Radial basis function kernel: Difference between revisions

Revision as of 14:05, 2 August 2016

Revision as of 21:59, 28 November 2016

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}}</ref>

:<math>

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

\begin{alignat}{2}

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

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

:<math>\exp\left(-\frac{1}{2}||\mathbf{x} - \mathbf{x'}||^2\right) &= \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}

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

\frac{x_1^{n_1}\cdots x_k^{n_k} }{\sqrt{n_1! \cdots n_k! }}

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

\frac{{x'}_1^{n_1}\cdots {x'}_k^{n_k} }{\sqrt{n_1! \cdots n_k! }}

\end{alignat}

</math>

==Approximations==