🔗 AMF

The AMF class implements a general alternating matrix factorization framework, allowing numerous types of matrix decompositions. The AMF class can decompose a large (potentially sparse) matrix V into two smaller matrices W and H, such that V ~= W * H, using a variety of strategies that involve iteratively updating first the W matrix, then the H matrix, and so forth. This technique may be used for dimensionality reduction, or as part of a recommender system.

The behavior of the AMF class is controlled entirely by its template parameters. Different choices of these template parameters lead to different algorithms for matrix decomposition. For instance, mlpack’s implementation of non-negative matrix factorization (NMF) is built on the AMF class with NMF-specific template parameters.

Simple usage example:

// Create a random sparse matrix (V) of size 10x100, with 15% nonzeros.
arma::sp_mat V;
V.sprandu(100, 100, 0.15);

// W and H will be low-rank matrices of size 100x10 and 10x100.
arma::mat W, H;

// Step 1: create object.  The choices of template parameters control the
// behavior of the decomposition.
mlpack::AMF<mlpack::SimpleResidueTermination   /* termination policy */,
            mlpack::RandomAcolInitialization<> /* policy to initialize W/H */,
            mlpack::SVDBatchLearning<>         /* alternating update rules */>
    amf;

// Step 2: apply alternating matrix factorization to decompose V.
double residue = amf.Apply(V, 10, W, H);

// Now print some information about the factorized matrices.
std::cout << "W has size: " << W.n_rows << " x " << W.n_cols << "."
    << std::endl;
std::cout << "H has size: " << H.n_rows << " x " << H.n_cols << "."
    << std::endl;
std::cout << "RMSE of reconstructed matrix: "
    << arma::norm(V - W * H, "fro") / std::sqrt(V.n_elem) << "." << std::endl;

More examples...

See also:

🔗 Template parameter overview

The behavior of the AMF class is controlled by its three template parameters. The full signature of the class is:

AMF<TerminationPolicyType, InitializationRuleType, UpdateRuleType>

The AMF class is most useful when each of these three parameters are intentionally chosen. The default template parameters simply configure the algorithm as non-negative matrix factorization (NMF), and so in that situation the NMF class can be used instead.

A number of convenient typedefs are possible to configure the AMF class as a predefined algorithm. It may be easier to use these than to manually specify each template parameter.

🔗 TerminationPolicyType

SimpleResidueTermination (default):

MaxIterationTermination:

SimpleToleranceTermination<MatType, WHMatType>:

CompleteIncrementalTermination<TerminationPolicy>

IncompleteIncrementalTermination<TerminationPolicy>

For custom termination policies, see Custom TerminationPolicyTypes.

🔗 InitializationRuleType

RandomAcolInitialization<N> (default):

NoInitialization:

GivenInitialization<MatType>:

RandomAMFInitialization:

AverageInitialization:

MergeInitialization<WRule, HRule>:

For custom initialization rules, see Custom InitializationRuleTypes.

🔗 UpdateRuleType

NMF updates

Non-negative matrix factorization (NMF) can be expressed with the AMF class using any of the following UpdateRuleTypes.

Note: when using these update rules, it may be more convenient to use the more specific NMF class. NMF is just a typedef for AMF<SimpleResidueTermination, RandomAcolInitialization<5>, NMFMultiplicativeDistanceUpdate>.

SVDBatchLearning

SVDCompleteIncrementalLearning

SVDIncompleteIncrementalLearning

For custom update rules, see Custom UpdateRuleTypes.

🔗 Constructors

🔗 Applying Decompositions

Notes:

Apply() Parameters:

name type description
V arma::sp_mat or arma::mat Input matrix to be factorized.
rank size_t Rank of decomposition; lower is smaller, higher is more accurate.
W arma::mat Output matrix in which W will be stored.
H arma::mat Output matrix in which H will be stored.

Note: Matrices with different element types can be used for V, W, and H; e.g., arma::fmat. While V can be sparse or dense, W and H must be dense matrices.

🔗 Simple Examples

See also the simple usage example for a trivial use of AMF.

Decompose a dense matrix with simple residue termination using custom parameters.

