How to prove this function related to absolute distance is a positive definite kernel?

Question

Prove $$K(x,y)=\frac{1}{1+||x-y||}$$ is a kernel for $x,y\in\mathbb{R}^d$.

I think one might use the fact $$K(x,y)=\int_0^{+\infty} e^{-t\left(1+||x-y||\right)}dt$$ like what we did in proving Hilbert matrix is positive definite. How do we proceed from here?

Answer 1

Let $X=(x_i)_{1\leq i\leq n} \in \cal{M}_{n,1}(\mathbb{R})$ and denote $A$ the $n \times n$ matrix having for coefficients $K(c_i,c_j)$ where $\{c_1, \dots, c_n\} \in \mathbb R^d$ are distinct. We have $$\begin{aligned} {}^tXAX&=\sum_{1\leq i,j\leq n}\frac{x_ix_j}{1 + \Vert c_i - c_j \Vert}\\ &=\sum_{1\leq i,j\leq n}x_ix_j\int_0^\infty e^{-t\left(1+\Vert c_i - c_j \Vert\right)}dt\\ &=\int_0^\infty \left(\sum_{i=1}^nx_ie^{-\frac{t}{2}\left(1+\Vert c_i - c_j \Vert\right)}\right)^2dt>0 \end{aligned} $$ for $X\neq0$, giving the announced result since

• $A$ is symmetric,
• the $c_i$ distinct,
• $(e^{\alpha x})_{\alpha \in \mathbb R}$ is a family of independent functions of $\mathcal C([0,\infty), \mathbb R)$, hence $t \mapsto \sum_{i=1}^nx_ie^{-\frac{t}{2}\left(1+\Vert c_i - c_j \Vert\right)}$ is a continuous function not always vanishing,
• the integral of a non negative continuous not always vanishing function is positive.