Definition. A Gaussian process is random function $f$ on a set $X$ such that for any $x_1,..,x_n \in X$, the vector $f(x_1),..,f(x_n)$ is multivariate Gaussian.

May refer to a random function / distribution, depending on the context.

Gaussian processes are 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.

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 $k$.

Useful for: Bayesian optimization, geostatistics, general decision making, etc.

$$ \htmlData{class=fragment fade-out,fragment-index=6}{ \footnotesize \mathclap{ k_{\nu, \kappa, \sigma^2}(x,x') = \sigma^2 \frac{2^{1-\nu}}{\Gamma(\nu)} \del{\sqrt{2\nu} \frac{\abs{x-x'}}{\kappa}}^\nu K_\nu \del{\sqrt{2\nu} \frac{\abs{x-x'}}{\kappa}} } } \htmlData{class=fragment d-print-none,fragment-index=6}{ \footnotesize \mathclap{ k_{\infty, \kappa, \sigma^2}(x,x') = \sigma^2 \exp\del{-\frac{\abs{x-x'}^2}{2\kappa^2}} } } $$ $\sigma^2$: variance $\kappa$: length scale $\nu$: smoothness
$\nu\to\infty$: RBF kernel (Gaussian, Heat, Diffusion)

$\nu = 1/2$

$\nu = 3/2$

$\nu = 5/2$

$\nu = \infty$

For manifolds. Not well-defined unless the manifold is isometric to a Euclidean space.
(Feragen et al. 2015)

For graphs. Not well-defined unless nodes can be isometrically embedded into a Hilbert space.
(Schonberg, 1930s)

For spaces of graphs. What is a space of graphs?!

Need a different generalization

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

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

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

Mesh the manifold, consider the discretized Laplace–Beltrami (a matrix).

For manifolds with rich group of symmetries, termed homogeneous spaces.

These include

• Lie groups, like rotations $\mathrm{SO}(n)$ and unitary matrices $\mathrm{SU}(n)$,
• «Proper» homogeneous spaces, e.g. spheres $\mathbb{S}_d$, projective spaces $\mathrm{RP}_d$, Stiefel manifolds $\mathrm{V}(k, n)$ or Grassmannians $\mathrm{Gr}(n, r)$.

$\lambda_n, f_n$ are connected to representation theory of the group of symmetries.

Sphere $\mathbb{S}_2$

Torus $\mathbb{T}^2 = \bb{S}_1 \x \bb{S}_1$

Projective plane $\mathrm{RP}^2$

Driven by generalized random phase Fourier features (GRPFF).

Take stationary $k$. Assume for simplicity $k(x, x) = 1$. Then

$k$ is RBF $\implies$ $S(\lambda)$ is Gaussian.   $k$ is Matérn $\implies$ $S(\lambda)$ is $t$ distributed.

$k$ is RBF $\implies$ $S(\lambda)$ is Gaussian.   $k$ is Matérn $\implies$ $S(\lambda)$ is $t$ distributed.

• $\pi^{(\lambda)}$ is an explicit integral. • $c(\lambda)$ has closed form.
• $c(\lambda)^{-2} S(\lambda)$ is a non-standard, potentially unnormalized, density.

Geometry-aware vs Euclidean

SPDE turns into a stochastic linear system. The solution has kernel $$ \htmlData{fragment-index=2,class=fragment}{ k_{\nu, \kappa, \sigma^2}(i, j) = \frac{\sigma^2}{C_{\nu, \kappa}} \sum_{n=0}^{\abs{V}-1} \Phi_{\nu, \kappa}(\lambda_n) \mathbf{f_n}(i)\mathbf{f_n}(j) } $$

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

https://geometric-kernels.github.io/

Since $e^{2 \pi i \innerprod{x - x'}{\lambda_l}} = e^{2 \pi i \innerprod{x}{\lambda_l}} \overline{e^{2 \pi i \innerprod{x'}{\lambda_l}}}$, the above is an inner product.

$$ \htmlData{class=fragment}{ f(x) \approx \frac{1}{\sqrt{L}} \sum_{l=1}^L w_l e^{2 \pi i \innerprod{x}{\lambda_l}} \qquad w_l \sim \mathrm{N}(0, 1) \qquad \lambda_l \sim S(\lambda) } $$

