Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 205 / 3

A positive-semidefinite kernel on $\mathcal{X}$ is a symmetric function $k:{\mathcal{X}}^2\to \mathbb{R}$ such that, for every $x_1,\dots ,x_n$ and $c_1,\dots ,c_n$,If $k(x,x^{\prime })=⟨\varphi (x),\varphi (x^{\prime })⟩$ for a feature map into an inner-product space, thenThus every feature-map inner product is a kernel. If ${\alpha }_1,{\alpha }_2\ge 0$, then each finite quadratic form for ${\alpha }_1{k}_1+{\alpha }_2{k}_2$ is the corresponding nonnegative linear combination, so nonnegative linear combinations of kernels are kernels.

For fixed $x_1,\dots ,x_n$ and coefficients $c_i$, every $k_m$ satisfiesThe sum is finite, so pointwise convergence permits passage to the limit:Symmetry also passes to the limit. Hence a pointwise limit of kernels is a kernel.

For a finite sample, the Gram matrix of $k_1{k}_2$ is the entrywise product of the two positive-semidefinite Gram matrices. The Schur product theorem makes it positive semidefinite, proving the product closure.

The Gaussian kernel with bandwidth ${\sigma }^2$ isFactor it asThe linear kernel $x^T{x}^{\prime }$ is positive semidefinite; products and nonnegative scalar multiples preserve positivity, as does multiplication by $a(x)a(x^{\prime })$. The partial sums are therefore kernels, and pointwise-limit closure proves that the Gaussian kernel is positive semidefinite.