Im a bit confused how 2 is a Kernel as the only solution I could think of is sqrt(2) x sqrt(2), this is also the one given in the solution. But as sqrt(2) is not a natural number, which seemed to be required for the kernel(as marked in the alterntives). Have I missundertstood the notation X=N somehow or maybe missed something else?
The restriction on natural numbers is only on \(k\), the number of dimensions in the output space of \(\Psi\). So if we focus on the 2-dimensional case of \(K(x, x') = (x^\top x')^2\), a solution for \(\Psi\) could be psi([a, b]) = [a, b, sqrt(2) a b].
The restriction on natural numbers is only on \(k\), the number of dimensions in the output space of \(\Psi\). So if we focus on the 2-dimensional case of \(K(x, x') = (x^\top x')^2\), a solution for \(\Psi\) could be psi([a, b]) = [a, b, sqrt(2) a b].
Ok so this expression
does not limit the values x and x' can take? Or what does the expression mean exactly in this case as the restriction on k is already given earlier in the question.
Ah, I thought you meant 'the second kernel in the list' with '2', but now I see you mean the first (with value 2).
This expression means that the inputs to the kernel are natural numbers. In this case \(\Psi(x) = \sqrt{2}\) is a valid function from \( \mathbb{N} \to \mathbb{R}^k\) for \(k=1 \in \mathbb{N}\), so it works out.
Question 9 2020
Im a bit confused how 2 is a Kernel as the only solution I could think of is sqrt(2) x sqrt(2), this is also the one given in the solution. But as sqrt(2) is not a natural number, which seemed to be required for the kernel(as marked in the alterntives). Have I missundertstood the notation X=N somehow or maybe missed something else?
Hope someone can help me! :)
1
The restriction on natural numbers is only on \(k\), the number of dimensions in the output space of \(\Psi\). So if we focus on the 2-dimensional case of \(K(x, x') = (x^\top x')^2\), a solution for \(\Psi\) could be psi([a, b]) = [a, b, sqrt(2) a b].
Wondering the same thing actually
The restriction on natural numbers is only on \(k\), the number of dimensions in the output space of \(\Psi\). So if we focus on the 2-dimensional case of \(K(x, x') = (x^\top x')^2\), a solution for \(\Psi\) could be psi([a, b]) = [a, b, sqrt(2) a b].
Ok so this expression
does not limit the values x and x' can take? Or what does the expression mean exactly in this case as the restriction on k is already given earlier in the question.
Thanks for the response btw!!
edit: why is this question marked as solved?
Ah, I thought you meant 'the second kernel in the list' with '2', but now I see you mean the first (with value 2).
This expression means that the inputs to the kernel are natural numbers. In this case \(\Psi(x) = \sqrt{2}\) is a valid function from \( \mathbb{N} \to \mathbb{R}^k\) for \(k=1 \in \mathbb{N}\), so it works out.
Yeah that makes the input doesnt matter at all in this case as is constant! I confused the value of the function with the input!
Thanks so much!!!!
Maybe you can explain to me, because I didn't get it… ^^'
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