// Create a low-rank V matrix by multiplying together two random matrices.
arma::mat V = arma::randu<arma::mat>(500, 25) *
              arma::randn<arma::mat>(25, 5000);

// Create the AMF object with a looser tolerance of 1e-3 and a maximum of 100
// iterations only.
// Since we have not specified the update rules, this will by default use the
// NMF multiplicative distance update.
mlpack::AMF<mlpack::SimpleResidueTermination> amf(
    mlpack::SimpleResidueTermination(1e-3, 500));

arma::mat W, H;

// Decompose with a rank of 25.
// W will have size 500 x 25, and H will have size 25 x 5000.
const double residue = amf.Apply(V, 25, W, H);

std::cout << "Residue of decomposition: " << residue << "." << std::endl;

// Compute RMSE of decomposition.
const double rmse = arma::norm(V - W * H, "fro") / std::sqrt(V.n_elem);
std::cout << "RMSE of decomposition: " << rmse << "." << std::endl;

Decompose the sparse MovieLens dataset using batch SVD learning, a rank-12 decomposition, and float element type.

// See https://datasets.mlpack.org/movielens-100k.csv.
arma::sp_fmat V;
mlpack::data::Load("movielens-100k.csv", V, true);

// Create the AMF object.  Use default parameters for the termination policy,
// initialization rule, and update rules.
mlpack::AMF<mlpack::SimpleResidueTermination,
            mlpack::RandomAcolInitialization<5>,
            mlpack::SVDBatchLearning<arma::fmat>> amf;

arma::fmat W, H;

// Decompose the Movielens dataset with rank 12.
const double residue = amf.Apply(V, 12, W, H);

std::cout << "Residue of MovieLens decomposition: " << residue << "."
    << std::endl;

// Compute RMSE of decomposition.
const double rmse = arma::norm(V - W * H, "fro") / std::sqrt(V.n_elem);
std::cout << "RMSE of decomposition: " << rmse << "." << std::endl;

Compare quality of decompositions of MovieLens with different update rules.

// See https://datasets.mlpack.org/movielens-100k.csv.
arma::sp_mat V;
mlpack::data::Load("movielens-100k.csv", V, true);

// Create four AMF objects using different update rules:
//  - SVDBatchLearning
//  - SVDCompleteIncrementalLearning
//  - SVDIncompleteIncrementalLearning
//  - NMFALSUpdate
// We use MaxIterationTermination for each, wrapped in incremental terminators
// if appropriate.

mlpack::AMF<mlpack::MaxIterationTermination,
            mlpack::RandomAcolInitialization<5>,
            mlpack::SVDBatchLearning<>>
    svdBatchAMF(mlpack::MaxIterationTermination(500),
                mlpack::RandomAcolInitialization<5>(),
                mlpack::SVDBatchLearning<>(0.0005 /* step size */));

mlpack::AMF<mlpack::CompleteIncrementalTermination<
                mlpack::MaxIterationTermination
            >,
            mlpack::RandomAcolInitialization<5>,
            mlpack::SVDCompleteIncrementalLearning<arma::sp_mat>>
    svdCompleteAMF(mlpack::CompleteIncrementalTermination<
                       mlpack::MaxIterationTermination>(500),
                   mlpack::RandomAcolInitialization<5>(),
                   mlpack::SVDCompleteIncrementalLearning<
                       arma::sp_mat>(0.0002));

mlpack::AMF<mlpack::IncompleteIncrementalTermination<
                mlpack::MaxIterationTermination>,
            mlpack::RandomAcolInitialization<5>,
            mlpack::SVDIncompleteIncrementalLearning<arma::sp_mat>>
    svdIncompleteAMF(mlpack::IncompleteIncrementalTermination<
                         mlpack::MaxIterationTermination>(500),
                     mlpack::RandomAcolInitialization<5>(),
                     mlpack::SVDIncompleteIncrementalLearning<
                         arma::sp_mat>(0.0002));

// NMFALSUpdate does not have any template parameters, so we don't need to pass
// it to the constructor.
mlpack::AMF<mlpack::MaxIterationTermination,
            mlpack::RandomAcolInitialization<5>,
            mlpack::NMFALSUpdate>
    nmf(mlpack::MaxIterationTermination(500));

// Decompose with the given rank.
arma::mat W, H;
const size_t rank = 15;

const double svdBatchResidue = svdBatchAMF.Apply(V, rank, W, H);
const double svdBatchRMSE = arma::norm(V - W * H, "fro") / std::sqrt(V.n_elem);
std::cout << "RMSE for SVD batch learning: " << svdBatchRMSE << "."
    << std::endl;

const double svdCompleteResidue = svdCompleteAMF.Apply(V, rank, W, H);
const double svdCompleteRMSE = arma::norm(V - W * H, "fro") /
    std::sqrt(V.n_elem);
std::cout << "RMSE for SVD complete incremental learning: " << svdCompleteRMSE
    << "." << std::endl;

const double svdIncompleteResidue = svdIncompleteAMF.Apply(V, rank, W, H);
const double svdIncompleteRMSE = arma::norm(V - W * H, "fro") /
    std::sqrt(V.n_elem);
std::cout << "RMSE for SVD incomplete incremental learning: "
    << svdIncompleteRMSE << "." << std::endl;

const double nmfResidue = nmf.Apply(V, rank, W, H);
const double nmfRMSE = arma::norm(V - W * H, "fro") / std::sqrt(V.n_elem);
std::cout << "RMSE for NMF with ALS update rules: " << nmfRMSE << "."
    << std::endl;

Use a pre-specified initialization for W and H.

