A kernel k is a function that for all x, z ∈ \(\mathbb{X}\): satisfies \(k({\bf{x}},{\bf{z}}) = \rm \Phi ({\bf{x}}) \cdot \rm \Phi ({\bf{z}})\), where Φ is a mapping from the input space \(\mathbb{X}\) to the feature space ℌ, i.e., \(\rm \Phi :\;x \mapsto\ \rm \Phi (x)\) ∈ ℌ. A kernel function can also be characterized as follows: Let \(\mathbb{X}\) be the input space. A function k: \(\mathbb{X}\) × \(\mathbb{X}\): ↦ ℝ (or ℂ) is kernel if and only if for any M ∈ ℕ and any finite data set \(\{ {\bf{x}}_1 , \cdots ,{\bf{x}}_M \}\) ⊂ \(\mathbb{X}\), the associated Gram matrix is positive semi-definite.
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(2009). Kernel. In: Li, S.Z., Jain, A. (eds) Encyclopedia of Biometrics. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-73003-5_583
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DOI: https://doi.org/10.1007/978-0-387-73003-5_583
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