# Matérn Gaussian Processes

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

$\nu = 1/2$

$\nu = 3/2$

$\nu = 5/2$

$\nu = \infty$

# Weighted Undirected Graphs

$$f : G \to \R$$

$$f\Big(\smash{\includegraphics[height=2.75em,width=1.5em]{figures/g1.svg}}\Big) \to \R$$

$$f\Big(\smash{\includegraphics[height=2.75em,width=1.5em]{figures/g2.svg}}\Big) \to \R$$

$$f\Big(\smash{\includegraphics[height=2.75em,width=1.5em]{figures/g3.svg}}\Big) \to \R$$

# Stochastic Partial Differential Equations

$$\htmlClass{fragment}{ \underset{\t{Matérn}}{\undergroup{\del{\frac{2\nu}{\kappa^2} - \Delta}^{\frac{\nu}{2}+\frac{d}{4}} f = \c{W}}} } \qquad \htmlClass{fragment}{ \underset{\t{squared exponential}}{\undergroup{\vphantom{\del{\frac{2\nu}{\kappa^2} - \Delta}^{\frac{\nu}{2}+\frac{d}{4}}} e^{-\frac{\kappa^2}{4}\Delta} f = \c{W}}} }$$ $\Delta$: Laplacian $\c{W}$: (rescaled) white noise
$e^{-\frac{\kappa^2}{4}\Delta}$: (rescaled) heat semigroup

# The Graph Laplacian

$$\htmlClass{fragment}{ (\m\Delta\v{f})(x) = \sum_{x' \~ x} w_{xx'} (f(x) - f(x')) }$$ $$\htmlClass{fragment}{ \m\Delta = \m{D} - \m{W} }$$ $\m{D}$: degree matrix $\m{W}$: (weighted) adjacency matrix

# Graph Matérn Gaussian Processes

$$\htmlClass{fragment}{ \underset{\t{Matérn}}{\undergroup{\del{\frac{2\nu}{\kappa^2} + \m\Delta}^{\frac{\nu}{2}} \v{f} = \c{W}\mathrlap{\hspace*{-2.42ex}\c{W}\hspace*{-2.42ex}\c{W}}}} } \qquad \htmlClass{fragment}{ \underset{\t{squared exponential}}{\undergroup{\vphantom{\del{\frac{2\nu}{\kappa^2} - \m\Delta}^{\frac{\nu}{2}+\frac{d}{4}}} e^{\frac{\kappa^2}{4}\m\Delta} \v{f} = \c{W}\mathrlap{\hspace*{-2.42ex}\c{W}\hspace*{-2.42ex}\c{W}}}} }$$ $\m\Delta$: graph Laplacian $\c{W}\mathrlap{\hspace*{-2.42ex}\c{W}\hspace*{-2.42ex}\c{W}}$: standard Gaussian

# Graph Matérn Gaussian Processes

$$\htmlClass{fragment}{ \underset{\t{Matérn}}{\undergroup{\vphantom{\v{f} \~\f{N}\del{\v{0},e^{-\frac{\kappa^2}{4}\m\Delta}}} \v{f} \~\f{N}\del{\v{0},{\textstyle\del{\frac{2\nu}{\kappa^2} + \m\Delta}^{-\nu}}}}} } \qquad \htmlClass{fragment}{ \underset{\t{squared exponential}}{\undergroup{\v{f} \~\f{N}\del{\v{0},e^{-\frac{\kappa^2}{2}\m\Delta}}}} }$$

# Graph Fourier Features

$$\htmlClass{fragment}{ k_\nu(x,x') = \frac{\sigma^2}{C_\nu} \sum_{n=1}^{|G|} \del{\frac{2\nu}{\kappa^2} + \lambda_n}^{-\nu} \v{f}_n(x)\v{f}_n(x') }$$ $\lambda_n,\v{f}_n$: eigenvalues and eigenvectors of graph Laplacian

# Thank you!

### https://avt.im/· @avt_im

V. Borovitskiy, I. Azangulov, A. Terenin, P. Mostowsky, M. P. Deisenroth. Matérn Gaussian Processes on Graphs. Artificial Intelligence and Statistics, 2021.

V. Borovitskiy, A. Terenin, P. Mostowsky, M. P. Deisenroth. Matérn Gaussian Processes on Riemannian Manifolds. Advances in Neural Information Processing Systems, 2020.