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Machine Learning: Difference between revisions
→Mercer's Theorem
| Line 137: | Line 137: | ||
Then <br> | Then <br> | ||
<math> | <math> | ||
\begin{ | \begin{align} | ||
\mathbf{v}^T \mathbf{K} \mathbf{v}&= v^T [\sum_j K_{ij}v_j]\\ | \mathbf{v}^T \mathbf{K} \mathbf{v}&= \mathbf{v}^T [\sum_j K_{ij}v_j]\\ | ||
&= \sum_i \sum_j v_{i}K_{ij}v_{j}\\ | &= \sum_i \sum_j v_{i}K_{ij}v_{j}\\ | ||
&= \sum_i \sum_j v_{i}\phi(\mathbf{x}^{(i)})^T\phi(\mathbf{x}^{(j)})v_{j}\\ | &= \sum_i \sum_j v_{i}\phi(\mathbf{x}^{(i)})^T\phi(\mathbf{x}^{(j)})v_{j}\\ | ||
| Line 146: | Line 146: | ||
&= \sum_k (\sum_i v_{i} \phi_k(\mathbf{x}^{(i)}))^2\\ | &= \sum_k (\sum_i v_{i} \phi_k(\mathbf{x}^{(i)}))^2\\ | ||
&\geq 0 | &\geq 0 | ||
\end{ | \end{align} | ||
</math> | </math> | ||
}} | }} | ||