K-Means

The K-Means algorithm solves clustering problem by partitioning \(n\) feature vectors into \(k\) clusters minimizing some criterion. Each cluster is characterized by a representative point, called a centroid.

Operation

Computational methods

Programming Interface

Training

Lloyd’s

train(…)

train_input

train_result

Inference

Lloyd’s

infer(…)

infer_input

infer_result

Mathematical formulation

Training

Given the training set \(X = \{ x_1, \ldots, x_n \}\) of \(p\)-dimensional feature vectors and a positive integer \(k\), the problem is to find a set \(C = \{ c_1, \ldots, c_k \}\) of \(p\)-dimensional centroids that minimize the objective function

\[\Phi_{X}(C) = \sum_{i = 1}^n d^2(x_i, C),\]

where \(d^2(x_i, C)\) is the squared Euclidean distance from \(x_i\) to the closest centroid in \(C\),

\[d^2(x_i, C) = \min_{1 \leq j \leq k} \| x_i - c_j \|^2, \quad 1 \leq i \leq n.\]

Expression \(\|\cdot\|\) denotes \(L_2\) norm.

Note

In the general case, \(d\) may be an arbitrary distance function. Current version of the oneDAL spec defines only Euclidean distance case.

Training method: Lloyd’s

The Lloyd’s method [Lloyd82] consists in iterative updates of centroids by applying the alternating Assignment and Update steps, where \(t\) denotes a index of the current iteration, e.g., \(C^{(t)} = \{ c_1^{(t)}, \ldots, c_k^{(t)} \}\) is the set of centroids at the \(t\)-th iteration. The method requires the initial centroids \(C^{(1)}\) to be specified at the beginning of the algorithm (\(t = 1\)).

(1) Assignment step: Assign each feature vector \(x_i\) to the nearest centroid. \(y_i^{(t)}\) denotes the assigned label (cluster index) to the feature vector \(x_i\).

\[y_i^{(t)} = \mathrm{arg}\min_{1 \leq j \leq k} \| x_i - c_j^{(t)} \|^2, \quad 1 \leq i \leq n.\]

Each feature vector from the training set \(X\) is assigned to exactly one centroid so that \(X\) is partitioned to \(k\) disjoint sets (clusters)

\[S_j^{(t)} = \big\{ \; x_i \in X : \; y_i^{(t)} = j \; \big\}, \quad 1 \leq j \leq k.\]

(2) Update step: Recalculate centroids by averaging feature vectors assigned to each cluster.

\[c_j^{(t + 1)} = \frac{1}{|S_j^{(t)}|} \sum_{x \in S_j^{(t)}} x, \quad 1 \leq j \leq k.\]

The steps (1) and (2) are performed until the following stop condition,

\[\sum_{j=1}^k \big\| c_j^{(t)} - c_j^{(t+1)} \big\|^2 < \varepsilon,\]

is satisfied or number of iterations exceeds the maximal value \(T\) defined by the user.

Inference

Given the inference set \(X' = \{ x_1', \ldots, x_m' \}\) of \(p\)-dimensional feature vectors and the set \(C = \{ c_1, \ldots, c_k \}\) of centroids produced at the training stage, the problem is to predict the index \(y_j' \in \{ 0, \ldots, k-1 \}\), \(1 \leq j \leq m\), of the centroid in accordance with a method-defined rule.

Inference method: Lloyd’s

Lloyd’s inference method computes the \(y_j'\) as an index of the centroid closest to the feature vector \(x_j'\),

\[y_j' = \mathrm{arg}\min_{1 \leq l \leq k} \| x_j' - c_l \|^2, \quad 1 \leq j \leq m.\]

Usage example

Training

kmeans::model<> run_training(const table& data,
                             const table& initial_centroids) {
   const auto kmeans_desc = kmeans::descriptor<float>{}
      .set_cluster_count(10)
      .set_max_iteration_count(50)
      .set_accuracy_threshold(1e-4);

   const auto result = train(kmeans_desc, data, initial_centroids);

   print_table("labels", result.get_labels());
   print_table("centroids", result.get_model().get_centroids());
   print_value("objective", result.get_objective_function_value());

   return result.get_model();
}

Inference

table run_inference(const kmeans::model<>& model,
                    const table& new_data) {
   const auto kmeans_desc = kmeans::descriptor<float>{}
      .set_cluster_count(model.get_cluster_count());

   const auto result = infer(kmeans_desc, model, new_data);

   print_table("labels", result.get_labels());
}

Programming Interface

All types and functions in this section shall be declared in the oneapi::dal::kmeans namespace and be available via inclusion of the oneapi/dal/algo/kmeans.hpp header file.

