= Laplace kernel
{c}
{title2=$k(x,y)=e^{-\lambda\|x-y\|_1}$}
For $\lambda\geq0$, the <Laplace kernel> on $\mathbb R^d$ is
$$
k(x,y)=\exp(-\lambda\|x-y\|_1).
$$
For $\lambda>0$, take <independent random variables> $\omega_j$ with <Cauchy distribution> of scale $\lambda$. Their <characteristic functions> give
$$
\mathbb E e^{i\omega\cdot(x-y)}=\prod_j e^{-\lambda|x_j-y_j|}=k(x,y).
$$
Each 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>.
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