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The rules I can use without a proof are the following ones:
enter image description here

The thing that I know that a kernel is valid if it is symmetric, e.g. ${f(x,y) = f(y,x)}$ and if ${c^TKc >= 0}$, for each vector ${c \neq 0}$, but I don't know how to prove it, and don't know how to get the matrix K (if this is the correct way to solve this exercise).

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2 Answers 2

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Let $\phi(\mathbf x)=\mathbf x\mathbf x^T\mathbf x$ be a nonlinear transformation for $\mathbf x$. Then, you can use the fourth rule in your screenshot, because $k(\mathbf x, \mathbf y)=\phi(\mathbf x)^T\phi(\mathbf y)$ is a valid kernel since $k_3(\mathbf x,\mathbf y)=\mathbf x^T\mathbf y$ is a valid kernel.

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  • $\begingroup$ But how do you know you have to start from a value ${\phi(x) = xx^Tx}$? By intuition? $\endgroup$
    – Ele975
    Oct 25, 2022 at 22:15
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    $\begingroup$ @Ele975 Because you generally want to write $\mathop{f}\left(x,y\right)$ as an inner product of the form $\langle \mathop{\phi}\left(x\right), \mathop{\phi}\left(y\right)\rangle$. Realizing that you have to take the transpose of $x^\top x x^\top$ to make it match the form of $y y^\top y$ is just a small step then. $\endgroup$
    – statmerkur
    Oct 25, 2022 at 23:17
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I think you can use R3 and get $xx^Tyy^T $and then mention that is $x^2y^2$

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