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Machine Learning: Difference between revisions
→Mercer's Theorem
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====Mercer's Theorem==== | ====Mercer's Theorem==== | ||
Let our kernel function be <math>K(z,x)</math>. | Let our kernel function be <math>K(z,x)</math>. | ||
Then for any sample S, the corresponding matrix <math>\mathbf{K}</math> where <math>K_{ij} = K(x^{(i)},x^{(j)}</math> is symmetric positive definite. | Then for any sample S, the corresponding matrix <math>\mathbf{K}</math> where <math>K_{ij} = K(x^{(i)},x^{(j)})</math> is symmetric positive definite. | ||
{{hidden | Proof | | {{hidden | Proof | | ||
Symmetry: <math>K_{ij} = K(\mathbf{x}^{(i)},\mathbf{x}^{(j)} = \phi(\mathbf{x}^{(i)})^T\phi(\mathbf{x}^{(j)}) = \phi(\mathbf{x}^{(j)})^T\phi(\mathbf{x}^{(i)}) = K_{ji}</math><br> | Symmetry: <math>K_{ij} = K(\mathbf{x}^{(i)},\mathbf{x}^{(j)} = \phi(\mathbf{x}^{(i)})^T\phi(\mathbf{x}^{(j)}) = \phi(\mathbf{x}^{(j)})^T\phi(\mathbf{x}^{(i)}) = K_{ji}</math><br> | ||