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Machine Learning: Difference between revisions
→Mercer's Theorem
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&= \sum_k \sum_i \sum_j v_{i} \phi_k(\mathbf{x}^{(i)}) \phi_k(\mathbf{x}^{(j)})v_{j}\\ | &= \sum_k \sum_i \sum_j v_{i} \phi_k(\mathbf{x}^{(i)}) \phi_k(\mathbf{x}^{(j)})v_{j}\\ | ||
&= \sum_k \sum_i v_{i} \phi_k(\mathbf{x}^{(i)}) \sum_j \phi_k(\mathbf{x}^{(j)})v_{j}\\ | &= \sum_k \sum_i v_{i} \phi_k(\mathbf{x}^{(i)}) \sum_j \phi_k(\mathbf{x}^{(j)})v_{j}\\ | ||
&= \sum_k | &= \sum_k (\sum_i v_{i} \phi_k(\mathbf{x}^{(i)}))^{2}\\ | ||
&\geq 0 \quad | &\geq 0 \quad | ||
\end{aligned} | \end{aligned} | ||