k-Nearest Neighbors Classification (k-NN)

\(k\)-NN classification algorithm infers the class for the new feature vector by computing majority vote of the \(k\) nearest observations from the training set.

Operation

Computational methods

Programming Interface

Training

Brute-force

k-d tree

train(…)

train_input

train_result

Inference

Brute-force

k-d tree

infer(…)

infer_input

infer_result

Mathematical formulation

Training

Let \(X = \{ x_1, \ldots, x_n \}\) be the training set of \(p\)-dimensional feature vectors, let \(Y = \{ y_1, \ldots, y_n \}\) be the set of class labels, where \(y_i \in \{ 0, \ldots, c-1 \}\), \(1 \leq i \leq n\). Given \(X\), \(Y\) and the number of nearest neighbors \(k\), the problem is to build a model that allows distance computation between the feature vectors in training and inference sets at the inference stage.

Training method: brute-force

The training operation produces the model that stores all the feature vectors from the initial training set \(X\).

Training method: k-d tree

The training operation builds a \(k\)-\(d\) tree that partitions the training set \(X\) (for more details, see k-d Tree).

Inference

Let \(X' = \{ x_1', \ldots, x_m' \}\) be the inference set of \(p\)-dimensional feature vectors. Given \(X'\), the model produced at the training stage and the number of nearest neighbors \(k\), the problem is to predict the label \(y_j'\) for each \(x_j'\), \(1 \leq j \leq m\), by performing the following steps:

  1. Identify the set \(N(x_j') \subseteq X\) of the \(k\) feature vectors in the training set that are nearest to \(x_j'\) with respect to the Euclidean distance.

  2. Estimate the conditional probability for the \(l\)-th class as the fraction of vectors in \(N(x_j')\) whose labels \(y_j\) are equal to \(l\):

    (1)\[P_{jl} = \frac{1}{| N(x_j') |} \Big| \big\{ x_r \in N(x_j') : y_r = l \big\} \Big|, \quad 1 \leq j \leq m, \; 0 \leq l < c.\]
  3. Predict the class that has the highest probability for the feature vector \(x_j'\):

    (2)\[y_j' = \mathrm{arg}\max_{0 \leq l < c} P_{jl}, \quad 1 \leq j \leq m.\]

Inference method: brute-force

Brute-force inference method determines the set \(N(x_j')\) of the nearest feature vectors by iterating over all the pairs \((x_j', x_i)\) in the implementation defined order, \(1 \leq i \leq n\), \(1 \leq j \leq m\). The final prediction is computed according to the equations (1) and (2).

Inference method: k-d tree

K-d tree inference method traverses the \(k\)-\(d\) tree to find feature vectors associated with a leaf node that are closest to \(x_j'\), \(1 \leq j \leq m\). The set \(\tilde{n}(x_j')\) of the currently-known nearest \(k\)-th neighbors is progressively updated during tree traversal. The search algorithm limits exploration of the nodes for which the distance between the \(x_j'\) and respective part of the feature space is not less than the distance between \(x_j'\) and the most distant feature vector from \(\tilde{n}(x_j')\). Once tree traversal is finished, \(\tilde{n}(x_j') \equiv N(x_j')\). The final prediction is computed according to the equations (1) and (2).

Usage example

Training

knn::model<> run_training(const table& data,
                          const table& labels) {
   const std::int64_t class_count = 10;
   const std::int64_t neighbor_count = 5;
   const auto knn_desc = knn::descriptor<float>{class_count, neighbor_count};

   const auto result = train(knn_desc, data, labels);

   return result.get_model();
}

Inference

table run_inference(const knn::model<>& model,
                    const table& new_data) {
   const std::int64_t class_count = 10;
   const std::int64_t neighbor_count = 5;
   const auto knn_desc = knn::descriptor<float>{class_count, neighbor_count};

   const auto result = infer(knn_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::knn namespace and be available via inclusion of the oneapi/dal/algo/knn.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 class_count,
                       std::int64_t neighbor_count);

   std::int64_t get_class_count() const;
   descriptor& set_class_count(std::int64_t);

   std::int64_t get_neighbor_count() const;
   descriptor& set_neighbor_count(std::int64_t);
};
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::bruteforce or method::kd_tree.

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

Constructors

descriptor(std::int64_t class_count, std::int64_t neighbor_count)

Creates a new instance of the class with the given class_count and neighbor_count property values.

Properties

std::int64_t neighbor_count

The number of neighbors \(k\).

Getter & Setter
std::int64_t get_neighbor_count() const
descriptor & set_neighbor_count(std::int64_t)
Invariants
std::int64_t class_count

The number of classes \(c\).

Getter & Setter
std::int64_t get_class_count() const
descriptor & set_class_count(std::int64_t)
Invariants

Method tags

namespace method {
   struct bruteforce {};
   struct kd_tree {};
   using by_default = bruteforce;
} // namespace method
struct bruteforce

Tag-type that denotes brute-force computational method.

struct kd_tree

Tag-type that denotes k-d tree computational method.

using by_default = bruteforce

Alias tag-type for brute-force computational method.

Task tags

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

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

using by_default = classification

Alias tag-type for classification task.

Model

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

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

Constructors

model()

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

Training train(...)

Input

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

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

   const table& get_labels() const;
   train_input& set_labels(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::classification.

Constructors

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

Creates a new instance of the class with the given data and labels property values.

Properties

const table &labels

Vector of labels \(y\) for the training set \(X\). Default value: table{}.

Getter & Setter
const table & get_labels() const
train_input & set_labels(const table &)
const table &data

The training set \(X\). Default value: table{}.

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;
};
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::classification.

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 \(k\)-NN model.

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 \(k\)-NN classifier. 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::bruteforce or method::kd_tree.

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

Parameters
  • desc – Descriptor of the algorithm.

  • input – Input data for the training operation.

Preconditions
input.data.has_data == true
input.labels.has_data == true
input.data.row_count == input.labels.row_count
input.labels.column_count == 1
input.labels[i] >= 0
input.labels[i] < desc.class_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&);

   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::classification.

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 property values.

Properties

const model<Task> &model

The trained \(k\)-NN model. Default value: model<Task>{}.

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

The dataset for inference \(X'\). 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;
};
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::classification.

Constructors

infer_result()

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

Public Methods

const table &get_labels() const

The predicted labels.

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 \(k\)-NN classifier. 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::bruteforce or method::kd_tree.

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

Parameters
  • desc – Descriptor of the algorithm.

  • input – Input data for the inference operation.

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