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Collaborative Filtering Tutorial

Collaborative filtering is an increasingly popular approach for recommender systems. A typical formulation of the problem is as follows: there are n users and m items, and each user has rated some of the items. We want to provide each user with a recommendation for an item they have not rated yet, which they are likely to rate highly. In another formulation, we may want to predict a user's rating of an item. This type of problem has been considered extensively, especially in the context of the Netflix prize. The winning approach for the Netflix prize was a collaborative filtering approach which utilized matrix decomposition. More information on their approach can be found in the following paper:

@article{koren2009matrix,
  title={Matrix factorization techniques for recommender systems},
  author={Koren, Yehuda and Bell, Robert and Volinsky, Chris},
  journal={Computer},
  number={8},
  pages={30--37},
  year={2009},
  publisher={IEEE}
}

The key to this approach is that the data is represented as an incomplete matrix V with size n x m, where V_ij represents user i's rating of item j, if that rating exists. The task, then, is to complete the entries of the matrix.

In the matrix factorization framework, the matrix V is assumed to be low-rank and decomposed into components as V ~ WH according to some heuristic.

In order to solve problems of this form, mlpack provides both an easy-to-use binding (detailed here as a command-line program), and a simple yet flexible C++ API that allows the implementation of new collaborative filtering techniques.

The mlpack_cf program

mlpack provides a command-line program, mlpack_cf, which is used to perform collaborative filtering on a given dataset. It can provide neighborhood-based recommendations for users. The algorithm used for matrix factorization is configurable, and the parameters of each algorithm are also configurable. Note that mlpack also provides the cf() function in other languages too; however, this tutorial focuses on the command-line program mlpack_cr. It is easy to adapt each example to each other language, though.

The following examples detail usage of the mlpack_cf program. Note that you can get documentation on all the possible parameters by typing:

$ mlpack_cf --help

Input format for mlpack_cf

The input file for the mlpack_cf program is specified with the --training_file or -t option. This file is a coordinate-format sparse matrix, similar to the Matrix Market (MM) format. The first coordinate is the user id; the second coordinate is the item id; and the third coordinate is the rating. So, for instance, a dataset with 3 users and 2 items, and ratings between 1 and 5, might look like the following:

$ cat dataset.csv
0, 1, 4
1, 0, 5
1, 1, 1
2, 0, 2

This dataset has four ratings: user 0 has rated item 1 with a rating of 4; user 1 has rated item 0 with a rating of 5; user 1 has rated item 1 with a rating of 1; and user 2 has rated item 0 with a rating of 2. Note that the user and item indices start from 0, and the identifiers must be numeric indices, and not names.

The type does not necessarily need to be a csv; it can be any supported storage format, assuming that it is a coordinate-format file in the format specified above. For more information on mlpack file formats, see the documentation for mlpack::data::Load().

mlpack_cf with default parameters

In this example, we have a dataset from MovieLens, and we want to use mlpack_cf with the default parameters, which will provide 5 recommendations for each user, and we wish to save the results in the file recommendations.csv. Assuming that our dataset is in the file MovieLens-100k.csv and it is in the correct format, we may use the mlpack_cf executable as below:

$ mlpack_cf -t MovieLens-100k.csv -v -o recommendations.csv

The -v option provides verbose output, and may be omitted if desired. Now, for each user, we have recommendations in recommendations.csv:

$ head recommendations.csv
317,422,482,356,495
116,120,180,6,327
312,49,116,99,236
312,116,99,236,285
55,190,317,194,63
171,209,180,175,95
208,0,94,87,57
99,97,0,203,172
257,99,180,287,0
171,203,172,209,88

So, for user 0, the top 5 recommended items that user 0 has not rated are items 317, 422, 482, 356, and 495. For user 5, the recommendations are on the sixth line: 171, 209, 180, 175, 95.

The mlpack_cf program can be built into a larger recommendation framework, with a preprocessing step that can turn user information and item information into numeric IDs, and a postprocessing step that can map these numeric IDs back to the original information.

Saving mlpack_cf models

The mlpack_cf program is able to save a particular model for later loading. Saving a model can be done with the --output_model_file or -M option. The example below builds a CF model on the MovieLens-100k.csv dataset, and then saves the model to the file cf-model.xml for later usage.

$ mlpack_cf -t MovieLens-100k.csv -M cf-model.xml -v

The models can also be saved as .bin or .txt; the .xml format provides a human-inspectable format (though the models tend to be quite complex and may be difficult to read). These models can then be re-used to provide specific recommendations for certain users, or other tasks.

