I am reading Tom Mitchell's Machine Learning. In section 8.2.3, he defines: Kernel function is the function of distance that is used to determine the weight of each training example. In other words, the kernel function is the function $K$ such that $w_i = K(d(x_i, x_q))$. However, when we talk about SVMs we have a kernel function $K(x,y) = \phi(x) \cdot \phi(y)$. Is kernel function (when talking about locally weighted regression or maybe k-NN) just a totally different thing from kernel function (SVMs)?
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1$\begingroup$ Unfortunately, many authors in ML community are too liberal with their use of terminology. Some, like Kevin Murphy in Chapter 14 of his ML book, bother to mention that the term kernel means different things depending on the context. In particular, he distinguishes a Mercer kernel, which by definition is symmetric and induces a positive semidefinite Gram matrix (Mercer's theorem says you always have a corresponding $\phi(\cdot)$, possibly infinitely dimensional). In contrast, weights arising in the Nadaraya-Watson model (also called "kernels" sometimes) are not even symmetric. $\endgroup$– paperskilltreesCommented Jun 21, 2021 at 21:44
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Yes, the two kinds of kernel functions are just distinct concepts. They do have related intuitions as similarity functions, but any connections beyond that are (to may knowledge) not particularly thorough.