Alternating matrix factorization decomposes a matrix V
in the form V ~ WH
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.
mlpack provides a simple C interface to perform Alternating Matrix Factorization.
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
template<typename MatType> double Apply(const MatType& V,
const size_t r,
arma::mat& W,
arma::mat& H);
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.
bool IsConverged(arma::mat& W, arma::mat& H)
Below is a list of all the termination policies that mlpack contains.
SimpleResidueTermination
SimpleToleranceTermination
ValidationRMSETermination
In SimpleResidueTermination
, the 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 the 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
divides the data into 2 sets, training set and
validation set. Entries of the 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
. A secondary termination criterion terminates
the algorithm when the 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 decrease in performance since it is computationally
expensive.
On the other hand, CompleteIncrementalTermination
and
IncompleteIncrementalTermination
are just wrapper classes for other
termination policies. These policies are used when AMF is applied with
SVDCompleteIncrementalLearning
and SVDIncompleteIncrementalLearning
,
respectively.
mlpack currently has 2 initialization policies implemented for AMF:
RandomInitialization
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 the case of
RandomAcolInitialization
, p
is a template parameter.
To implement their own initialization policy, users need to define the following function in their class.
template<typename MatType>
inline static void Initialize(const MatType& V,
const size_t r,
arma::mat& W,
arma::mat& H)
mlpack implements the following update rules for the AMF class:
NMFALSUpdate
NMFMultiplicativeDistanceUpdate
NMFMultiplicativeDivergenceUpdate
SVDBatchLearning
SVDIncompleteIncrementalLearning
SVDCompleteIncrementalLearning
Non-Negative Matrix factorization can be achieved with NMFALSUpdate
,
NMFMultiplicativeDivergenceUpdate
or NMFMultiplicativeDivergenceUpdate
.
NMFALSUpdate
implements a simple Alternating Least Squares optimization while
the other rules implement algorithms given in the paper 'Algorithms for
Non-negative Matrix Factorization'.
The remaining update rules perform the singular value decomposition of the
matrix V
. This SVD factorization is optimized for use by mlpack's
collaborative filtering code (see the collaborative filtering
tutorial). This use of SVD factorizers for collaborative filtering is
described in the paper 'A Guide to Singular Value Decomposition for
Collaborative Filtering' by Chih-Chao Ma. For further details about the
algorithms refer to the respective class documentation.
The use of AMF
for Non-Negative Matrix factorization is simple. The AMF module
defines NMFALSFactorizer
which can be used directly without knowing the
internal structure of AMF
. For example:
#include <mlpack.hpp>
using namespace std;
using namespace arma;
using namespace mlpack;
int main()
{
NMFALSFactorizer nmf;
mat W, H;
mat V = randu<mat>(100, 100);
size_t r = 10;
double residue = nmf.Apply(V, r, W, H);
}
NMFALSFactorizer
uses SimpleResidueTermination
, which is most preferred with
Non-Negative Matrix factorizers. The 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
.
mlpack has the following SVD factorizers implemented for AMF:
SVDBatchFactorizer
SVDIncompleteIncrementalFactorizer
SVDCompleteIncrementalFactorizer
Each of these factorizers takes a template parameter MatType
, which specifies
the type of the matrix V
(dense or sparse---these have types arma::mat
and
arma::sp_mat
, respectively). When the matrix to be factorized is relatively
sparse, specifying MatType = arma::sp_mat
can provide a runtime boost.
#include <mlpack.hpp>
using namespace std;
using namespace arma;
using namespace mlpack;
int main()
{
sp_mat V = sprandu<sp_mat>(100,100,0.1);
size_t r = 10;
mat W, H;
SVDBatchFactorizer<sp_mat> svd;
double residue = svd.Apply(V, r, W, H);
}
For further documentation on the AMF
class, consult the AMF
source code comments.