mlpack

NeighborSearch tutorial (k-nearest-neighbors)

Nearest-neighbors search is a common machine learning task. In this setting, we have a query and a reference dataset. For each point in the query dataset, we wish to know the k points in the reference dataset which are closest to the given query point.

Alternately, if the query and reference datasets are the same, the problem can be stated more simply: for each point in the dataset, we wish to know the k nearest points to that point.

mlpack provides:

🔗 Command-line mlpack_knn

The simplest way to perform nearest-neighbors search in mlpack is to use the mlpack_knn executable. (Note that mlpack also provides bindings to other languages, so, e.g., the knn() function is available in Python and Julia and has the same options. So, any example here can be readily adapted to another language that mlpack provides bindings for.)

The mlpack_knn program will perform nearest-neighbors search and place the resultant neighbors into one file and the resultant distances into another. The output files are organized such that the first row corresponds to the nearest neighbors of the first query point, with the first column corresponding to the nearest neighbor, and so forth.

Below are several examples of simple usage (and the resultant output). The -v option is used so that output is given. Further documentation on each individual option can be found by typing

$ mlpack_knn --help

🔗 One dataset, 5 nearest neighbors

$ mlpack_knn -r dataset.csv -n neighbors_out.csv -d distances_out.csv -k 5 -v
[INFO ] Loading 'dataset.csv' as CSV data.  Size is 3 x 1000.
[INFO ] Loaded reference data from 'dataset.csv' (3 x 1000).
[INFO ] Building reference tree...
[INFO ] Tree built.
[INFO ] Searching for 5 nearest neighbors with dual-tree kd-tree search...
[INFO ] 18412 node combinations were scored.
[INFO ] 54543 base cases were calculated.
[INFO ] Search complete.
[INFO ] Saving CSV data to 'neighbors_out.csv'.
[INFO ] Saving CSV data to 'distances_out.csv'.
[INFO ]
[INFO ] Execution parameters:
[INFO ]   distances_file: distances_out.csv
[INFO ]   help: false
[INFO ]   info: ""
[INFO ]   input_model_file: ""
[INFO ]   k: 5
[INFO ]   leaf_size: 20
[INFO ]   naive: false
[INFO ]   neighbors_file: neighbors_out.csv
[INFO ]   output_model_file: ""
[INFO ]   query_file: ""
[INFO ]   random_basis: false
[INFO ]   reference_file: dataset.csv
[INFO ]   seed: 0
[INFO ]   single_mode: false
[INFO ]   tree_type: kd
[INFO ]   verbose: true
[INFO ]   version: false
[INFO ]
[INFO ] Program timers:
[INFO ]   computing_neighbors: 0.108968s
[INFO ]   loading_data: 0.006495s
[INFO ]   saving_data: 0.003843s
[INFO ]   total_time: 0.126036s
[INFO ]   tree_building: 0.003442s

Convenient program timers are given for different parts of the calculation at the bottom of the output, as well as the parameters the simulation was run with. Now, if we look at the output files:

$ head neighbors_out.csv
862,344,224,43,885
703,499,805,639,450
867,472,972,380,601
397,319,277,443,323
840,827,865,38,438
732,876,751,492,616
563,222,569,985,940
361,97,928,437,79
547,695,419,961,716
982,113,689,843,634

$ head distances_out.csv
5.986076164057e-02,7.664920518084e-02,1.116050961847e-01,1.155595474371e-01,1.169810085522e-01
7.532635022982e-02,1.012564715841e-01,1.127846944644e-01,1.209584396720e-01,1.216543647014e-01
7.659571546879e-02,1.014588981948e-01,1.025114621511e-01,1.128082429187e-01,1.131659758673e-01
2.079405647909e-02,4.710724516732e-02,7.597622408419e-02,9.171977778898e-02,1.037033340864e-01
7.082206779700e-02,9.002355499742e-02,1.044181406406e-01,1.093149568834e-01,1.139700558608e-01
5.688056488896e-02,9.478072514474e-02,1.085637706630e-01,1.114177921451e-01,1.139370265105e-01
7.882260880455e-02,9.454474078041e-02,9.724494179950e-02,1.023829575445e-01,1.066927013814e-01
7.005321598247e-02,9.131417221561e-02,9.498248889074e-02,9.897964162308e-02,1.121202216165e-01
5.295654132754e-02,5.509877761894e-02,8.108227366619e-02,9.785461174861e-02,1.043968140367e-01
3.992859920333e-02,4.471418646159e-02,7.346053904990e-02,9.181982339584e-02,9.843075910782e-02

So, the nearest neighbor to point 0 is point 862, with a distance of 5.986076164057e-02. The second nearest neighbor to point 0 is point 344, with a distance of 7.664920518084e-02. The third nearest neighbor to point 5 is point 751, with a distance of 1.085637706630e-01.

