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I have a question regarding my machine learning lecture where we had to decide whether $$K(x,y)=x_1y_1-x_2y_2$$ is a valid kernel (e.g. for a SVM). My intuition would say that it is a valid kernel since we can display it with: $$\Phi(x)=(x_1, ix_2)\implies K(x,y)=\Phi(x)\Phi(y)$$ with $i$ being the imaginary number. Is that right?

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Nope. Definition of inner product requires positive definiteness, that is $\langle x, x \rangle > 0$ if $x \ne 0$.

For complex numbers, notice that $\langle x, y \rangle = \sum_{i=1}^n x_i y_i $ is not a valid inner product since $i^2=-1$.

A typical choice of inner product is $\langle x, y \rangle = \sum_{i=1}^n x_i \bar{y}_i $.

Since $$K((0,1) , (0, 1))= 0\cdot 0 - 1\cdot 1 = -1 < 0$$ it is not a valid kernel.

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