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================
Kernel Functions
================

A kernel function takes input vectors from the original (:math:`p`-dimensional) space
and returns the dot product of the vectors in the :math:`s`-dimensional feature space.
Thus, having :math:`x,y \in \mathbb{R}^p` and
:math:`\phi \in \mathbb{R}^p \leftarrow \mathbb{R}^s`, the kernel function is

.. math::
   K(x, y) \equiv <\phi (x), \phi (y)>
.. _kernel_func_def:

Where :math:`\phi(x)` is a vector-valued function:
:math:`\phi(x) = \phi_1(x), \phi_2(x), \ldots, \phi_s(x)`.

If :math:`\phi (x) = x`, the kernel is linear.
Kernels are used in the SVM model, but for some tasks they could be used separately to transform vectors
from one space to another.

The following table describes current device support:

+--------------+------+------+
| Kernel type  | CPU  | GPU  |
+==============+======+======+
| Linear       | Yes  | Yes  |
+--------------+------+------+
| Polynomial   | Yes  | No   |
+--------------+------+------+
| RBF          | Yes  | Yes  |
+--------------+------+------+
| Sigmoid      | Yes  | No   |
+--------------+------+------+


.. toctree::
   :titlesonly:

   linear-kernel.rst
   polynomial-kernel.rst
   rbf-kernel.rst
   sigmoid-kernel.rst

.. rubric:: Examples: Linear Kernel

.. include::  ../../../includes/kernel-functions/linear-kernel-examples.rst

.. rubric:: Examples: Polynomial Kernel

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

.. rubric:: Examples: RBF Kernel

.. include:: ../../../includes/kernel-functions/rbf-kernel-examples.rst

.. rubric:: Examples: Sigmoid Kernel

.. include:: ../../../includes/kernel-functions/sigmoid-kernel-examples.rst