🔗 Query and reference dataset, 10 nearest neighbors

$ mlpack_knn -q query_dataset.csv -r reference_dataset.csv \
> -n neighbors_out.csv -d distances_out.csv -k 10 -v
[INFO ] Loading 'reference_dataset.csv' as CSV data.  Size is 3 x 1000.
[INFO ] Loaded reference data from 'reference_dataset.csv' (3 x 1000).
[INFO ] Building reference tree...
[INFO ] Tree built.
[INFO ] Loading 'query_dataset.csv' as CSV data.  Size is 3 x 50.
[INFO ] Loaded query data from 'query_dataset.csv' (3x50).
[INFO ] Searching for 10 nearest neighbors with dual-tree kd-tree search...
[INFO ] Building query tree...
[INFO ] Tree built.
[INFO ] Search complete.
[INFO ] Saving CSV data to 'neighbors_out.csv'.
[INFO ] Saving CSV data to 'distances_out.csv'.
[INFO ]
[INFO ] Execution parameters:
[INFO ]   distances_file: distances_out.csv
[INFO ]   help: false
[INFO ]   info: ""
[INFO ]   input_model_file: ""
[INFO ]   k: 10
[INFO ]   leaf_size: 20
[INFO ]   naive: false
[INFO ]   neighbors_file: neighbors_out.csv
[INFO ]   output_model_file: ""
[INFO ]   query_file: query_dataset.csv
[INFO ]   random_basis: false
[INFO ]   reference_file: reference_dataset.csv
[INFO ]   seed: 0
[INFO ]   single_mode: false
[INFO ]   tree_type: kd
[INFO ]   verbose: true
[INFO ]   version: false
[INFO ]
[INFO ] Program timers:
[INFO ]   computing_neighbors: 0.022589s
[INFO ]   loading_data: 0.003572s
[INFO ]   saving_data: 0.000755s
[INFO ]   total_time: 0.032197s
[INFO ]   tree_building: 0.002590s

🔗 One dataset, 3 nearest neighbors, leaf size of 15 points

$ mlpack_knn -r dataset.csv -n neighbors_out.csv -d distances_out.csv -k 3 -l 15 -v
[INFO ] Loading 'dataset.csv' as CSV data.  Size is 3 x 1000.
[INFO ] Loaded reference data from 'dataset.csv' (3 x 1000).
[INFO ] Building reference tree...
[INFO ] Tree built.
[INFO ] Searching for 3 nearest neighbors with dual-tree kd-tree search...
[INFO ] 19692 node combinations were scored.
[INFO ] 36263 base cases were calculated.
[INFO ] Search complete.
[INFO ] Saving CSV data to 'neighbors_out.csv'.
[INFO ] Saving CSV data to 'distances_out.csv'.
[INFO ]
[INFO ] Execution parameters:
[INFO ]   distances_file: distances_out.csv
[INFO ]   help: false
[INFO ]   info: ""
[INFO ]   input_model_file: ""
[INFO ]   k: 3
[INFO ]   leaf_size: 15
[INFO ]   naive: false
[INFO ]   neighbors_file: neighbors_out.csv
[INFO ]   output_model_file: ""
[INFO ]   query_file: ""
[INFO ]   random_basis: false
[INFO ]   reference_file: dataset.csv
[INFO ]   seed: 0
[INFO ]   single_mode: false
[INFO ]   tree_type: kd
[INFO ]   verbose: true
[INFO ]   version: false
[INFO ]
[INFO ] Program timers:
[INFO ]   computing_neighbors: 0.059020s
[INFO ]   loading_data: 0.002791s
[INFO ]   saving_data: 0.002369s
[INFO ]   total_time: 0.069277s
[INFO ]   tree_building: 0.002713s

Further documentation on options should be found by using the --help option.

🔗 The KNN class

The KNN class is, specifically, a typedef of the more extensible NeighborSearch class, querying for nearest neighbors using the Euclidean distance.

typedef NeighborSearch<NearestNeighborSort, EuclideanDistance> KNN;

Using the KNN class is particularly simple; first, the object must be constructed and given a dataset. Then, the method is run, and two matrices are returned: one which holds the indices of the nearest neighbors, and one which holds the distances of the nearest neighbors. These are of the same structure as the output --neighbors_file and --distances_file for the command-line program (see above). A handful of examples of simple usage of the KNN class are given below.

🔗 5 nearest neighbors on a single dataset

#include <mlpack.hpp>

using namespace mlpack;

// Our dataset matrix, which is column-major.
extern arma::mat data;

KNN a(data);

// The matrices we will store output in.
arma::Mat<size_t> resultingNeighbors;
arma::mat resultingDistances;

a.Search(5, resultingNeighbors, resultingDistances);

The output of the search is stored in resultingNeighbors and resultingDistances.

