Radial basis function kernel: Difference between revisions

Revision as of 05:53, 11 December 2020

Revision as of 19:40, 3 March 2021

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\begin{alignat}{2}

\exp\left(-\frac{1}{2}\|\mathbf{x} - \mathbf{x'}\|^2\right)

&= \exp(\frac{2}{2}\mathbf{x}^\top \mathbf{x'} - \frac{1}{2}\|\mathbf{x}\|^2 - \frac{1}{2}\|\mathbf{x'}\|^2)\\

&= \exp(\mathbf{x}^\top \mathbf{x'}) \exp( - \frac{1}{2}\|\mathbf{x}\|^2) \exp( - \frac{1}{2}\|\mathbf{x'}\|^2) \\

&= \sum_{j=0}^\infty \frac{(\mathbf{x}^\top \mathbf{x'})^j}{j!} \exp\left(-\frac{1}{2}\|\mathbf{x}\|^2\right) \exp\left(-\frac{1}{2}\|\mathbf{x'}\|^2\right)\\

&= \sum_{j=0}^\infty \sum_{\sum n_i=j}