man nstutorial (3): NeighborSearch tutorial (k-nearest-neighbors)

Introduction

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:

a simple command-line executable to run nearest-neighbors search (and furthest-neighbors search)
a simple C++ interface to perform nearest-neighbors search (and furthest-neighbors search)
a generic, extensible, and powerful C++ class (NeighborSearch) for complex usage

A list of all the sections this tutorial contains.

•: Introduction
•: Table of Contents
•: Command-Line 'allknn'
•: One dataset, 5 nearest neighbors
•: Query and reference dataset, 10 nearest neighbors
•: One dataset, 3 nearest neighbors, leaf size of 15 points

•: The 'AllkNN' class
•: 5 nearest neighbors on a single dataset
•: 10 nearest neighbors on a query and reference dataset
•: Naive (exhaustive) search for 6 nearest neighbors on one dataset

•: The extensible 'NeighborSearch' class
•: SortPolicy policy class
•: MetricType policy class
•: TreeType policy class

Further documentation

Command-Line 'allknn'

The simplest way to perform nearest-neighbors search in mlpack is to use the allknn executable. This 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

$ allknn --help

One dataset, 5 nearest neighbors

$ allknn -r dataset.csv -n neighbors_out.csv -d distances_out.csv -k 5 -v
[INFO ] Loading 'dataset.csv' as CSV data.
[INFO ] Loaded reference data from 'dataset.csv'.
[INFO ] Building reference tree...
[INFO ] Trees built.
[INFO ] Computing 5 nearest neighbors...
[INFO ] Neighbors computed.
[INFO ] Re-mapping indices...
[INFO ] Saving CSV data to 'distances_out.csv'.
[INFO ] Saving CSV data to 'neighbors_out.csv'.
[INFO ]
[INFO ] Execution parameters:
[INFO ]   distances_file: distances_out.csv
[INFO ]   help: false
[INFO ]   info: ""
[INFO ]   k: 5
[INFO ]   leaf_size: 20
[INFO ]   naive: false
[INFO ]   neighbors_file: neighbors_out.csv
[INFO ]   query_file: ""
[INFO ]   reference_file: dataset.csv
[INFO ]   single_mode: false
[INFO ]   verbose: true
[INFO ]
[INFO ] Program timers:
[INFO ]   computing_neighbors: 0.152495s
[INFO ]   total_time: 0.201274s
[INFO ]   tree_building: 0.005050s

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
14,5,13,16,27
90,79,80,15,10
39,84,10,123,1
81,43,109,12,37
15,1,79,90,10
0,14,16,13,27
90,79,11,1,15
41,45,12,37,49
11,81,13,6,15
41,7,45,49,47
$ head distances_out.csv
7.09614421e-04,2.05940173e-03,4.05346068e-03,4.66175278e-03,1.09757665e-02
8.92190948e-04,1.69442242e-03,2.82750475e-03,4.06590850e-03,7.54169243e-03
5.91539406e-03,6.83482612e-03,8.02877800e-03,9.04907425e-03,1.61458442e-02
7.15652913e-03,9.18228524e-03,1.00540941e-02,1.07541171e-02,1.28892864e-02
5.37535983e-03,9.05721409e-03,9.89017184e-03,1.01457735e-02,1.14021593e-02
2.05940173e-03,5.14437192e-03,9.97483954e-03,1.02463627e-02,1.44355783e-02
4.27355419e-03,6.36750547e-03,6.72478577e-03,8.77323532e-03,1.04530549e-02
1.99935847e-03,3.88240331e-03,4.19118273e-03,9.30693568e-03,1.21237481e-02
2.15454276e-03,8.18895210e-03,1.18360450e-02,1.25135454e-02,1.27783327e-02
8.43087996e-03,1.22946325e-02,1.60472209e-02,1.88661413e-02,1.89727686e-02

So, the nearest neighbor to point 0 is point 14, with a distance of 7.096144e-4. The second nearest neighbor to point 0 is point 5, with a distance of 2.059402e-3. The third nearest neighbor to point 5 is point 16, with a distance of 9.9748395e-3.

