kernel

Definition

Let $\mathcal{X}$ be a non-empty set. A function $k : \mathcal{X} \times \mathcal{X} \to \mathbb{R}$ is a kernel if there exists a $\mathbb{R}$ -Hilbert space and a map $\phi: \mathcal{X} \to \mathcal{H}$ such that $\forall x,x' \in \mathcal{X}$ ,

k(x,x') := \langle \phi(x),\phi(x')\rangle_\mathcal{H}

Notes

theorem sums of kernels $k_1 + k_2$ are kernels, and $\alpha k$ is a kernel for $\alpha >0$

References

http://mlss.tuebingen.mpg.de/2015/slides/gretton/part_1.pdf