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Introduction
The popular kmeans algorithm for clustering has been around since the late 1950s, and the standard algorithm was proposed by Stuart Lloyd in 1957. Given a set of points , kmeans clustering aims to partition each point into a cluster (where and , the number of clusters, is a parameter). The partitioning is done to minimize the objective function
where is the centroid of cluster . The standard algorithm is a twostep algorithm:
 Assignment step. Each point in is assigned to the cluster whose centroid it is closest to.
 Update step. Using the new cluster assignments, the centroids of each cluster are recalculated.
The algorithm has converged when no more assignment changes are happening with each iteration. However, this algorithm can get stuck in local minima of the objective function and is particularly sensitive to the initial cluster assignments. Also, situations can arise where the algorithm will never converge but reaches steady state – for instance, one point may be changing between two cluster assignments.
There is vast literature on the kmeans algorithm and its uses, as well as strategies for choosing initial points effectively and keeping the algorithm from converging in local minima. mlpack does implement some of these, notably the BradleyFayyad algorithm (see the reference below) for choosing refined initial points. Importantly, the C++ KMeans
class makes it very easy to improve the kmeans algorithm in a modular way.
mlpack provides:
 a simple commandline executable to run kmeans
 a simple C++ interface to run kmeans
 a generic, extensible, and powerful C++ class for complex usage
Table of Contents
A list of all the sections this tutorial contains.
 Introduction
 Table of Contents
 CommandLine 'kmeans'
 The 'KMeans' class
 Template parameters for the 'KMeans' class
 Further documentation
CommandLine 'kmeans'
mlpack provides a commandline executable, mlpack_kmeans
, to allow easy execution of the kmeans algorithm on data. Complete documentation of the executable can be found by typing
As of October 2014, support for overclustering has been removed due to bugs and lack of usage. If this is support you were using, or are interested, please file a bug or get in touch with the mlpack developers in some way so that the support can be reimplemented.
Below are several examples demonstrating simple use of the mlpack_kmeans
executable.
Simple kmeans clustering
We want to find 5 clusters using the points in the file dataset.csv. By default, if any of the clusters end up empty, that cluster will be reinitialized to contain the point furthest from the cluster with maximum variance. The cluster assignments of each point will be stored in assignments.csv. Each row in assignments.csv will correspond to the row in dataset.csv.
Saving the resulting centroids
Sometimes it is useful to save the centroids of the clusters found by kmeans; one example might be for plotting the points. The C
(–centroid_file
) option allows specification of a file into which the centroids will be saved (one centroid per line, if it is a CSV or other text format).
Allowing empty clusters
If you would like to allow empty clusters to exist, instead of reinitializing them, simply specify the e
(–allow_empty_clusters
) option. Note that when you save your clusters, some of the clusters may be filled with NaNs. This is expected behavior – if a cluster has no points, the concept of a centroid makes no sense.
Limiting the maximum number of iterations
As mentioned earlier, the kmeans algorithm can often fail to converge. In such a situation, it may be useful to stop the algorithm by way of limiting the maximum number of iterations. This can be done with the m
(–max_iterations
) parameter, which is set to 1000 by default. If the maximum number of iterations is 0, the algorithm will run until convergence – or potentially forever. The example below sets a maximum of 250 iterations.
Using BradleyFayyad "refined start"
The method proposed by Bradley and Fayyad in their paper "Refining initial points for kmeans clustering" is implemented in mlpack. This strategy samples points from the dataset and runs kmeans clustering on those points multiple times, saving the resulting clusters. Then, kmeans clustering is run on those clusters, yielding the original number of clusters. The centroids of those resulting clusters are used as initial centroids for kmeans clustering on the entire dataset.
This technique generally gives better initial points than the default random partitioning, but depending on the parameters, it can take much longer. This initialization technique is enabled with the r
(–refined_start
) option. The S
(–samplings
) parameter controls how many samplings of the dataset are performed, and the p
(–percentage
) parameter controls how much of the dataset is randomly sampled for each sampling (it must be between 0.0 and 1.0). For more information on the refined start technique, see the paper referenced in the introduction of this tutorial.
The example below performs kmeans clustering, giving 5 clusters, using the refined start technique, sampling 10% of the dataset 25 times to produce the initial centroids.
