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Matrices in mlpack

mlpack uses Armadillo matrices for linear algebra support. Armadillo is a fast C++ matrix library which uses advanced template metaprogramming techniques to provide the fastest possible linear algebra operations.

Armadillo logo

Detailed documentation on Armadillo can be found on the Armadillo website.

Nonetheless, there are a few further caveats for mlpack Armadillo usage.

An Armadillo primer

The Armadillo syntax is straightforward and is aimed at ease-of-use and readability. To give a flavor of what a linear algebra program using Armadillo looks like, see the trivial (contrived) program below that performs some basic matrix operations.

// Create a 10x15 matrix with random elements.
arma::mat m(10, 15, arma::fill::randu);

std::cout << "Size of m: " << m.n_rows << " x " << m.n_cols << "." << std::endl;

// Sum all elements in the matrix.
const double sumVal = arma::accu(m);
std::cout << "Sum of all elements: " << sumVal << "." << std::endl;

// Sum the elements in each column.
arma::rowvec sums = arma::sum(m, 0);
std::cout << "Sums in each column: " << sums;

// Add 1 to all elements.
m += 1;

// Subtract sums from each row.
m.each_row() -= sums;

// Print an individual element.
std::cout << "m(3, 4) is: " << m(3, 4) << "." << std::endl;

For more information on Armadillo, see the following resources:

Representing data in mlpack

Armadillo matrices, unlike numpy and some other toolkits, store data in a column-major format. This means that each column is located in contiguous memory; i.e., x(0, 0) is adjacent to x(1, 0) in memory.

This means that, for the vast majority of machine learning methods, it is faster to store observations as columns and dimensions as rows. This is counter to most standard machine learning texts! It also has some implications for linear algebra operations; for instance, computing the Gram matrix of a matrix X is typically expressed as X^T X, but when using column-major matrices, the expression must be X X^T.

In general, the following Armadillo types are commonly used inside mlpack:

  • arma::mat: datasets and general-purpose matrices
  • arma::Row<size_t>: integer response data, e.g., labels for classification datasets
  • arma::rowvec: floating-point response data, e.g., responses for regression datasets
  • arma::vec: general-purpose column vectors
  • arma::sp_mat, arma::fmat: alternate types for representing data; see Alternate matrix types

Loading data

mlpack provides two simple functions for loading and saving data matrices in a column-major form:

  • data::Load(filename, matrix, fatal=false, transpose=true, type=FileType::AutoDetect) (full documentation)
  • data::Save(filename, matrix, fatal=false, transpose=true, type=FileType::AutoDetect) (full documentation)

As an example, consider the following CSV file:

$ cat data.csv
3,3,3,3,0
3,4,4,3,0
3,4,4,3,0
3,3,4,3,0
3,6,4,3,0
2,4,4,3,0
2,4,4,1,0
3,3,3,2,0
3,4,4,2,0
3,4,4,2,0
3,3,4,2,0
3,6,4,2,0
2,4,4,2,0

The following program will load the data, print information about it, and save a modified dataset to disk.

// Load data from `data.csv` into `m`.  Throw an exception on failure (i.e. set
// `fatal` to `true`).
arma::mat m;
mlpack::data::Load("data.csv", m, true);

// Since mlpack uses column-major data,
//
//  - each column corresponds to a data point!
//  - each row corresponds to a dimension!
//
std::cout << "The matrix in 'data.csv' has: " << std::endl;
std::cout << " - " << m.n_cols << " points." << std::endl;
std::cout << " - " << m.n_rows << " dimensions." << std::endl;

std::cout << "The second point in the dataset: " << std::endl;
std::cout << m.col(1).t();

// Now modify the matrix and save to a different format (space-separated
// values).
m += 3;
mlpack::data::Save("data-mod.txt", m);

Although Armadillo does provide a .load() and .save() member function for matrices, the data::Load() and data::Save() functions offer additional flexibility, and ensure that data is saved and loaded in a column-major format.