// See https://datasets.mlpack.org/movielens-100k.csv.
arma::sp_mat V;
mlpack::data::Load("movielens-100k.csv", V, true);

arma::mat W, H;

// Pre-initialize W and H.
// W will be filled with random values from a normal distribution.
// H will be filled with 1s.
W.randn(V.n_rows, 15);
H.set_size(15, V.n_cols);
H.fill(0.2);

mlpack::AMF<mlpack::MaxIterationTermination,
            mlpack::NoInitialization,
            mlpack::SVDBatchLearning<>>
    amf(mlpack::MaxIterationTermination(1000));
const double residue = amf.Apply(V, 15, W, H);
const double rmse = arma::norm(V - W * H, "fro") / std::sqrt(V.n_elem);

std::cout << "RMSE of SVDBatchLearning decomposition with pre-specified W and "
    << "H: " << rmse << "." << std::endl;

Use MergeInitialization to specify different strategies to initialize the W and H matrices.

// See https://datasets.mlpack.org/movielens-100k.csv.
arma::sp_mat V;
mlpack::data::Load("movielens-100k.csv", V, true);

arma::mat W, H;

// This will initialize the W matrix.
mlpack::RandomAcolInitialization<5> initW;

// This will initialize the H matrix.
mlpack::RandomAMFInitialization initH;

// Combine the two initializations so we can pass it to the AMF class.
typedef mlpack::MergeInitialization<mlpack::RandomAcolInitialization<5>,
                                    mlpack::RandomAMFInitialization> InitType;
InitType init(initW, initH);

// Create an AMF object with the custom initialization.
mlpack::AMF<mlpack::CompleteIncrementalTermination<
                mlpack::SimpleResidueTermination
            >,
            InitType,
            mlpack::SVDCompleteIncrementalLearning<arma::sp_mat>>
    amf(mlpack::CompleteIncrementalTermination<
            mlpack::SimpleResidueTermination>(), init);

// Perform AMF with a rank of 10.
const double residue = amf.Apply(V, 10, W, H);

std::cout << "Residue after training: " << residue << "." << std::endl;

Use ValidationRMSETermination to decompose the MovieLens dataset until the RMSE of the held-out validation set is sufficiently low.

// See https://datasets.mlpack.org/movielens-100k.csv.
arma::sp_mat V;
mlpack::data::Load("movielens-100k.csv", V, true);

arma::mat W, H;

// Create a ValidationRMSETermination class that will hold out 3k points from V.
// This will remove 3000 nonzero entries from V.
mlpack::ValidationRMSETermination<arma::sp_mat> t(V, 3000);

// Create the AMF object with the instantiated termination policy.
mlpack::AMF<mlpack::ValidationRMSETermination<arma::sp_mat>,
            mlpack::RandomAcolInitialization<5>,
            mlpack::SVDBatchLearning<>> amf(t);

// Perform AMF with a rank of 20.
// Note the RMSE returned here is the RMSE on the validation set.
const double rmse = amf.Apply(V, 20, W, H);
const double rmseTrain = arma::norm(V - W * H, "fro") / std::sqrt(V.n_elem);

std::cout << "Training RMSE:   " << rmseTrain << "." << std::endl;
std::cout << "Validation RMSE: " << rmse << "." << std::endl;

🔗 Custom TerminationPolicyTypes

See also the list of available TerminationPolicyTypes.

If custom functionality is desired for controlling the termination of the AMF algorithm, a custom class may be implemented that must implement the following functions:

// You can use this as a starting point for implementation.
class CustomTerminationPolicy
{
 public:
  // Initialize the termination policy for the given matrix V.  (It is okay to
  // do nothing.)  This function is called at the beginning of Apply().
  //
  // If the termination policy requires V to compute convergence, store a
  // reference or pointer to it in this function.
  template<typename MatType>
  void Initialize(const MatType& V);

  // Check if convergence has occurred for the given W and H matrices.  Return
  // `true` if so.
  //
  // Note that W and H may have different types than V (i.e. V may be sparse,
  // and W and H must be dense.)
  template<typename WHMatType>
  bool IsConverged(const MatType& H, const MatType& W);

  // Return the value that should be returned for the `amf.Apply()` function
  // when convergence has been reached.  This is called at the end of
  // `amf.Apply()`.
  const double Index();

  // Return the number of iterations that have been completed.  This is called
  // at the end of `amf.Apply()`.
  const size_t Iteration();
};

🔗 Custom InitializationRuleTypes

See also the list of available InitializationRuleTypes.

