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?


1 Answer 1


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.