Loading and using categorical data

Some mlpack techniques support mixed categorical data, e.g., data where some dimensions take only categorical values (e.g. 0, 1, 2, etc.). String data and other non-numerical data can be represented as categorical values, and mlpack has support to load mixed categorical data:

  • The data::DatasetInfo auxiliary class stores information about whether each dimension is numeric or categorical. (full documentation)
  • data::Load(filename, matrix, info, fatal=false, transpose=true) (full documentation)

For example, consider the following CSV file that contains strings:

$ cat mixed_string_data.csv
3,"hello",3,"f",0
3,"goodbye",4,"f",0
3,"goodbye",4,"e",0
3,"hello",4,"d",0
3,"hello",4,"d",0
2,"hello",4,"d",0
2,"hello",4,"d",0
3,"goodbye",3,"f",0
3,"goodbye",4,"f",0
3,"hello",4,"f",0
3,"hello",4,"c",0
3,"hello",4,"f",0
2,"hello",4,"c",0

The following program will load the data file, print information about categorical dimensions, and prepare the data for use with an mlpack algorithm that supports mixed categorical data.

// Load data from `mixed_string_data.csv` into `m`.  Throw an exception on
// failure (i.e. set `fatal` to `true`).  This populates the `info` object.
arma::mat m;
mlpack::data::DatasetInfo info;
mlpack::data::Load("mixed_string_data.csv", m, info, true);

// Print information about the data.
std::cout << "The matrix in 'mixed_string_data.csv' has: " << std::endl;
std::cout << " - " << m.n_cols << " points." << std::endl;
std::cout << " - " << info.Dimensionality() << " dimensions." << std::endl;

// Print which dimensions are categorical.
for (size_t d = 0; d < info.Dimensionality(); ++d)
{
  if (info.Type(d) == mlpack::data::Datatype::categorical)
  {
    std::cout << " - Dimension " << d << " is categorical with "
        << info.NumMappings(d) << " distinct categories." << std::endl;
  }
}

// Modify the third point to be 4,"wonderful",1,"c",0.
// Note that we manually map the string values; MapString() returns the category
// for a given value.
m(0, 2) = 4;
m(1, 2) = info.MapString<double>("wonderful", 1); // Create new third category.
m(2, 2) = 1;
m(3, 2) = info.MapString<double>("c", 1);
m(4, 2) = 0;

// `m` can now be used with any mlpack algorithm that supports categorical data.

Not every mlpack method supports categorical data. Below are the list of methods that do have categorical data support:

Alternate matrix types

mlpack's documentation focuses on the use of the arma::mat, arma::vec, and arma::rowvec types, with an underlying double numeric type, e.g., 64-bit floating-point. But, many mlpack algorithms and support utilities have support for alternate matrix types and element types:

  • Many methods, such as LogisticRegression, allow specifying the matrix types as a template parameter.

  • Some methods, such as DecisionTree, accept different matrix types to Train(), Classify(), or Predict(), without needing to specify an explicit template parameter.

  • In general, any matrix type that supports the Armadillo API can be used; this includes:

    • Single-precision floating point matrices (arma::fmat, arma::frowvec, arma::fvec)
    • Sparse matrices (arma::sp_mat, arma::sp_fmat)
    • GPU matrices via Bandicoot (coot::mat, coot::fmat) --- (note: support is under development and still experimental)

A simple example of using single-precision floating point data to train an AdaBoost model is below.

// 1000 random points in 10 dimensions, using 32-bit precision (float).
arma::fmat dataset(10, 1000, arma::fill::randu);
// Random labels for each point, totaling 5 classes.
arma::Row<size_t> labels =
    arma::randi<arma::Row<size_t>>(1000, arma::distr_param(0, 4));

// Train in the constructor, using floating-point data.
// The weak learner type is now a floating-point Perceptron.
using PerceptronType = mlpack::Perceptron<
    mlpack::SimpleWeightUpdate,
    mlpack::ZeroInitialization,
    arma::fmat>;
mlpack::AdaBoost<PerceptronType, arma::fmat> ab(dataset, labels, 5);

// Create test data (500 points).
arma::fmat testDataset(10, 500, arma::fill::randu);
arma::Row<size_t> predictions;
ab.Classify(testDataset, predictions);
// Now `predictions` holds predictions for the test dataset.

