Gaussian RBF (Radial Basis Function) kernel

`RBF()` computes the RBF kernel between all possible pairs of rows of a matrix or data.frame with dimension NxD.

Usage

RBF(X, g = NULL)

Arguments

X

Matrix or data.frame that contains real numbers ("integer", "float" or "double").

g

Gamma hyperparameter. If g=0 or NULL, `RBF()` returns the matrix of squared Euclidean distances instead of the RBF kernel matrix. (Defaults=NULL).

Value

Kernel matrix (dimension: NxN).

Details

Let \(x_i,x_j\) be two real vectors. Then, the RBF kernel is defined as: $$K_{RBF}(x_i,x_j)=\exp(-\gamma \|x_i - x_j \|^2)$$

Sometimes the RBF kernel is given a hyperparameter called sigma. In that case: \(\gamma = 1/\sigma^2\).

Examples

dat <- matrix(rnorm(250),ncol=50,nrow=5)
RBF(dat,g=0.1)
#>              [,1]         [,2]         [,3]         [,4]         [,5]
#> [1,] 1.000000e+00 2.601581e-05 7.283653e-04 0.0015929600 6.536862e-05
#> [2,] 2.601581e-05 1.000000e+00 6.283656e-05 0.0002441566 5.776624e-05
#> [3,] 7.283653e-04 6.283656e-05 1.000000e+00 0.0008188197 3.386474e-04
#> [4,] 1.592960e-03 2.441566e-04 8.188197e-04 1.0000000000 3.185288e-04
#> [5,] 6.536862e-05 5.776624e-05 3.386474e-04 0.0003185288 1.000000e+00