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Suppose we have feature transformation $\Phi(x) = [1, x_1, x_2, x_1x_2]$. Now we want to find the kernel corresponding to $\Phi$.

What I have done is using kernel decomposition, we have: $$ K(x, y) = \Phi(x) .\Phi(y)\\ K(x, y) = 1.1 + x_1y_1 + x_2y_2 + x_1x_2y_1y_2 \\ K(x, y) = 1 + \sum_{i=1}^{N}x_iy_i + \prod_{i=1}^{N}x_iy_i $$

$x, y$ are in $\Bbb{R}^2$

Is that kernel valid?

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Yes, it is valid.

Kernel is just an inner product in the feature space. In fact, here you have explicitly started from the feature transformation and we understand the feature transformation.

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