

Definition at line 276 of file sparse_coding.hpp.

Referenced by NewtonTolerance().

double mlpack::sparse_coding::SparseCoding::objTolerance

private

Tolerance for main objective.

Definition at line 274 of file sparse_coding.hpp.

Referenced by ObjTolerance().

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

src/mlpack/methods/sparse_coding/sparse_coding.hpp

Public Member Functions

Private Attributes

Detailed Description

Constructor & Destructor Documentation

Member Function Documentation

Member Data Documentation