Definition. A Gaussian process is random function $f : X \to \R$ such that for any $x_1,..,x_n$, the vector $f(x_1),..,f(x_n)$ is multivariate Gaussian.

The distribution of a Gaussian process is characterized by

• a mean function $m(x) = \E(f(x))$,
• a kernel (covariance) function $k(x, x') = \Cov(f(x), f(x'))$,

Notation: $f \~ \f{GP}(m, k)$.

The kernel $k$ must be positive (semi-)definite, i.e. for all $x_1, .., x_n \in X$
the matrix $K_{\v{x} \v{x}} := \cbr{k(x_i, x_j)}_{\substack{1 \leq i \leq n \\ 1 \leq j \leq n}}$ must be positive (semi-)definite.

Gaussian Process Regression takes

• a prior Gaussian Process $\f{GP}(m, k)$
• and data $(x_1, y_1), .., (x_n, y_n) \in X \x \R$,

giving the posterior (conditional) Gaussian Process $\f{GP}(\hat{m}, \hat{k})$.

The functions $\hat{m}$ and $\hat{k}$ may be explicitly expressed in terms of $m$ and $k$.

Also

• geostatistics,
• robotics,
• more...

$$ \htmlData{class=fragment fade-out,fragment-index=9}{ \footnotesize \mathclap{ k_{\nu, \kappa, \sigma^2}(x,x') = \sigma^2 \frac{2^{1-\nu}}{\Gamma(\nu)} \del{\sqrt{2\nu} \frac{\norm{x-x'}}{\kappa}}^\nu K_\nu \del{\sqrt{2\nu} \frac{\norm{x-x'}}{\kappa}} } } \htmlData{class=fragment d-print-none,fragment-index=9}{ \footnotesize \mathclap{ k_{\infty, \kappa, \sigma^2}(x,x') = \sigma^2 \exp\del{-\frac{\norm{x-x'}^2}{2\kappa^2}} } } $$ $\sigma^2$: variance $\kappa$: length scale $\nu$: smoothness
$\nu\to\infty$: Gaussian kernel (RBF)

$\nu = 1/2$

$\nu = 3/2$

$\nu = 5/2$

$\nu = \infty$

Manifolds

Graphs

How should Matérn kernels generalize to this setting?

$$ k_{\infty, \kappa, \sigma^2}^{(d_g)}(x,x') = \sigma^2\exp\del{-\frac{d_g(x,x')^2}{2\kappa^2}} $$

Theorem. (Feragen et al.) Let $M$ be a complete Riemannian manifold without boundary. If $k_{\infty, \kappa, \sigma^2}^{(d_g)}$ is positive semi-definite for all $\kappa$, then $M$ is isometric to a Euclidean space.

Need a different candidate generalization

$$ \htmlData{class=fragment,fragment-index=0}{ \underset{\t{Matérn}}{\undergroup{\del{\frac{2\nu}{\kappa^2} - \Delta}^{\frac{\nu}{2}+\frac{d}{4}} f = \c{W}}} } \qquad \htmlData{class=fragment,fragment-index=1}{ \underset{\t{Gaussian}}{\undergroup{\vphantom{\del{\frac{2\nu}{\kappa^2} - \Delta}^{\frac{\nu}{2}+\frac{d}{4}}} e^{-\frac{\kappa^2}{4}\Delta} f = \c{W}}} } $$ $\Delta$: Laplacian $\c{W}$: Gaussian white noise

Generalizes well to the Riemannian/graph setting

$$ \htmlData{class=fragment,fragment-index=0}{ \del{\frac{2\nu}{\kappa^2} - \Delta}^{\frac{\nu}{2}+\frac{d}{4}} f = \c{W} } $$

The solution is a Gaussian process with kernel $$ \htmlData{class=fragment}{ k_{\nu, \kappa, \sigma^2}(x,x') = \frac{\sigma^2}{C_\nu} \sum_{n=0}^\infty \del{\frac{2\nu}{\kappa^2} - \lambda_n}^{-\nu-\frac{d}{2}} f_n(x) f_n(x') } $$

$\lambda_n, f_n$: eigenvalues and eigenfunctions of the Laplace–Beltrami

$k_{1/2}(\htmlStyle{color:rgb(255, 19, 0)!important}{\bullet},\.)$

The solution is a Gaussian process with kernel $$ \htmlData{class=fragment}{ k_{\nu, \kappa, \sigma^2}(i, j) = \frac{\sigma^2}{C_{\nu}} \sum_{n=0}^{\abs{V}-1} \del{\frac{2\nu}{\kappa^2} + \mathbf{\lambda_n}}^{-\nu} \mathbf{f_n}(i)\mathbf{f_n}(j) } $$

$\lambda_n, \mathbf{f_n}$: eigenvalues and eigenvectors of the Laplacian matrix

$k_{5/2}(\htmlStyle{color:rgb(255, 19, 0)!important}{\circ},\.)$

A naive Euclidean model

Wind speed extrapolation on a map

The corresponding model on a globe

A geometry-aware model

Wind speed extrapolation on a map

The corresponding model on a globe