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For $\lambda \ge 0$, the Laplace kernel on ${\mathbb{R}}^d$ isFor $\lambda >0$, take independent random variables ${\omega }_j$ with Cauchy distribution of scale $\lambda$. Their characteristic functions giveEach sampled frequency has a two-coordinate feature map $(\cos (\omega \cdot x),\sin (\omega \cdot x))$. Its inner product gives the cosine of the difference, so averaging proves the positive-semidefinite kernel property and yields Random Fourier features. At $\lambda =0$, the Laplace kernel is the constant-one positive-semidefinite kernel. Negative $\lambda$ would violate the two-point Cauchy-Schwarz inequality for positive-semidefinite kernels.

Let $V_1,\dots ,V_d$ be independent standard Cauchy random variables. Their characteristic functions and independence giveTaking real parts yieldsFor 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.