// Print some information about the test predictions.
std::cout << arma::accu(predictions == 3) << " test points classified as class "
    << "3." << std::endl;

A simple example of using sparse 32-bit floating point data to train a LogisticRegression model is below.

// Create random, sparse 100-dimensional data.
arma::sp_fmat dataset;
dataset.sprandu(100, 5000, 0.3);
arma::Row<size_t> labels =
    arma::randi<arma::Row<size_t>>(5000, arma::distr_param(0, 1));

// Train with L2 regularization penalty parameter of 0.1.
mlpack::LogisticRegression<arma::sp_fmat> lr(dataset, labels, 0.1);

// Now classify a test point.
arma::sp_fvec point;
point.sprandu(100, 1, 0.3);

size_t prediction;
arma::fvec probabilitiesVec;
lr.Classify(point, prediction, probabilitiesVec);

std::cout << "Prediction for random test point: " << prediction << "."
    << std::endl;
std::cout << "Class probabilities for random test point: "
    << probabilitiesVec.t();

Adapting from other toolkits (Eigen, etc.)

In general, C++ linear algebra toolkits store data in a column-major representation, and transitioning between toolkits is a matter of getting access to the underlying memory.

Copy an Eigen matrix into an Armadillo matrix.

// Note: this will only work if the Eigen matrix is stored in column-major
// order.  See https://eigen.tuxfamily.org/dox/group__TopicStorageOrders.html
// for more details.
Eigen::MatrixXd m;
const size_t rows = 10;
const size_t cols = 20;
m.setRandom(rows, cols); // 10x20 random matrix.

// Copy into an Armadillo matrix.
arma::mat mCopy(&m(0, 0), rows, cols);

Copy an XTensor matrix into an Armadillo matrix.

// Note: this will only work correctly if the layout_type of the XTensor matrix
// is column-major (i.e. xt::layout_type::column_major).  See
// https://xtensor.readthedocs.io/en/latest/container.html for more details.
const size_t rows = 10;
const size_t cols = 20;

// Create a random 10 x 20 matrix with normally distributed values.
// Note that we must ensure that the matrix is laid out in column-major form.
xt::xarray<double, xt::layout_type::column_major> m =
    xt::random::randn<double>({ rows, cols });

// Copy into an Armadillo matrix.
arma::mat mCopy(m.data(), rows, cols);

Make an Armadillo matrix that is an alias of an Eigen matrix. Note that changes to the Eigen matrix will be reflected in the Armadillo matrix, and vice versa.

If the Eigen matrix is deallocated, the Armadillo matrix will become invalid. Be careful! More details here.

// Note: this will only work if the Eigen matrix is stored in column-major
// order.  See https://eigen.tuxfamily.org/dox/group__TopicStorageOrders.html
// for more details.
Eigen::MatrixXd m;
const size_t rows = 10;
const size_t cols = 20;
m.setRandom(rows, cols); // 10x20 random matrix.

// Make an Armadillo matrix that is an alias of the Eigen matrix.  This avoids
// the copy, but is potentially dangerous: be careful that the Eigen matrix is
// not deleted while the Armadillo matrix is in use!
//
// See https://arma.sourceforge.net/docs.html#adv_constructors_mat
arma::mat mAlias(&m(0, 0), rows, cols, false, true);

Copy an Armadillo matrix to an Eigen matrix.

const size_t rows = 10;
const size_t cols = 20;
arma::mat m(10, 20, arma::fill::randu);

// Construct the Eigen matrix by using a map of the Armadillo memory.
Eigen::MatrixXd eigenM(Eigen::Map<Eigen::MatrixXd>(m.memptr(), rows, cols));