Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 205 / 1 / b / ii

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- Solution ii

Solution

Let

V_{1}, \dots, V_{d}

be independent standard Cauchy random variables. Their characteristic functions and independence give

E e^{iλ V^{T} (x - y)} = \prod_{j = 1}^{d} e^{- λ ∣ x_{j} - y_{j} ∣} = e^{- λ ∥ x - y ∥_{1}} .

Taking real parts yields

e^{- λ ∥ x - y ∥_{1}} = E [cos (λ V^{T} x) cos (λ V^{T} y) + sin (λ V^{T} x) sin (λ V^{T} y)] .

For each realization of

V

, both products are rank-one positive-semidefinite kernels. Their sum and then their expectation remain positive semidefinite by the closure property of positive-semidefinite kernels. This is a Random Fourier feature representation of the Laplace kernel.

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