Query and reference dataset, 10 nearest neighbors

$ allknn -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.
[INFO ] Loaded reference data from 'reference_dataset.csv'.
[INFO ] Building reference tree...
[INFO ] Loading 'query_dataset.csv' as CSV data.
[INFO ] Query data loaded from 'query_dataset.csv'.
[INFO ] Building query tree...
[INFO ] Tree built.
[INFO ] Computing 10 nearest neighbors...
[INFO ] Neighbors computed.
[INFO ] Re-mapping indices...
[INFO ] Saving CSV data to 'distances_out.csv'.
[INFO ] Saving CSV data to 'neighbors_out.csv'.
[INFO ]
[INFO ] Execution parameters:
[INFO ]   distances_file: distances_out.csv
[INFO ]   help: false
[INFO ]   info: ""
[INFO ]   k: 10
[INFO ]   leaf_size: 20
[INFO ]   naive: false
[INFO ]   neighbors_file: neighbors_out.csv
[INFO ]   query_file: query_dataset.csv
[INFO ]   reference_file: reference_dataset.csv
[INFO ]   single_mode: false
[INFO ]   verbose: true
[INFO ]
[INFO ] Program timers:
[INFO ]   computing_neighbors: 0.000081s
[INFO ]   total_time: 0.062828s
[INFO ]   tree_building: 0.004949s

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

$ allknn -r dataset.csv -n neighbors_out.csv -d distances_out.csv -k 3 -l 15 -v
[INFO ] Loading 'dataset.csv' as CSV data.
[INFO ] Loaded reference data from 'dataset.csv'.
[INFO ] Building reference tree...
[INFO ] Trees built.
[INFO ] Computing 3 nearest neighbors...
[INFO ] Neighbors computed.
[INFO ] Re-mapping indices...
[INFO ] Saving CSV data to 'distances_out.csv'.
[INFO ] Saving CSV data to 'neighbors_out.csv'.
[INFO ]
[INFO ] Execution parameters:
[INFO ]   distances_file: distances_out.csv
[INFO ]   help: false
[INFO ]   info: ""
[INFO ]   k: 3
[INFO ]   leaf_size: 15
[INFO ]   naive: false
[INFO ]   neighbors_file: neighbors_out.csv
[INFO ]   query_file: ""
[INFO ]   reference_file: dataset.csv
[INFO ]   single_mode: false
[INFO ]   verbose: true
[INFO ]
[INFO ] Program timers:
[INFO ]   computing_neighbors: 0.105119s
[INFO ]   total_time: 0.145321s
[INFO ]   tree_building: 0.005690s

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

The 'AllkNN' class

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

typedef NeighborSearch<NearestNeighborSort, metric::EuclideanDistance>
    AllkNN;

Using the AllkNN 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 CLI interface (see above). A handful of examples of simple usage of the AllkNN class are given below.

5 nearest neighbors on a single dataset

#include <mlpack/methods/neighbor_search/neighbor_search.hpp>
using namespace mlpack::neighbor;
// Our dataset matrix, which is column-major.
extern arma::mat data;
AllkNN 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/methods/neighbor_search/neighbor_search.hpp>
using namespace mlpack::neighbor;
// Our dataset matrices, which are column-major.
extern arma::mat queryData, referenceData;
AllkNN a(referenceData, queryData);
// The matrices we will store output in.
arma::Mat<size_t> resultingNeighbors;
arma::mat resultingDistances;
a.Search(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/methods/neighbor_search/neighbor_search.hpp>
using namespace mlpack::neighbor;
// Our dataset matrix, which is column-major.
extern arma::mat dataset;
AllkNN 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 = mlpack::metric::EuclideanDistance,
  typename TreeType = mlpack::tree::BinarySpaceTree<bound::HRectBound<2>,
                                                    QueryStat<SortPolicy> >
>
class NeighborSearch;

By choosing different components for each of these template classes, a very arbitrary neighbor searching object can be constructed.

SortPolicy policy class

The SortPolicy template parameter allows specification of how the NeighborSearch object will decide which points are to be searched for. The mlpack::neighbor::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 mlpack::neighbor::FurthestNeighborSort class is another implementation, which is used to create the 'AllkFN' 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 mlpack::metric::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 (mlpack::metric::MahalanobisDistance), which must store state (the covariance matrix). Therefore, you can write a non-static MetricType class and use it seamlessly with NeighborSearch.

TreeType policy class

The NeighborSearch class also allows a custom tree to be used. The standard MLPACK tree, mlpack::tree::BinarySpaceTree, is also highly extensible in its own right, and its documentation should be consulted for more information. Currently, the NeighborSearch tree requires a tree which only has left and right children, and no points in nodes (only in leaves), but this support is planned to be extended.

A simple usage of the TreeType policy could be to use a different type of bound with the tree. For instance, you could use a ball bound instead of a rectangular bound:

// Construct a NeighborSearch object with ball bounds.
NeighborSearch<
  NearestNeighborSort,
  metric::EuclideanDistance,
  tree::BinarySpaceTree<bound::BallBound<2>,
                        QueryStat<SortPolicy> >
> neighborSearch(dataset);

It is important to note that the NeighborSearch class requires use of the QueryStat tree statistic to function properly. Therefore, if you write a custom tree, be sure it can accept the QueryStat type. See the mlpack::tree::BinarySpaceTree documentation for more information on tree statistics.

Further documentation

For further documentation on the NeighborSearch class, consult the complete API documentation.