Descriptor

template <typename Float = float,
          typename Method = method::by_default,
          typename Task = task::by_default>
class descriptor {
public:
   explicit descriptor(std::int64_t cluster_count = 2);

   int64_t get_cluster_count() const;
   descriptor& set_cluster_count(int64_t);

   int64_t get_max_iteration_count() const;
   descriptor& set_max_iteration_count(int64_t);

   double get_accuracy_threshold() const;
   descriptor& set_accuracy_threshold(double);
};
template<typename Float = float, typename Method = method::by_default, typename Task = task::by_default>
class descriptor
Template Parameters
  • Float – The floating-point type that the algorithm uses for intermediate computations. Can be float or double.

  • Method – Tag-type that specifies an implementation of algorithm. Can be method::lloyd.

  • Task – Tag-type that specifies the type of the problem to solve. Can be task::clustering.

Constructors

descriptor(std::int64_t cluster_count = 2)

Creates a new instance of the class with the given cluster_count.

Properties

int64_t max_iteration_count

The maximum number of iterations \(T\). Default value: 100.

Getter & Setter
int64_t get_max_iteration_count() const
descriptor & set_max_iteration_count(int64_t)
Invariants
int64_t cluster_count

The number of clusters \(k\). Default value: 2.

Getter & Setter
int64_t get_cluster_count() const
descriptor & set_cluster_count(int64_t)
Invariants
double accuracy_threshold

The threshold \(\varepsilon\) for the stop condition. Default value: 0.0.

Getter & Setter
double get_accuracy_threshold() const
descriptor & set_accuracy_threshold(double)
Invariants

Method tags

namespace method {
   struct lloyd {};
   using by_default = lloyd;
} // namespace method
struct lloyd

Tag-type that denotes Lloyd’s computational method.

using by_default = lloyd

Alias tag-type for Lloyd’s computational method.

Task tags

namespace task {
   struct clustering {};
   using by_default = clustering;
} // namespace task
struct clustering

Tag-type that parameterizes entities used for solving clustering problem.

using by_default = clustering

Alias tag-type for the clustering task.

Model

template <typename Task = task::by_default>
class model {
public:
   model();

   const table& get_centroids() const;

   int64_t get_cluster_count() const;
};
template<typename Task = task::by_default>
class model
Template Parameters

Task – Tag-type that specifies type of the problem to solve. Can be task::clustering.

Constructors

model()

Creates a new instance of the class with the default property values.

Public Methods

const table &get_centroids() const

A \(k \times p\) table with the cluster centroids. Each row of the table stores one centroid.

int64_t get_cluster_count() const

Number of clusters \(k\) in the trained model.

Training train(...)

Input

template <typename Task = task::by_default>
class train_input {
public:
   train_input(const table& data = table{},
               const table& initial_centroids = table{});

   const table& get_data() const;
   train_input& set_data(const table&);

   const table& get_initial_centroids() const;
   train_input& set_initial_centroids(const table&);
};
template<typename Task = task::by_default>
class train_input
Template Parameters

Task – Tag-type that specifies type of the problem to solve. Can be task::clustering.

Constructors

train_input(const table &data = table{}, const table &initial_centroids = table{})

Creates a new instance of the class with the given data and initial_centroids.

Properties

const table &initial_centroids

A \(k \times p\) table with the initial centroids, where each row stores one centroid.

Getter & Setter
const table & get_initial_centroids() const
train_input & set_initial_centroids(const table &)
const table &data

An \(n \times p\) table with the data to be clustered, where each row stores one feature vector.

Getter & Setter
const table & get_data() const
train_input & set_data(const table &)

Result

template <typename Task = task::by_default>
class train_result {
public:
   train_result();

   const model<Task>& get_model() const;

   const table& get_labels() const;

   int64_t get_iteration_count() const;

   double get_objective_function_value() const;
};
template<typename Task = task::by_default>
class train_result
Template Parameters

Task – Tag-type that specifies type of the problem to solve. Can be task::clustering.

Constructors

train_result()

Creates a new instance of the class with the default property values.

