Radial Basis Function (RBF) kernel

The Radial Basis Function (RBF) kernel is a popular kernel function used in kernelized learning algorithms.

Operation

Computational methods

Programming Interface

dense

dense

compute(…)

compute_input

compute_result

Mathematical formulation

Computing

Given a set \(X\) of \(n\) feature vectors \(x_1 = (x_{11}, \ldots, x_{1p}), \ldots, x_n = (x_{n1}, \ldots, x_{np})\) of dimension \(p\) and a set \(Y\) of \(m\) feature vectors \(y_1 = (y_{11}, \ldots, y_{1p}), \ldots, y_m = (y_{m1}, \ldots, x_{mp})\), the problem is to compute the RBF kernel function \(K(x_i,, y_i)\) for any pair of input vectors:

\[K\left({x}_{i},{y}_{j}\right)=exp\left(-\frac{{\left(\|{x}_{i}-{y}_{j}\|\right)}^{2}}{2{\sigma }^{2}}\right)\]

Programming Interface

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

Descriptor

template<typename Task = task::by_default>
class descriptor_base

Constructors

descriptor_base()

Properties

double sigma = 1.0

The coefficient \(\sigma\) of the RBF kernel.

Getter & Setter
double get_sigma() const
template<typename Float = detail::descriptor_base<>::float_t, typename Method = detail::descriptor_base<>::method_t, typename Task = detail::descriptor_base<>::task_t>
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::v1::dense.

  • Task – Tag-type that specifies the type of the problem to solve. Can be task::v1::compute.

Public Methods

auto &set_sigma(double value)

Method tags

struct dense
using by_default = dense

Task tags

struct compute

Tag-type that parameterizes entities that are used to compute statistics, distance, and so on.

using by_default = compute

Alias tag-type for the dense method.

Training compute(...)

Input

template<typename Task = task::by_default>
class compute_input
Template Parameters

Task – Tag-type that specifies the type of the problem to solve. Can be task::v1::compute.

Constructors

compute_input(const table &x, const table &y)

Creates a new instance of the class with the given x and y.

Properties

const table &x = table{}

An \(n1 \times p\) table with the data x, where each row stores one feature vector.

Getter & Setter
const table & get_x() const
auto & set_x(const table &data)
const table &y = table{}

An \(n1 \times p\) table with the data x, where each row stores one feature vector.

Getter & Setter
const table & get_y() const
auto & set_y(const table &data)

Result

template<typename Task = task::by_default>
class compute_result
Template Parameters

Task – Tag-type that specifies the type of the problem to solve. Can be task::v1::compute.

Constructors

compute_result()

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

Properties

const table &values = table{}

A \(n1 \times n2\) table with the result kernel functions.

Getter & Setter
const table & get_values() const
auto & set_values(const table &value)

Operation

template<typename Descriptor>
rbf_kernel::compute_result compute(const Descriptor &desc, const rbf_kernel::compute_input &input)
Template Parameters

Descriptor – RBF Kernel algorithm descriptor rbf_kernel::desc.

Preconditions
input.data.is_empty == false