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Section 6.2: Complex

Gaussian kernel (6.23)

[[latex($k(\mathbf{x},\mathbf{x^{p}) = exp(-||\mathbf{x} - \mathbf{x}p}||^{2 / 2\sigma}2) $)]] (6.23)

is this valid? (6.24)

[[latex($ ||\mathbf{x} - \mathbf{x^p}||2 = \mathbf{x}^{T\mathbf{x} + (\mathbf{x}p})^T\mathbf{xp} - 2\mathbf{x}^T\mathbf{xp} $)]]

substitute into 6.23 to give 6.25

use 6.14 and 6.16, ie latex($ f(\mathbf{x})k_{1}(\mathbf{x},\mathbf{x^p})f(\mathbf{x^p}) $) and latex($ exp(k_{1}(\mathbf{x},\mathbf{x^p})) $)

to see that it is valid.

We can also replace latex($\mathbf{x}^T,\mathbf{x^p} $) in the above with a nonlinear Kernel latex($\kappa(\mathbf{x},\mathbf{x^p}) $)