What happens if we run a support vector machine model using a kernel that does not satisfy requirements such as non-positive semi definite?

This is my flow of thought: In kernel methods $w.x$ is replaced by $\sum_i \alpha_i k(x, x_i)$. Now if k is not positive semi definite, $\exists y$ such that $k(y,y)<0$. This means there exists y such that $k(y, y) < 0$. So I feel that the classifier will become noisy, because it will keep misclassifying y. Is this correct? If so, how do I make my arguments more rigorous?

  • $\begingroup$ If the Kernel matrix it's non-positive semi definite then we have that there is no feature mapping $\phi(x)$ such that $k(x_i,x_j)=\phi(x_i)^T\phi(x_j)$. Thereby we can't apply it to our equation, because we aren't computing new valid features from our initial features $x$. $\endgroup$
    – Javier TG
    Nov 3, 2020 at 21:33


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