Radial Basis Function (RBF) kernel¶
The Radial Basis Function (RBF) kernel is a popular kernel function used in kernelized learning algorithms.
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Computational methods |
Programming Interface |
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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 \(Z=K(X, Y), Z \in \mathbb{R}^{n \times m}\) for dense \(X\) and \(Y\) matrices.