Since $ \htmlData{fragment-index=3,class=fragment fade-out disappearing-fragment}{ e^{2 \pi i \innerprod{x - x'}{\lambda_l}} } \htmlData{fragment-index=3,class=fragment fade-in appearing-fragment}{ {\color{blue}\pi^{(\lambda_l)}(x, x')} } \!\,= \htmlData{fragment-index=4,class=fragment fade-out disappearing-fragment}{ e^{2 \pi i \innerprod{x}{\lambda_l}} \overline{e^{2 \pi i \innerprod{x'}{\lambda_l}}} } \htmlData{fragment-index=4,class=fragment fade-in appearing-fragment}{ {\color{blue} \pi^{(\lambda_l)}(x, ?) \overline{\pi^{(\lambda_l)}(x', ?)} } } $, the above is an inner product. $ \vphantom{ e^{2 \pi i \innerprod{x - x'}{\lambda_l}} {\color{blue}\pi^{(\lambda_l)}(x, x')} \!\,= e^{2 \pi i \innerprod{x}{\lambda_l}} \overline{e^{2 \pi i \innerprod{x'}{\lambda_l}}} {\color{blue} \pi^{(\lambda_l)}(x, ?) \overline{\pi^{(\lambda_l)}(x', ?)} } } \sout{ {\color{blue}\pi^{(\lambda_l)}(x, x')} = {\color{blue} \pi^{(\lambda_l)}(x, ?) \overline{\pi^{(\lambda_l)}(x', ?)} } } $, $ \vphantom{ e^{2 \pi i \innerprod{x - x'}{\lambda_l}} {\color{blue}\pi^{(\lambda_l)}(x, x')} \!\,= e^{2 \pi i \innerprod{x}{\lambda_l}} \overline{e^{2 \pi i \innerprod{x'}{\lambda_l}}} {\color{blue} \pi^{(\lambda_l)}(x, ?) \overline{\pi^{(\lambda_l)}(x', ?)} } } {\color{blue}\pi^{(\lambda_l)}(x, x')} = \E_{h \sim \mu_H} e^{\innerprod{i \lambda + \rho}{\,a(h, x)}} \overline{ e^{\innerprod{i \lambda + \rho}{\,a(h, x')}} }, $ where

$$ \begin{aligned} \htmlData{fragment-index=1,class=fragment}{k(x,y)} &\htmlData{fragment-index=2,class=fragment}{ = \frac{\sigma^2}{C_{\nu, \kappa}} \sum_{n=0}^\infty \Phi_{\nu, \kappa}(\lambda_n) f_n(x) \mathrlap{f_n(y)} \htmlData{fragment-index=3,class=fragment}{ \obr{\phantom{f_n(y)}}{\hspace{-0.5cm}\text{spherical harmonics}\hspace{-0.5cm}} } } \\ &\htmlData{fragment-index=4,class=fragment}{= \frac{\sigma^2}{C_{\nu, \kappa}} \sum_{n=0}^\infty \Phi_{\nu, \kappa}(\lambda_n) \mathrlap{\del{\sum_{k=1}^{d_n} f_{n, k}(x) f_{n, k}(y)}} \htmlData{fragment-index=5,class=fragment}{ \ubr{\phantom{\del{\sum_{k=1}^{d_n} f_{n, k}(x) f_{n, k}(y)}}}{\hspace{-0.7cm} C_{n, d} \cdot \phi_{n, d}(x, y) \text{ --- zonal spherical harmonics}\hspace{-0.7cm}} } } \end{aligned} $$

The last equation

• is much more efficient for computing $k$,
• generalizes: zonal spherical harmonics → zonal ??????? ???????spherical functions,
• unfortunately, is unfit for sampling. Or is it?

Zonal spherical harmonics satisfy reproducing property:

Hence $\sqrt{C_{n, d}/L} \cdot \phi_{n, d}(x, \v{u})$ forms an approximate feature transform.

This enables efficient sampling without knowing $f_n$. And this generalizes!