Schoenberg's theorem on negative definite kernels

Remarks

Proof idea: from 2 to 1, if $\sum_i c_i = 0$, then $\sum_{i,j} c_i c_j \frac{1 - e^{-t\Psi(x_i, x_j)}}{t} = -\frac{1}{t}\sum_{i,j} c_i c_j e^{-t\Psi(x_i, x_j)} \le 0$, and the left side tends to $\sum_{i,j} c_i c_j \Psi(x_i, x_j)$ as $t \to 0$. From 1 to 2, fix $x_0 \in X$; then $\Phi(x, y) = \Psi(x, x_0) + \Psi(y, x_0) - \Psi(x, y)$ is a positive-definite kernel. By the Schur product theorem, $e^{t\Phi}$ is positive-definite, and $e^{-t\Psi(x, y)} = e^{t\Phi(x, y)}\, e^{-t\Psi(x, x_0)}\, e^{-t\Psi(y, x_0)}$ is again positive-definite.

Example: $\Psi(x, y) = \abs{x - y}^2$ on $\R^n$ is conditionally negative definite, so every Gaussian kernel $e^{-t\abs{x - y}^2}$ is positive-definite.

For a group $G$, applied to $\Psi(g, h) = \psi(h^{-1}g)$, the theorem says that $\psi$ is a conditionally negative definite function exactly when $e^{-t\psi}$ is a positive-definite function for every $t > 0$.

Going deeper

For a student

Schoenberg's theorem is the standard way to produce positive-definite kernels, such as the Gaussian kernel, from conditionally negative definite ones. For groups, it links affine isometric actions on Hilbert space, which give conditionally negative definite functions, with unitary representations, which give positive-definite functions. This is the mechanism behind the Delorme–Guichardet theorem on property (T) and behind the Haagerup property. In probability, a complex-valued version of the theorem underlies the Lévy–Khintchine description of the characteristic exponents of Lévy processes.