mlpack::lcc::LocalCoordinateCoding Class Reference

An implementation of Local Coordinate Coding (LCC) that codes data which approximately lives on a manifold using a variation of l1-norm regularized sparse coding; in LCC, the penalty on the absolute value of each point's coefficient for each atom is weighted by the squared distance of that point to that atom. More...

#include <lcc.hpp>

Public Member Functions
template<typename DictionaryInitializer = sparse_coding::DataDependentRandomInitializer>
	LocalCoordinateCoding (const arma::mat &data, const size_t atoms, const double lambda, const size_t maxIterations=0, const double tolerance=0.01, const DictionaryInitializer &initializer=DictionaryInitializer())
	Set the parameters to LocalCoordinateCoding, and train the dictionary. More...
	LocalCoordinateCoding (const size_t atoms=0, const double lambda=0.0, const size_t maxIterations=0, const double tolerance=0.01)
	Set the parameters to LocalCoordinateCoding. More...
template<typename DictionaryInitializer = sparse_coding::DataDependentRandomInitializer>
double	Train (const arma::mat &data, const DictionaryInitializer &initializer=DictionaryInitializer())
	Run local coordinate coding. More...
void	Encode (const arma::mat &data, arma::mat &codes)
	Code each point via distance-weighted LARS. More...
void	OptimizeDictionary (const arma::mat &data, const arma::mat &codes, const arma::uvec &adjacencies)
	Learn dictionary by solving linear system. More...
double	Objective (const arma::mat &data, const arma::mat &codes, const arma::uvec &adjacencies) const
	Compute objective function given the list of adjacencies. More...
size_t	Atoms () const
	Get the number of atoms.
size_t &	Atoms ()
	Modify the number of atoms.
const arma::mat &	Dictionary () const
	Accessor for dictionary.
arma::mat &	Dictionary ()
	Mutator for dictionary.
double	Lambda () const
	Get the L1 regularization parameter.
double &	Lambda ()
	Modify the L1 regularization parameter.
size_t	MaxIterations () const
	Get the maximum number of iterations.
size_t &	MaxIterations ()
	Modify the maximum number of iterations.
double	Tolerance () const
	Get the objective tolerance.
double &	Tolerance ()
	Modify the objective tolerance.
template<typename Archive >
void	serialize (Archive &ar, const uint32_t)
	Serialize the model.

Detailed Description

An implementation of Local Coordinate Coding (LCC) that codes data which approximately lives on a manifold using a variation of l1-norm regularized sparse coding; in LCC, the penalty on the absolute value of each point's coefficient for each atom is weighted by the squared distance of that point to that atom.

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} ||X - D Z||_{Fro}^2

• lambda sum_{i=1}^m sum_{j=1}^k dist(X_i,D_j)^2 Z_i^j where lambda > 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 by solving a linear system (note that the objective is a positive definite quadratic program). The sparse coding step involves solving a large number of weighted l1-norm regularized linear regression problems problems; this can be done efficiently using LARS, an algorithm that can solve the LASSO (paper below).

The papers are listed below.

@incollection{NIPS2009_0719,
  title = {Nonlinear Learning using Local Coordinate Coding},
  author = {Kai Yu and Tong Zhang and Yihong Gong},
  booktitle = {Advances in Neural Information Processing Systems 22},
  editor = {Y. Bengio and D. Schuurmans and J. Lafferty and C. K. I. Williams
      and A. Culotta},
  pages = {2223--2231},
  year = {2009}
}

@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}
}

Constructor & Destructor Documentation

LocalCoordinateCoding() [1/2]

template<typename DictionaryInitializer >

mlpack::lcc::LocalCoordinateCoding::LocalCoordinateCoding	(	const arma::mat &	data,
		const size_t	atoms,
		const double	lambda,
		const size_t	maxIterations = 0,
		const double	tolerance = 0.01,
		const DictionaryInitializer &	initializer = DictionaryInitializer()
	)

Set the parameters to LocalCoordinateCoding, and train the dictionary.

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 sparse_coding::NothingInitializer (i.e. Train<sparse_coding::NothingInitializer>(data)).

Parameters

data	Data matrix.
atoms	Number of atoms in dictionary.
lambda	Regularization parameter for weighted l1-norm penalty.
maxIterations	Maximum number of iterations for training (0 runs until convergence).
tolerance	Tolerance for the objective function.
initializer	Intializer to use.

LocalCoordinateCoding() [2/2]

mlpack::lcc::LocalCoordinateCoding::LocalCoordinateCoding	(	const size_t	atoms = 0,
		const double	lambda = 0.0,
		const size_t	maxIterations = 0,
		const double	tolerance = 0.01
	)

Set the parameters to LocalCoordinateCoding.

This constructor will not train the model, and a subsequent call to Train() will be required before the model can encode points with Encode(). The default values for atoms and lambda should be changed if you intend to train the model!

Parameters

atoms	Number of atoms in dictionary.
lambda	Regularization parameter for weighted l1-norm penalty.
maxIterations	Maximum number of iterations for training (0 runs until convergence).
tolerance	Tolerance for the objective function.

Member Function Documentation

Encode()

void mlpack::lcc::LocalCoordinateCoding::Encode	(	const arma::mat &	data,
		arma::mat &	codes
	)

Code each point via distance-weighted LARS.

Parameters

data	Matrix containing points to encode.
codes	Output matrix to store codes in.

Objective()

double mlpack::lcc::LocalCoordinateCoding::Objective	(	const arma::mat &	data,
		const arma::mat &	codes,
		const arma::uvec &	adjacencies
	)		const

Compute objective function given the list of adjacencies.

Parameters

data	Matrix containing points to encode.
codes	Output matrix to store codes in.
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)

OptimizeDictionary()

void mlpack::lcc::LocalCoordinateCoding::OptimizeDictionary	(	const arma::mat &	data,
		const arma::mat &	codes,
		const arma::uvec &	adjacencies
	)

Learn dictionary by solving linear system.

Parameters

data	Matrix containing points to encode.
codes	Output matrix to store codes in.
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)

Train()

template<typename DictionaryInitializer >

double mlpack::lcc::LocalCoordinateCoding::Train	(	const arma::mat &	data,
		const DictionaryInitializer &	initializer = DictionaryInitializer()
	)

Run local coordinate coding.

Parameters

data	Data matrix.
initializer	Intializer to use.

Returns

The final objective value.

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

• src/mlpack/methods/local_coordinate_coding/lcc.hpp
• src/mlpack/methods/local_coordinate_coding/lcc.cpp
• src/mlpack/methods/local_coordinate_coding/lcc_impl.hpp