Loading mlpack_cf models

Instead of training a model, the mlpack_cf model can also load a model to provide recommendations, using the --input_model_file or -m option. For instance, the example below will load the model from cf-model.xml and then generate 3 recommendations for each user in the dataset, saving the results to recommendations.csv.

$ mlpack_cf -m cf-model.xml -v -o recommendations.csv

Specifying rank of mlpack_cf decomposition

By default, the matrix factorizations in the mlpack_cf program decompose the data matrix into two matrices W and H with rank two. Often, this default parameter is not correct, and it makes sense to use a higher-rank decomposition. The rank can be specified with the --rank or -R parameter:

$ mlpack_cf -t MovieLens-100k.csv -R 10 -v

In the example above, the data matrix will be decomposed into two matrices of rank 10. In general, higher-rank decompositions will take longer, but will give more accurate predictions.

mlpack_cf with single-user recommendation

In the previous two examples, the output file recommendations.csv contains one line for each user in the input dataset. But often, recommendations may only be desired for a few users. In that case, we can assemble a file of query users, with one user per line:

$ cat query.csv
0
17
31

Now, if we run the mlpack_cf executable with this query file, we will obtain recommendations for users 0, 17, and 31:

$ mlpack_cf -i MovieLens-100k.csv -R 10 -q query.csv -o recommendations.csv
$ cat recommendations.csv
474,356,317,432,473
510,172,204,483,182
0,120,236,257,126

mlpack_cf with non-default factorizer

The --algorithm (or -a) parameter controls the factorizer that is used. Several options are available:

  • NMF: non-negative matrix factorization; see AMF
  • SVDBatch: SVD batch factorization
  • SVDIncompleteIncremental: incomplete incremental SVD
  • SVDCompleteIncremental: complete incremental SVD
  • RegSVD: regularized SVD; see RegularizedSVD

The default factorizer is NMF. The example below uses the RegSVD factorizer:

$ mlpack_cf -i MovieLens-100k.csv -R 10 -q query.csv -a RegSVD -o recommendations.csv

mlpack_cf with non-default neighborhood size

The mlpack_cf program produces recommendations using a neighborhood: similar users in the query user's neighborhood will be averaged to produce predictions. The size of this neighborhood is controlled with the --neighborhood (or -n) option. An example using a neighborhood with 10 similar users is below:

$ mlpack_cf -i MovieLens-100k.csv -R 10 -q query.csv -a RegSVD -n 10

The CF class

The CF class in mlpack offers a simple, flexible API for performing collaborative filtering for recommender systems within C++ applications. In the constructor, the CF class takes a coordinate-list dataset and decomposes the matrix according to the specified FactorizerType template parameter.

Then, the GetRecommendations() function may be called to obtain recommendations for certain users (or all users), and the W() and H() matrices may be accessed to perform other computations.

The data which the CF constructor takes should be an Armadillo matrix (arma::mat) with three rows. The first row corresponds to users; the second row corresponds to items; the third column corresponds to the rating. This is a coordinate list format, like the format the mlpack_cf executable takes. The data::Load() function can be used to load data.

The following examples detail a few ways that the CF class can be used.

CF with default parameters

This example constructs the CF object with default parameters and obtains recommendations for each user, storing the output in the recommendations matrix.

#include <mlpack.hpp>

using namespace mlpack;

// The coordinate list of ratings that we have.
extern arma::mat data;
// The size of the neighborhood to use to get recommendations.
extern size_t neighborhood;
// The rank of the decomposition.
extern size_t rank;

// Build the CF object and perform the decomposition.
// The constructor takes a default-constructed factorizer, which, by default,
// is of type NMFALSFactorizer.
CF cf(data, NMFPolicy(), neighborhood, rank);

// Store the results in this object.
arma::Mat<size_t> recommendations;

// Get 5 recommendations for all users.
cf.GetRecommendations(5, recommendations);

CF with other factorizers

mlpack provides a number of existing factorizers which can be used in place of the default NMFALSFactorizer (which is non-negative matrix factorization with alternating least squares update rules). These include:

  • BatchSVDPolicy
  • SVDCompletePolicy
  • SVDIncompletePolicy
  • NMFPolicy
  • RegSVDPolicy
  • QuicSVDPolicy
  • BiasSVDPolicy
  • SVDPlusPlusPolicy
  • RandomizedSVDPolicy
  • BlockKrylovSVDPolicy

The AMF class has many other possibilities than those listed here; it is a framework for alternating matrix factorization techniques. See the AMF class documentation or tutorial on AMF for more information.