🔗 10 nearest neighbors on a query and reference dataset

#include <mlpack.hpp>

using namespace mlpack;

// Our dataset matrices, which are column-major.
extern arma::mat queryData, referenceData;

KNN a(referenceData);

// The matrices we will store output in.
arma::Mat<size_t> resultingNeighbors;
arma::mat resultingDistances;

a.Search(queryData, 10, resultingNeighbors, resultingDistances);

🔗 Naive (exhaustive) search for 6 nearest neighbors on one dataset

This example uses the O(n^2) naive search (not the tree-based search).

#include <mlpack.hpp>

using namespace mlpack;

// Our dataset matrix, which is column-major.
extern arma::mat dataset;

KNN a(dataset, true);

// The matrices we will store output in.
arma::Mat<size_t> resultingNeighbors;
arma::mat resultingDistances;

a.Search(6, resultingNeighbors, resultingDistances);

Needless to say, naive search can be very slow…

🔗 The extensible NeighborSearch class

The NeighborSearch class is very extensible, having the following template arguments:

template<
  typename SortPolicy = NearestNeighborSort,
  typename MetricType = EuclideanDistance,
  typename MatType = arma::mat,
  template<typename TreeMetricType,
           typename TreeStatType,
           typename TreeMatType> class TreeType = KDTree,
  template<typename RuleType> class TraversalType =
      TreeType<MetricType, NeighborSearchStat<SortPolicy>,
               MatType>::template DualTreeTraverser>
>
class NeighborSearch;

By choosing different components for each of these template classes, a very arbitrary neighbor searching object can be constructed. Note that each of these template parameters have defaults, so it is not necessary to specify each one.

🔗 SortPolicy policy class

The SortPolicy template parameter allows specification of how the NeighborSearch object will decide which points are to be searched for. The NearestNeighborSort class is a well-documented example. A custom SortPolicy class must implement the same methods which NearestNeighborSort does:

static size_t SortDistance(const arma::vec& list, double newDistance);

static bool IsBetter(const double value, const double ref);

template<typename TreeType>
static double BestNodeToNodeDistance(const TreeType* queryNode,
                                     const TreeType* referenceNode);

template<typename TreeType>
static double BestPointToNodeDistance(const arma::vec& queryPoint,
                                      const TreeType* referenceNode);

static const double WorstDistance();

static const double BestDistance();

The FurthestNeighborSort class is another implementation, which is used to create the KFN typedef class, which finds the furthest neighbors, as opposed to the nearest neighbors.

🔗 MetricType policy class

The MetricType policy class allows the neighbor search to take place in any arbitrary metric space. The LMetric class is a good example implementation. A MetricType class must provide the following functions:

// Empty constructor is required.
MetricType();

// Compute the distance between two points.
template<typename VecType>
double Evaluate(const VecType& a, const VecType& b);

Internally, the NeighborSearch class keeps an instantiated MetricType class (which can be given in the constructor). This is useful for a metric like the Mahalanobis distance (MahalanobisDistance), which must store state (the covariance matrix). Therefore, you can write a non-static MetricType class and use it seamlessly with NeighborSearch.

For more information on the MetricType policy, see the documentation for MetricType.

🔗 MatType policy class

The MatType template parameter specifies the type of data matrix used. This type must implement the same operations as an Armadillo matrix, and so standard choices are arma::mat and arma::sp_mat.

🔗 TreeType policy class

The NeighborSearch class allows great extensibility in the selection of the type of tree used for search. This type must follow the typical mlpack TreeType policy, documented here.

Typical choices might include KDTree, BallTree, StandardCoverTree, RTree, or RStarTree. It is easily possible to make your own tree type for use with NeighborSearch; consult the TreeType documentation for more details.

An example of using the NeighborSearch class with a ball tree is given below.

// Construct a NeighborSearch object with ball bounds.
NeighborSearch<
    NearestNeighborSort,
    EuclideanDistance,
    arma::mat,
    BallTree
> neighborSearch(dataset);

🔗 TraverserType policy class

The last template parameter the NeighborSearch class offers is the TraverserType class. The TraverserType class holds the strategy used to traverse the trees in either single-tree or dual-tree search mode. By default, it is set to use the default traverser of the given TreeType (which is the member TreeType::DualTreeTraverser).

This class must implement the following two methods:

// Instantiate with a given RuleType.
TraverserType(RuleType& rule);

// Traverse with two trees.
void Traverse(TreeType& queryNode, TreeType& referenceNode);

The RuleType class provides the following functions for use in the traverser:

// Evaluate the base case between two points.
double BaseCase(const size_t queryIndex, const size_t referenceIndex);

// Score the two nodes to see if they can be pruned, returning DBL_MAX if so.
double Score(TreeType& queryNode, TreeType& referenceNode);

Note also that any traverser given must satisfy the definition of a pruning dual-tree traversal given in the paper “Tree-independent dual-tree algorithms”.

🔗 Further documentation

For further documentation on the NeighborSearch class, consult the comments in the source code, found in mlpack/methods/neighbor_search/.