If custom functionality is desired for initializing W and H, a custom class may be implemented that must implement the following functions:

// You can use this as a starting point for implementation.
class CustomInitialization
{
 public:
  // Initialize the W and H matrices, given V and the rank of the decomposition.
  // This is called at the start of `Apply()`.
  //
  // Note that `MatType` may be different from `WHMatType`; e.g., `V` could be
  // sparse, but `W` and `H` must be dense.
  template<typename MatType, typename WHMatType>
  void Initialize(const MatType& V,
                  const size_t rank,
                  WHMatType& W,
                  WHMatType& H);

  // Initialize one of the W or H matrices, given V and the rank of the
  // decomposition.
  //
  // If `isW` is `true`, then `M` should be treated as though it is `W`;
  // if `isW` is `false`, then `M` should be treated as thought it is `H`.
  //
  // This function only needs to be implemented if it is intended to use the
  // custom initialization strategy with `MergeInitialization`.
  template<typename MatType, typename WHMatType>
  void InitializeOne(const MatType& V,
                     const size_t rank,
                     WHMatType& M,
                     const bool isW);
};

For example, the code below implements a custom termination policy that sets a limit on how long AMF is allowed to take:

class CustomTimeTermination
{
 public:
  CustomTimeTermination(const double totalAllowedTime) :
      totalAllowedTime(totalAllowedTime) { }

  template<typename MatType>
  void Initialize(const MatType& /* V */)
  {
    totalTime = 0.0;
    iteration = 0;
    c.tic();
  }

  template<typename WHMatType>
  bool IsConverged(const WHMatType& /* W */, const WHMatType& /* H */)
  {
    totalTime += c.toc();
    c.tic();
    ++iteration;
    return (totalTime > totalAllowedTime);
  }

  const double Index() const { return totalTime; }
  const size_t Iteration() const { return iteration; }

 private:
  double totalAllowedTime;
  double totalTime;
  size_t iteration;
  arma::wall_clock c; // used for convenient timing
};

Then we can use it in a program:

// See https://datasets.mlpack.org/movielens-100k.csv.
arma::sp_fmat V;
mlpack::data::Load("movielens-100k.csv", V, true);

CustomTimeTermination t(5 /* seconds */);
mlpack::AMF<CustomTimeTermination,
            mlpack::RandomAcolInitialization<5>,
            mlpack::SVDBatchLearning<arma::fmat>> amf(t);

arma::fmat W, H;
const double actualTime = amf.Apply(V, 10, W, H);
const double rmse = arma::norm(V - W * H, "fro") / std::sqrt(V.n_elem);

std::cout << "Actual time used for decomposition: " << actualTime << "."
    << std::endl;
std::cout << "RMSE after ~5 seconds: " << rmse << "." << std::endl;

🔗 Custom UpdateRuleTypes

See also the list of available UpdateRuleTypes.

If custom functionality is desired for the update rules to be applied to W and H, a custom class may be implemented that must implement the following functions:

// You can use this as a starting point for implementation.
class CustomUpdateRule
{
 public:
  // Set initial values for the factorization.  This is called at the beginning
  // of Apply(), before WUpdate() or HUpdate() are called.
  //
  // `MatType` will be the type of `V` (an Armadillo dense or sparse matrix).
  //
  template<typename MatType>
  void Initialize(const MatType& V, const size_t rank);

  // Update the `W` matrix given `V` and the current `H` matrix.
  //
  // `MatType` will be the type of `V`, and `WHMatType` will be the type of `W`
  // and `H`.  Both will be matrix types matching the Armadillo API.
  template<typename MatType, typename WHMatType>
  void WUpdate(const MatType& V, WHMatType& W, const WHMatType& H);

  // Update the `H` matrix given `V` and the current `W` matrix.
  //
  // `MatType` will be the type of `V`, and `WHMatType` will be the type of `W`
  // and `H`.  Both will be matrix types matching the Armadillo API.
  template<typename MatType, typename WHMatType>
  void HUpdate(const MatType& V, const WHMatType& W, WHMatType& H);

  // Serialize the update rule using the cereal library.
  // This is only necessary if the update rule will be used with an AMF object
  // that is saved or loaded with data::Save() or data::Load().
  template<typename Archive>
  void serialize(Archive& ar, const uint32_t version);
};