Public Methods

const model<Task> &get_model() const

The trained K-means model.

const table &get_labels() const

An \(n \times 1\) table with the labels \(y_i\) assigned to the samples \(x_i\) in the input data, \(1 \leq 1 \leq n\).

int64_t get_iteration_count() const

The number of iterations performed by the algorithm.

double get_objective_function_value() const

The value of the objective function \(\Phi_X(C)\), where \(C\) is model.centroids (see kmeans::model::centroids).

Operation

template <typename Float, typename Method, typename Task>
train_result<Task> train(const descriptor<Float, Method, Task>& desc,
                         const train_input<Task>& input);
template<typename Float, typename Method, typename Task>
train_result<Task> train(const descriptor<Float, Method, Task> &desc, const train_input<Task> &input)

Runs the training operation for K-Means clustering. For more details see oneapi::dal::train.

Template Parameters
  • Float – The floating-point type that the algorithm uses for intermediate computations. Can be float or double.

  • Method – Tag-type that specifies an implementation of algorithm. Can be method::lloyd.

  • Task – Tag-type that specifies type of the problem to solve. Can be task::clustering.

Parameters
  • desc – Descriptor of the algorithm.

  • input – Input data for the training operation.

Preconditions
input.data.has_data == true
input.initial_centroids.row_count == desc.cluster_count
input.initial_centroids.column_count == input.data.column_count
Postconditions
result.labels.row_count == input.data.row_count
result.labels.column_count == 1
result.labels[i] >= 0
result.labels[i] < desc.cluster_count
result.iteration_count <= desc.max_iteration_count
result.model.centroids.row_count == desc.cluster_count
result.model.centroids.column_count == input.data.column_count

Inference infer(...)

Input

template <typename Task = task::by_default>
class infer_input {
public:
   infer_input(const model<Task>& m = model<Task>{},
               const table& data = table{});

   const model<Task>& get_model() const;
   infer_input& set_model(const model<Task>&);

   const table& get_data() const;
   infer_input& set_data(const table&);
};
template<typename Task = task::by_default>
class infer_input
Template Parameters

Task – Tag-type that specifies type of the problem to solve. Can be task::clustering.

Constructors

infer_input(const model<Task> &m = model<Task>{}, const table &data = table{})

Creates a new instance of the class with the given model and data.

Properties

const model<Task> &model

An \(n \times p\) table with the data to be assigned to the clusters, where each row stores one feature vector. Default value: model<Task>{}.

Getter & Setter
const model< Task > & get_model() const
infer_input & set_model(const model< Task > &)
const table &data

The trained K-Means model. Default value: table{}.

Getter & Setter
const table & get_data() const
infer_input & set_data(const table &)

Result

template <typename Task = task::by_default>
class infer_result {
public:
   infer_result();

   const table& get_labels() const;

   double get_objective_function_value() const;
};
template<typename Task = task::by_default>
class infer_result
Template Parameters

Task – Tag-type that specifies type of the problem to solve. Can be task::clustering.

Constructors

infer_result()

Creates a new instance of the class with the default property values.

Public Methods

const table &get_labels() const

An \(n \times 1\) table with assignments labels to feature vectors in the input data.

double get_objective_function_value() const

The value of the objective function \(\Phi_X(C)\), where \(C\) is defined by the corresponding infer_input::model::centroids.

Operation

template <typename Float, typename Method, typename Task>
infer_result<Task> infer(const descriptor<Float, Method, Task>& desc,
                         const infer_input<Task>& input);
template<typename Float, typename Method, typename Task>
infer_result<Task> infer(const descriptor<Float, Method, Task> &desc, const infer_input<Task> &input)

Runs the inference operation for K-Means clustering. For more details see oneapi::dal::infer.

Template Parameters
  • Float – The floating-point type that the algorithm uses for intermediate computations. Can be float or double.

  • Method – Tag-type that specifies an implementation of algorithm. Can be method::lloyd.

  • Task – Tag-type that specifies type of the problem to solve. Can be task::clustering.

Parameters
  • desc – Descriptor of the algorithm.

  • input – Input data for the inference operation.

Preconditions
input.data.has_data == true
input.model.centroids.has_data == true
input.model.centroids.row_count == desc.cluster_count
input.model.centroids.column_count == input.data.column_count
Postconditions
result.labels.row_count == input.data.row_count
result.labels.column_count == 1
result.labels[i] >= 0
result.labels[i] < desc.cluster_count