The Gaussian kernel is positive definite

ProofIntermediate⏱ 30 minClassical MLkernelpositive definiteschurconstruction

Theory: Kernels, RKHS and the Representer Theorem →

Problem

Prove that $K(x,x') = \exp(-\|x-x'\|_2^2/2\sigma^2)$ is a positive definite kernel, using only the closure properties (sums, products, rank-one, limits) and the fact that $\langle x,x'\rangle$ is positive definite.

Write a complete argument. Every inequality you use should be one you could state precisely.

Solution

Opening this before you have a real attempt on paper is the fastest way to feel like you understand something you cannot reproduce.

Related problems

All problems

Problem	Field	Type	Difficulty	Time
Implement kernel ridge regression run	Classical ML	Coding	Intermediate	1 hour
Kernel ridge regression in closed form	Classical ML	Proof	Intermediate	30 min
The reproducing property and its consequences	Classical ML	Proof	Advanced	30 min
The Schur product theorem	Classical ML	Proof	Advanced	30 min
Prove the representer theorem abyss	Classical ML	Proof	Graduate	1 hour