kernel methods
Questions from past exams
Give the definition of valid kernel and describe how valid kernels can be built. Provide an example of a methods that uses kernels and specify the advantages of using them in this specific method. (3)
A valid kernel $k(x, x')$ is any function that can be expressed as $k(x,x') = \phi^T(x)\phi(x')$. This also means that a necessary and sufficient condition for a kernel to be valid is that the Gram matrix $K$, whose elements are given by $k(x_n,x_m)$, is positive semi-definite for all possible choices of set ${x_n}$.
A valid kernel can be built, mainly, in three different ways:
- We choose a feature space mapping $\phi(x)$ and we define the kernel as $k(x,x') = \phi(x)^T \phi(x')$.
- We choose directly the kernel function $k(x,x')$ and we check that it can be written in the form $\phi^T(x) \phi(x)$
- We know some valid kernels and we combine or apply functions to them that we know will keep the validity of the kernel. For example if we know that $k_1(x,x')$ is a valid kernel then we also know that $ck_1(x,x')$ is valid as well.
One example of method which uses kernel functions is support vector machines for binary classification. The advantage of using the kernel function is that ...
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