.. ******************************************************************************
.. * Copyright 2021 Intel Corporation
.. *
.. * Licensed under the Apache License, Version 2.0 (the "License");
.. * you may not use this file except in compliance with the License.
.. * You may obtain a copy of the License at
.. *
.. *     http://www.apache.org/licenses/LICENSE-2.0
.. *
.. * Unless required by applicable law or agreed to in writing, software
.. * distributed under the License is distributed on an "AS IS" BASIS,
.. * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
.. * See the License for the specific language governing permissions and
.. * limitations under the License.
.. *******************************************************************************/

.. default-domain:: cpp

.. _alg_polynomial_kernel:

===================================
Polynomial kernel
===================================

.. include::  ../../../includes/kernel-functions/polynomial-kernel-introduction.rst

------------------------
Mathematical formulation
------------------------

.. _polynomial_kernel_c_math:

Computing
---------

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

.. math::
   K(x_i, y_j) = (k {x_i}^T y_j + b)^d,

where :math:`k\in\mathbb{R},\ b\in\mathbb{R},\ d\in{\{0,\ 1,\ 2,\ ...\}}, \quad 1 \leq i \leq n, \quad 1 \leq j \leq m`.

.. _polynomial_kernel_c_math_dense:

Computation method: *dense*
---------------------------
The method computes the polynomial kernel function :math:`Z=K(X, Y), Z \in \mathbb{R}^{n \times m}` for
dense :math:`X` and :math:`Y` matrices.

---------------------
Programming Interface
---------------------

Refer to :ref:`API Reference: Polynomial kernel <api_polynomial_kernel>`.