The use of another factorizer is straightforward; the example from the previous section is adapted below to use RegularizedSVD:

#include <mlpack.hpp>

using namespace mlpack;

// The coordinate list of ratings that we have.
extern arma::mat data;
// The size of the neighborhood to use to get recommendations.
extern size_t neighborhood;
// The rank of the decomposition.
extern size_t rank;

// Build the CF object and perform the decomposition.
CFType cf(data, RegSVDPolicy(), neighborhood, rank);

// Store the results in this object.
arma::Mat<size_t> recommendations;

// Get 5 recommendations for all users.
cf.GetRecommendations(5, recommendations);

Predicting individual user/item ratings

The Predict() method can be used to predict the rating of an item by a certain user, using the same neighborhood-based approach as the GetRecommendations() function or the mlpack_cf executable. Below is an example of the use of that function.

The example below will obtain the predicted rating for item 50 by user 12.

#include <mlpack.hpp>

using namespace mlpack;

// The coordinate list of ratings that we have.
extern arma::mat data;
// The size of the neighborhood to use to get recommendations.
extern size_t neighborhood;
// The rank of the decomposition.
extern size_t rank;

// Build the CF object and perform the decomposition.
// The constructor takes a default-constructed factorizer, which, by default,
// is of type NMFALSFactorizer.
CF cf(data, NMFPolicy(), neighborhood, rank);

const double prediction = cf.Predict(12, 50); // User 12, item 50.

Other operations with the W and H matrices

Sometimes, the raw decomposed W and H matrices can be useful. The example below obtains these matrices, and multiplies them against each other to obtain a reconstructed data matrix with no missing values.

#include <mlpack.hpp>

using namespace mlpack;

// The coordinate list of ratings that we have.
extern arma::mat data;
// The size of the neighborhood to use to get recommendations.
extern size_t neighborhood;
// The rank of the decomposition.
extern size_t rank;

// Build the CF object and perform the decomposition.
// The constructor takes a default-constructed factorizer, which, by default,
// is of type NMFALSFactorizer.
CF cf(data, NMFPolicy(), neighborhood, rank);

// References to W and H matrices.
const arma::mat& W = cf.W();
const arma::mat& H = cf.H();

// Multiply the matrices together.
arma::mat reconstructed = W * H;

Template parameters for the CF class

The CF class takes the FactorizerType as a template parameter to some of its constructors and to the Train() function. The FactorizerType class defines the algorithm used for matrix factorization. There are a number of existing factorizers that can be used in mlpack; these were detailed in the 'other factorizers' example of the previous section.

The FactorizerType class must implement one of the two following methods:

  • Apply(arma::mat& data, const size_t rank, arma::mat& W, arma::mat& H);
  • Apply(arma::sp_mat& data, const size_t rank, arma::mat& W, arma::mat& H);

The difference between these two methods is whether arma::mat or arma::sp_mat is used as input. If arma::mat is used, then the data matrix is a coordinate list with three columns, as in the constructor to the CF class. If arma::sp_mat is used, then a sparse matrix is passed with the number of rows equal to the number of items and the number of columns equal to the number of users, and each nonzero element in the matrix corresponds to a non-missing rating.

The method that the factorizer implements is specified via the FactorizerTraits class, which is a template metaprogramming traits class:

template<typename FactorizerType>
struct FactorizerTraits
{
  /**
   * If true, then the passed data matrix is used for factorizer.Apply().
   * Otherwise, it is modified into a form suitable for factorization.
   */
  static const bool UsesCoordinateList = false;
};

If FactorizerTraits<MyFactorizer>::UsesCoordinateList is true, then CF will try to call Apply() with an arma::mat object. Otherwise, CF will try to call Apply() with an arma::sp_mat object. Specifying the value of UsesCoordinateList is straightforward; provide this specialization of the FactorizerTraits class:

template<>
struct FactorizerTraits<MyFactorizer>
{
  static const bool UsesCoordinateList = true; // Set your value here.
};

The Apply() function also takes a reference to the matrices W and H. When the Apply() function returns, the input data matrix should be decomposed into these two matrices. W should have number of rows equal to the number of items and number of columns equal to the rank parameter, and H should have number of rows equal to the rank parameter, and number of columns equal to the number of users.

The AMF class can be used as a base for factorizers that alternate between updating W and updating H. A useful reference is the AMF tutorial.

Further documentation

Further documentation for the CF class may be found in the comments in the source code of the files in src/mlpack/methods/cf/. In addition, more information on the AMF class of factorizers may be found in the sources for AMF, in src/mlpack/methods/amf/.