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title: Untitled

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## Deep GP (nullification)

$cov = k_{**} - k_{*}(\frac{1}{\sigma^2}I - \frac{1}{\sigma^4}g^T(I+\frac{1}{\sigma^2}gg^T)^{-1}g)k_{*}^T$


Если раскрыть $k$ через $g$


$cov = g_{*}^Tg_{*} - g_{*}^Tg(\frac{1}{\sigma^2}I - \frac{1}{\sigma^4}g^T(I+\frac{1}{\sigma^2}gg^T)^{-1}g)g^Tg_{*}$


$cov = g_{*}^T(I - g(\frac{1}{\sigma^2}I - \frac{1}{\sigma^4}g^T(I+\frac{1}{\sigma^2}gg^T)^{-1}g)g^T)g_{*}$
$cov = g_{*}^T(I - \frac{1}{\sigma^2}gg^T + \frac{1}{\sigma^4}gg^T(I+\frac{1}{\sigma^2}gg^T)^{-1}gg^T)g_{*}$

Обозначим
$Q = \frac{1}{\sigma^2}gg^T$


$cov = g_{*}^T(I - Q + Q(I+Q)^{-1}Q)g_{*}$

**При условии** $I << Q$ 

$(Q+I)^{-1} = (Q(I+Q^{-1}))^{-1} \approx (I-Q^{-1})Q^{-1}$


$cov \approx g_{*}^T(I - Q + Q(I-Q^{-1})Q^{-1}Q)g_{*} = g_{*}^T(I - Q + Q(I-Q^{-1}))g_{*}$ = 0




