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Machine Learning: Difference between revisions
→Mercer's Theorem
| Line 135: | Line 135: | ||
Positive Definite:<br> | Positive Definite:<br> | ||
Let <math>\mathbf{v} \in \mathbb{R}^n</math>.<br> | Let <math>\mathbf{v} \in \mathbb{R}^n</math>.<br> | ||
Then | Then <br> | ||
<math> | <math> | ||
\begin{aligned} | \begin{aligned} | ||
\mathbf{v}^T \mathbf{K} \mathbf{v} | \mathbf{v}^T \mathbf{K} \mathbf{v}&= v^T [\sum_j K_{ij}v_j]\\ | ||
&= 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}\\ | ||