Goal: minimize unknown function $\phi$ in as few evaluations as possible.

1. Build posterior $f \given \v{y}$ using data $(x_1,\phi(x_1)),..,(x_n,\phi(x_n))$.
2. Choose $$ \htmlClass{fragment}{ x_{n+1} = \argmin_{x\in\c{X}} \alpha_n(x) } $$ to optimize the acquisition function $\alpha_n$, for instance Thompson sampling $\alpha_n \~ f\given\v{y}$.

Automatic explore-exploit tradeoff.

$\nu = 1/2$

$\nu = 3/2$

$\nu = 5/2$

$\nu = \infty$

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

How should Matérn kernels generalize to this setting?

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

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

Need a different candidate generalization

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

Generalizes well to the Riemannian setting

$$ k_\nu(x,x') = \frac{\sigma^2}{C_\nu} \sum_{n=0}^\infty \del{\frac{2\nu}{\kappa^2} - \lambda_n}^{\nu-\frac{d}{2}} f_n(x) f_n(x') $$

$\lambda_n, f_n$: Laplace–Beltrami eigenpairs

Analytic expressions for circle, sphere, ..

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

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

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

Note: different sign convention, analogous to Euclidean $-\Delta$

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

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

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

(a) Complete graph

(b) Star graph

(c) Random graph

(d) Random graph

Prior variance depends on geometry

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Same vector field: represented differently in different frames

Construct vector fields by embedding and flattening scalar fields