Laplace kernel ($k(x,y)=e^{-\lambda \Vert x-y{\Vert }_1}$)

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.