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

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- Solution i

Solution

Take the real L2 space

H = L^{2} ((0, \infty), d t)

and the feature map

φ (x) (t) = 1_{(0, x]} (t) .

Each indicator function belongs to

H

because its squared L2 norm is

x < \infty

. Their inner product is the length of the overlapping intervals:

⟨ φ (x), φ (x^{'})⟩ = \int_{0}^{\infty} 1_{(0, x]} (t) 1_{(0, x^{'}]} (t) d t = min (x, x^{'}) = k_{1} (x, x^{'}) .

This realizes

k_{1}

as a positive-semidefinite kernel; it is also the Brownian covariance kernel.

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