Past exam of the advanced topics in statistical machine learning course of the University of Oxford / 2025 / 2 / a / iii / Solution

The closure property of positive-semidefinite kernels gives three constructions, each seen directly from the feature maps ${\phi }_1,{\phi }_2$.

Another useful consequence is pullback: if $h:\mathcal{X}\to \mathcal{Z}$ and $k_0$ is a positive-semidefinite kernel on $\mathcal{Z}$, then $k_0(h(x),h(x^{\prime }))$ has feature map ${\phi }_0\circ h$.