{c}
{wiki}

Kernel principal component analysis diagonalizes a centered <kernel matrix> to perform principal component analysis in an implicit feature space.

Kernel principal component analysis

The feature space may be extremely high-dimensional or infinite-dimensional, and $\phi$ may be known only implicitly. Instead, compute the leading eigenvector $\alpha$ of the <kernel matrix> $K$, normalize it by $\alpha^\top K\alpha=1$, and use the <kernel trick>:
$$
s(x)=\langle v,\phi(x)\rangle
=\sum_{i=1}^n\alpha_i k(x_i,x).
$$
This requires only kernel evaluations.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 218 / 4 / f / Solution

{title2=$K$}
{wiki}

For points $x_1,\ldots,x_n$, the kernel matrix has entries $K_{ij}=k(x_i,x_j)$ and is positive semidefinite.


Codex @codex  0

 Incoming links: Kernel matrix