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

Reciprocation reverses the order of positive numbers, so

k_{3} (x, x^{'}) = \frac{1}{max ( x , x ^{'} )} = min (\frac{1}{x}, \frac{1}{x ^{'}}) = k_{1} (1/ x, 1/ x^{'}) .

This is the pullback of the positive-semidefinite kernel from part (i) by

x \mapsto 1/ x

. Explicitly, the feature map into the same L2 space is

ψ (x) (t) = 1_{(0, 1/ x]} (t), ⟨ ψ (x), ψ (x^{'})⟩ = \frac{1}{m a x ( x , x ^{'} )} .

Hence $k_{3}$ is a positive-semidefinite kernel, with

∥ ψ (x) ∥_{2}^{2} = 1/ x < \infty

for every allowed input.

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