Using different kmeans algorithms
The mlpack_kmeans
program implements six different strategies for clustering; each of these gives the exact same results, but will have different runtimes. The particular algorithm to use can be specified with the a
or –algorithm
option. The choices are:
naive:
the standard Lloyd iteration; takes time per iteration.pellegmoore
: the 'blacklist' algorithm, which builds a kdtree on the data. This can be fast when k is small and the dimensionality is reasonably low.elkan:
Elkan's algorithm for kmeans, which maintains upper and lower distance bounds between each point and each centroid. This can be very fast, but it does not scale well to the case of large N or k, and uses a lot of memory.hamerly:
Hamerly's algorithm is a variant of Elkan's algorithm that handles memory usage much better and thus can operate with much larger datasets than Elkan's algorithm.dualtree:
The dualtree algorithm for kmeans builds a kdtree on both the centroids and the points in order to prune away as much work as possible. This algorithm is most effective when both N and k are large.dualtreecovertree
: This is the dualtree algorithm using cover trees instead of kdtrees. It satisfies the runtime guarantees specified in the dualtree kmeans paper.
In general, the naive
algorithm will be much slower than the others on datasets that are larger than tiny.
The example below uses the dualtree
algorithm to perform kmeans clustering with 5 clusters on the dataset in dataset.csv
, using the initial centroids in initial_centroids.csv
, saving the resulting cluster assignments to assignments.csv:
The 'KMeans' class
The KMeans<>
class (with default template parameters) provides a simple way to run kmeans clustering using mlpack in C++. The default template parameters for KMeans<>
will initialize cluster assignments randomly and disallow empty clusters. When an empty cluster is encountered, the point furthest from the cluster with maximum variance is set to the centroid of the empty cluster.
Running kmeans and getting cluster assignments
The simplest way to use the KMeans<>
class is to pass in a dataset and a number of clusters, and receive the cluster assignments in return. Note that the dataset must be columnmajor – that is, one column corresponds to one point. See the matrices guide for more information.
Now, the vector assignments
holds the cluster assignments of each point in the dataset.
Running kmeans and getting centroids of clusters
Often it is useful to not only have the cluster assignments, but the centroids of each cluster. Another overload of Cluster()
makes this easily possible:
Note that the centroids matrix has columns equal to the number of clusters and rows equal to the dimensionality of the dataset. Each column represents the centroid of the according cluster – centroids.col(0)
represents the centroid of the first cluster.
Limiting the maximum number of iterations
The first argument to the constructor allows specification of the maximum number of iterations. This is useful because often, the kmeans algorithm does not converge, and is terminated after a number of iterations. Setting this parameter to 0 indicates that the algorithm will run until convergence – note that in some cases, convergence may never happen. The default maximum number of iterations is 1000.
Then you can run Cluster()
as normal.
Setting initial cluster assignments
If you have an initial guess for the cluster assignments for each point, you can fill the assignments vector with the guess and then pass an extra boolean (initialAssignmentGuess) as true to the Cluster()
method. Below are examples for either overload of Cluster()
.
 Note
 If you have a heuristic or algorithm which makes initial guesses, a more elegant solution is to create a new class fulfilling the InitialPartitionPolicy template policy. See the section about changing the initial partitioning strategy for more details.
 Note
 If you set the InitialPartitionPolicy parameter to something other than the default but give an initial cluster assignment guess, the InitialPartitionPolicy will not be used to initialize the algorithm. See the section about changing the initial partitioning strategy for more details.
Setting initial cluster centroids
An equally important option to being able to make initial cluster assignment guesses is to make initial cluster centroid guesses without having to assign each point in the dataset to an initial cluster. This is similar to the previous section, but now you must pass two extra booleans – the first (initialAssignmentGuess) as false, indicating that there are not initial cluster assignment guesses, and the second (initialCentroidGuess) as true, indicating that the centroids matrix is filled with initial centroid guesses.
This, of course, only works with the overload of Cluster()
that takes a matrix to put the resulting centroids in. Below is an example.
 Note
 If you have a heuristic or algorithm which makes initial guesses, a more elegant solution is to create a new class fulfilling the InitialPartitionPolicy template policy. See the section about changing the initial partitioning strategy for more details.
 Note
 If you set the InitialPartitionPolicy parameter to something other than the default but give an initial cluster centroid guess, the InitialPartitionPolicy will not be used to initialize the algorithm. See the section about changing the initial partitioning strategy for more details.
Running sparse kmeans
The Cluster()
function can work on both sparse and dense matrices, so all of the above examples can be used with sparse matrices instead, if the fifth template parameter is modified. Below is a simple example. Note that the centroids are returned as a dense matrix, because the centroids of collections of sparse points are not generally sparse.
Template parameters for the 'KMeans' class
The KMeans<>
class also takes three template parameters, which can be modified to change the behavior of the kmeans algorithm. There are three template parameters:
MetricType:
controls the distance metric used for clustering (by default, the squared Euclidean distance is used)InitialPartitionPolicy:
the method by which initial clusters are set; by default, SampleInitialization is usedEmptyClusterPolicy:
the action taken when an empty cluster is encountered; by default, MaxVarianceNewCluster is usedLloydStepType:
this defines the strategy used to make a single Lloyd iteration; by default this is the typical Lloyd iteration specified in NaiveKMeansMatType:
type of data matrix to use for clustering
The class is defined like below:
In the following sections, each policy is described further, with examples of how to modify them.
