mlpack/doc/tutorials/amf/amf.txt

/*!

@file amf.txt
@author Sumedh Ghaisas
@brief Tutorial for how to use the AMF class.

@page amftutorial Alternating Matrix Factorization tutorial.

@section intro_amftut Introduction

Alternating Matrix Factorization

Alternating matrix factorization decomposes matrx V in the form \f$ V \approx WH \f$
where W is called the basis matrix and H is called the encoding matrix. V is
taken to be of size n x m and the obtained W is n x r and H is r x m. The size
r is called the rank of the factorization. Factorization is done by alternately
calculating W and H respectively while holding the other matrix constant.

\b mlpack provides:

 - a \ref amf_amftut "simple C++ interface" to perform Alternating Matrix Factorization

@section toc_amftut Table of Contents

A list of all the sections this tutorial contains.

 - \ref intro_amftut
 - \ref toc_amftut
 - \ref amf_amftut
   - \ref t_policy_amftut
   - \ref init_rule_amftut
   - \ref update_rule_amftut
   - \ref nmf_amftut
   - \ref svd_amftut
 - \ref further_doc_amftut

@section amf_amftut The 'AMF' class

The AMF class is templatized with 3 parameters; the first contains the policy
used to determine when the algorithm has converged; the second contains the
initialization rule for the W and H matrix; the last contains the update rule
to be used during each iteration. This templatization allows the user to try
various update rules, initialization rules, and termination policies (including
ones not supplied with mlpack) for factorization.

The class provides the following method that performs factorization
@code
template<typename MatType> double Apply(const MatType& V,
                                        const size_t r,
                                        arma::mat& W,
                                        arma::mat& H);
@endcode

@subsection t_policy_amftut Using different termination policies

The AMF implementation comes with different termination policies to support many
implemented algorithms. Every termination policy implements the following method
which returns the status of convergence.
@code
bool IsConverged(arma::mat& W, arma::mat& H)
@endcode

list of all the termination policies

 - \ref mlpack::amf::SimpleResidueTermination
 - \ref mlpack::amf::SimpleToleranceTermination
 - \ref mlpack::amf::ValidationRMSETermination

In SimpleResidueTermination, termination decision depends on two factors, value
of residue and number of iteration. If the current value of residue drops below
the threshold or the number of iterations goes beyond the threshold, positive
termination signal is passed to AMF.

In SimpleToleranceTermination, termination criterion is met when increase in
residue value drops below the given tolerance. To accommodate spikes, certain
number of successive residue drops are accepted. Secondary termination criterion
terminates algorithm when iteration count goes beyond the threshold.

ValidationRMSETermination divids the data into 2 sets, training set and
validation set. Entries of validation set are nullifed in the input matrix.
Termination criterion is met when increase in validation set RMSe value drops
below the given tolerance. To accommodate spikes certain number of successive
validation RMSE drops are accepted. This upper imit on successive drops can be
adjusted with reverseStepCount. Secondary termination criterion terminates
algorithm when iteration count goes above the threshold. Though this termination
policy is better measure of convergence than the above 2 termination policies,
it may cause a overhead in performance.

On the other hand \ref mlpack::amf::CompleteIncrementalTermination
"CompleteIncrementalTermination" and \ref mlpack::amf::IncompleteIncrementalTermination
are just wrapper classes for other termination policies. These policies are used
when AMF is applied with \ref mlpack::amf::SVDCompleteIncrementalLearning
"SVDCompleteIncrementalLearning" and \ref mlpack::amf::SVDIncompleteIncrementalLearning
"SVDIncompleteIncrementalLearning" respectively.

@subsection init_rule_amftut Using different initialization policies

The AMF class comes with 2 initialization policies
 - \ref mlpack::amf::RandomInitialization "RandomInitialization"
 - \ref mlpack::amf::RandomAcolInitialization "RandomAcolInitialization"

RandomInitialization initializes matrices W and H with random uniform distribution
while RandomAcolInitialization initializes the W matrix by averaging p randomly
chosen columns of V.  In case of RandomAcolInitialization, p is a template parameter.

