# Polynomial kernel¶

The Polynomial kernel is a popular kernel function used in kernelized learning algorithms. It represents the similarity of training samples in a feature space of polynomials of the original data and allows to fit non-linear models.

 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 $$y_1 = (y_{11}, \ldots, y_{1p}), \ldots, y_m = (y_{m1}, \ldots, y_{mp})$$, the problem is to compute the polynomial kernel function $$K(x_i, y_j)$$ for any pair of input vectors:

$K(x_i, y_j) = (k {x_i}^T y_j + b)^d,$

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

### Computation method: dense¶

The method computes the polynomial kernel function $$K(X, Y)$$ for $$X$$ and $$Y$$ matrices.

## Programming Interface¶

Refer to API Reference: Polynomial kernel.