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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...

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...

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

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

 
const arma::mat & Dictionary () const
 Accessor for dictionary. More...

 
arma::mat & Dictionary ()
 Mutator for dictionary. More...

 
void Encode (const arma::mat &data, arma::mat &codes)
 Code each point via distance-weighted LARS. More...

 
double Lambda () const
 Get the L1 regularization parameter. More...

 
double & Lambda ()
 Modify the L1 regularization parameter. More...

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

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

 
void OptimizeDictionary (const arma::mat &data, const arma::mat &codes, const arma::uvec &adjacencies)
 Learn dictionary by solving linear system. More...

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

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

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

 
template
<
typename
DictionaryInitializer
=
sparse_coding::DataDependentRandomInitializer
>
void Train (const arma::mat &data, const DictionaryInitializer &initializer=DictionaryInitializer())
 Run local coordinate coding. More...

 

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}
}

Definition at line 79 of file lcc.hpp.

Constructor & Destructor Documentation

◆ LocalCoordinateCoding() [1/2]

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
dataData matrix.
atomsNumber of atoms in dictionary.
lambdaRegularization parameter for weighted l1-norm penalty.
maxIterationsMaximum number of iterations for training (0 runs until convergence).
toleranceTolerance for the objective function.

◆ LocalCoordinateCoding() [2/2]

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
atomsNumber of atoms in dictionary.
lambdaRegularization parameter for weighted l1-norm penalty.
maxIterationsMaximum number of iterations for training (0 runs until convergence).
toleranceTolerance for the objective function.

Member Function Documentation

◆ Atoms() [1/2]

size_t Atoms ( ) const
inline

Get the number of atoms.

Definition at line 174 of file lcc.hpp.

◆ Atoms() [2/2]

size_t& Atoms ( )
inline

Modify the number of atoms.

Definition at line 176 of file lcc.hpp.

◆ Dictionary() [1/2]

const arma::mat& Dictionary ( ) const
inline

Accessor for dictionary.

Definition at line 179 of file lcc.hpp.

◆ Dictionary() [2/2]

arma::mat& Dictionary ( )
inline

Mutator for dictionary.

Definition at line 181 of file lcc.hpp.

◆ Encode()

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

Code each point via distance-weighted LARS.

Parameters
dataMatrix containing points to encode.
codesOutput matrix to store codes in.

◆ Lambda() [1/2]

double Lambda ( ) const
inline

Get the L1 regularization parameter.

Definition at line 184 of file lcc.hpp.

◆ Lambda() [2/2]

double& Lambda ( )
inline

Modify the L1 regularization parameter.

Definition at line 186 of file lcc.hpp.

◆ MaxIterations() [1/2]

size_t MaxIterations ( ) const
inline

Get the maximum number of iterations.

Definition at line 189 of file lcc.hpp.

◆ MaxIterations() [2/2]

size_t& MaxIterations ( )
inline

Modify the maximum number of iterations.

Definition at line 191 of file lcc.hpp.

◆ Objective()

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

Compute objective function given the list of adjacencies.

◆ OptimizeDictionary()

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

Learn dictionary by solving linear system.

Parameters
adjacenciesIndices 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)

◆ serialize()

void serialize ( Archive &  ar,
const unsigned  int 
)

Serialize the model.

Referenced by LocalCoordinateCoding::Tolerance().

◆ Tolerance() [1/2]

double Tolerance ( ) const
inline

Get the objective tolerance.

Definition at line 194 of file lcc.hpp.

◆ Tolerance() [2/2]

double& Tolerance ( )
inline

Modify the objective tolerance.

Definition at line 196 of file lcc.hpp.

References LocalCoordinateCoding::serialize().

◆ Train()

void Train ( const arma::mat &  data,
const DictionaryInitializer &  initializer = DictionaryInitializer() 
)

Run local coordinate coding.

Parameters
nIterationsMaximum number of iterations to run algorithm.
objToleranceTolerance of objective function. When the objective function changes by a value lower than this tolerance, the optimization terminates.

The documentation for this class was generated from the following file:
  • src/mlpack/methods/local_coordinate_coding/lcc.hpp