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

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

### Computation method: dense¶

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