In this work we present kernel functions and their use in kernel methods. We state the general problem of binary classification and its solution in case of homogeneously linearly separable dataset. Through Cover's theorem we recognize the need for kernel functions. We present reproducing kernel Hilbert spaces and prove the Moore–Aronszajn and representer theorems. We take a look at a few examples of kernel functions and the rules for constructing new ones. We cover three important kernel methods in depth: support vector machines, principal component analysis and Gaussian processes. We also present the link between Gaussian processes and other machine learning methods.
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