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Let's suppose we have a kernel function

$k(x,x')=10 $

In order to prove that this a valid kernel function there are generally two conditions

  • It is symmetric
  • There exists a map $\varphi:R^d \rightarrow H$ called kernel feature map into some high dimensional feature space $H$ such that $\forall x,x' \ in \ R^d :k(x,x') = \ <\varphi(x),\varphi(x')> $.

How to formally approach to prove these two conditions for this kernel function?

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If I get you correctly, you simply want $k(x, x')$ to be the constant function $10$? In that case

  • It is symmetric because for all $x, x'$: $k(x, x')=10=k(x',x)$
  • You can take $H = \mathbb{R}$ and $\varphi(x) = \sqrt{10}$. Then for any $x, x'$: $\langle\varphi(x),\varphi(x')\rangle = \sqrt{10}^2 = 10 = k(x,x')$
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