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I am having a problem comprehending with the relation of kernel, weight and linear separation.

I have a case where I am given a kernel $k_1$. that has a corresponding mapping $\phi_1$. And we know that it's linear separated in that space.

I need to show that given some $a,b > 0$: $$k=a*k_1+b*k_2$$ to show if we have linear separation in the higher dimension.

How do I approach that?

Danica
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ohad edelstain
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  • If b=-a and k2=k1 then everything is being mapped to zero and there is no separation at all. It seems to me as if there is some piece of information about k2 you are not telling us ;-) I think that there is an abstract definition of “kernel”. Maybe it makes sense to study the proof that shows that sums of kernels are (under some suitable conditions) kernels again... ? Maybe you somehow figure out properties about the embedding that corresponds to the sum from the single ones? – Fabian Werner May 17 '19 at 15:37
  • as you can see I mentioned that a,b are greater then 0 – ohad edelstain May 18 '19 at 08:34
  • My bad, I overlooked this. Ok, so if $\phi$ is the map for k1 and $\psi$ is the one for k2, can’t we simply write down the scalar product space for ak1+bk2? – Fabian Werner May 18 '19 at 11:26
  • Found the answer in the thread https://stats.stackexchange.com/questions/26465/behavior-of-a-sum-of-kernel-functions – ohad edelstain May 25 '19 at 13:29

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