Changing the distance metric used for kmeans
Most machine learning algorithms in mlpack support modifying the distance metric, and KMeans<>
is no exception. Similar to NeighborSearch (see the section in the NeighborSearch tutorial), any class in mlpack::metric can be given as an argument. The mlpack::metric::LMetric class is a good example implementation.
A class fulfilling the MetricType policy must provide the following two functions:
Most of the standard metrics that could be used are stateless and therefore the Evaluate()
method is implemented statically. However, there are metrics, such as the Mahalanobis distance (mlpack::metric::MahalanobisDistance), that store state. To this end, an instantiated MetricType object is stored within the KMeans
class. The example below shows how to pass an instantiated MahalanobisDistance in the constructor.
 Note
 While the MetricType policy only requires two methods, one of which is an empty constructor, more can always be added. mlpack::metric::MahalanobisDistance also has constructors with parameters, because it is a stateful metric.
Changing the initial partitioning strategy used for kmeans
There have been many initial cluster strategies for kmeans proposed in the literature. Fortunately, the KMeans<>
class makes it very easy to implement one of these methods and plug it in without needing to modify the existing algorithm code at all.
By default, the KMeans<>
class uses mlpack::kmeans::SampleInitialization, which randomly samples points as initial centroids. However, writing a new policy is simple; it needs to only implement the following functions:
The templatization of the Cluster()
function allows both dense and sparse matrices to be passed in. If the desired policy does not work with sparse (or dense) matrices, then the method can be written specifically for one type of matrix – however, be warned that if you try to use KMeans
with that policy and the wrong type of matrix, you will get many ugly compilation errors!
Note that only one of the two possible Cluster()
functions are required. This is because sometimes it is easier to express an initial partitioning policy as something that returns point assignments, and sometimes it is easier to express the policy as something that returns centroids. The KMeans<> class will use whichever of these two functions is given; if both are given, the overload that returns centroids will be preferred.
One alternate to the default SampleInitialization policy is the RefinedStart policy, which is an implementation of the Bradley and Fayyad approach for finding initial points detailed in "Refined initial points for kmeans clustering" and other places in this document. Another option is the RandomPartition class, which randomly assigns points to clusters, but this may not work very well for most settings. See the documentation for mlpack::kmeans::RefinedStart and mlpack::kmeans::RandomPartition for more information.
If the Cluster()
method returns point assignments instead of centroids, then valid initial assignments must be returned for every point in the dataset.
As with the MetricType template parameter, an initialized InitialPartitionPolicy can be passed to the constructor of KMeans
as a fourth argument.
Changing the action taken when an empty cluster is encountered
Sometimes, during clustering, a situation will arise where a cluster has no points in it. The KMeans
class allows easy customization of the action to be taken when this occurs. By default, the point furthest from the centroid of the cluster with maximum variance is taken as the centroid of the empty cluster; this is implemented in the mlpack::kmeans::MaxVarianceNewCluster class. Another alternate choice is the mlpack::kmeans::AllowEmptyClusters class, which simply allows empty clusters to persist.
A custom policy can be written and it must implement the following methods:
The EmptyCluster()
function is called for each cluster that is empty at each iteration of the algorithm. As with InitialPartitionPolicy, the EmptyCluster()
function does not need to be generalized to support both dense and sparse matrices – but usage with the wrong type of matrix will cause compilation errors.
Like the other template parameters to KMeans
, EmptyClusterPolicy implementations that have state can be passed to the constructor of KMeans
as a fifth argument. See the kmeans::KMeans documentation for further details.
The LloydStepType template parameter
The internal algorithm used for a single step of the kmeans algorithm can easily be changed; mlpack implements several existing classes that satisfy the LloydStepType
policy:
 mlpack::kmeans::NaiveKMeans
 mlpack::kmeans::ElkanKMeans
 mlpack::kmeans::HamerlyKMeans
 mlpack::kmeans::PellegMooreKMeans
 mlpack::kmeans::DualTreeKMeans
Note that the LloydStepType
policy is itself a template template parameter, and must accept two template parameters of its own:
MetricType:
the type of metric to useMatType:
the type of data matrix to use
The LloydStepType
policy also mandates three functions:
 a constructor:
"LloydStepType(const MatType& dataset, MetricType& metric);"
 an
Iterate()
function:
 a function to get the number of distance calculations:
Note that Iterate()
does not need to return valid centroids if the cluster is empty. This is because EmptyClusterPolicy
will handle the empty centroid. This behavior can be used to avoid small amounts of computation.
For examples, see the five aforementioned implementations of classes that satisfy the LloydStepType
policy.
Further documentation
For further documentation on the KMeans class, consult the complete API documentation.
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