To implement their own initialization policy, users need to define the following
function in their class.
@code
template<typename MatType>
inline static void Initialize(const MatType& V,
                              const size_t r,
                              arma::mat& W,
                              arma::mat& H)
@endcode

@subsection update_rule_amftut Using different update rules

AMF supports following update rules
 - \ref mlpack::amf::NMFALSUpdate "AMFALSUpdate"
 - \ref mlpack::amf::NMFMultiplicativeDistanceUpdate "NMFMultiplicativeDistanceUpdate"
 - \ref mlpack::amf::NMFMultiplicativeDivergenceUpdate "NMFMultiplicativeDivergenceUpdate"
 - \ref mlpack::amf::SVDBatchLearning "SVDBatchLearning"
 - \ref mlpack::amf::SVDIncompleteIncrementalLearning "SVDIncompleteIncrementalLearning"
 - \ref mlpack::amf::SVDCompleteIncrementalLearning "SVDCompleteIncrementalLearning"

Non-Negative Matrix factorization can be achieved with NMFALSUpdate,
NMFMultiplicativeDivergenceUpdate or NMFMultiplicativeDivergenceUpdate.
NMFALSUpdate implements simple Alternating Least Square optimization while
the other rules implement algorithms given in paper 'Algorithms for Non-negative
Matrix Factorization'.

The remaining update rules perform Singular Value Decomposition of matrix V.
This SVD factorization is optimized for the use by Collaborative Filtering. This
use of SVD factorizers for Collaborative Filtering is described in the paper
'A Guide to singular Value Decomposition' by Chih-Chao Ma. For further details
about the algorithms refer to the respective class documentation.

@subsection nmf_amftut Using Non-Negative Matrix Factorization with AMF

The use of AMF for Non-Negative Matrix factorization is simple. The AMF module
defines \ref mlpack::amf::NMFALSFactorizer "NMFALSFactorizer" which can be used
directly without knowing the internal structure of AMF. For example -

@code
#include <iostream>
#include <mlpack/core.hpp>
#include <mlpack/methods/amf/amf.hpp>

using namespace std;
using namespace arma;
using namespace mlpack::amf;

int main()
{
  NMFALSFactorizer nmf;
  mat W, H;
  mat V = randu<mat>(100, 100);
  double residue = nmf.Apply(V, W, H);
  return 1;
}
@endcode

NMFALSFactorizer uses SimpleResidueTermination which is most preferred with
Non-Negative Matrix factorizers. Initialization of W and H in NMFALSFactorizer
is random. The Apply function returns the residue obtained by comparing the
constructed matrix W * H with the original matrix V.

@subsection svd_amftut Using Singular Value Decomposition with AMF

AMF implementation supports following SVD factorizers
 - \ref mlpack::amf::SVDBatchFactorizer "SVDBatchFactorizer"
 - \ref mlpack::amf::SparseSVDBatchFactorizer "SparseSVDBatchFactorizer"
 - \ref mlpack::amf::SVDIncompleteIncrementalFactorizer "SVDIncompleteIncrementalFactorizer"
 - \ref mlpack::amf::SparseSVDIncompleteIncrementalFactorizer "SparseSVDIncompleteIncrementalFactorizer"
 - \ref mlpack::amf::SVDCompleteIncrementalFactorizer "SVDCompleteIncrementalFactorizer"
 - \ref mlpack::amf::SparseSVDCompleteIncrementalFactorizer "SparseSVDCompleteIncrementalFactorizer"

The sparse version of factorizers can be used with Armadillo's sparse matrix
support. These  specialized implementations boost runtime performance when the
matrix to be factorized is relatively sparse.

@code
#include <mlpack/core.hpp>
#include <mlpack/methods/amf/amf.hpp>

using namespace std;
using namespace arma;
using namespace mlpack::amf;

int main()
{
  sp_mat V = randu<sp_mat>(100,100);
  mat W, H;

  SparseSVDBatchFactorizer svd;
  double residue = svd.Apply(V, W, H);
}
@endcode

@section further_doc_amftut Further documentation

For further documentation on the AMF class, consult the \ref mlpack::amf::AMF
"complete API documentation".

*/