Clearly, the latter approach is more elegant and more general, but it is also also more effortful and potentially less constructive.
Since we are ultimately interested in working with Gaussian vector fields algorithmically, we adopt the former approach, but keep the latter view in mind to ensure soundness of our overall strategy.
Recall that the vector direct sum is denoted by $\oplus$.
We proceed as follows.

\begin{definition}[Gaussian vector field]
Let $X$ be a smooth manifold.
We say that a random section $f : \Omega \x X \-> TX$ is \emph{Gaussian} if for any finite set of locations $x_1, \ldots, x_n \in X$, the random vector $(f(\.,x_1),\ldots,f(\.,x_n)) \in T_{x_1}X \oplus \ldots \oplus T_{x_n}X$ is Gaussian in the sense of duality.
\end{definition}

Thus, we see that in spite of the fact that a tangent bundle is a manifold, and not a vector space, it possesses enough linear structure to admit a sensible notion of finite-dimensional marginals.
Here, we immediately see the value of the duality-based approach to Gaussianity originally developed in \Cref{ch:intro}: by using this, rather than working with bases, we avoid the need for cumbersome change-of-basis consistency checks.
We now examine this notion's view from the vantage point of an embedding into Euclidean space.

\begin{proposition}
Let $i : X \-> \R^p$ be a smooth embedding. Then
\[
f_i : \Omega \x i(X) &\-> \R^p
f_i(\omega,x) &= i_* f(\omega, i^{-1}(x))&
\]
is a vector-valued Gaussian process in the standard sense.
\end{proposition}

\begin{proof}
Since the embedding $i$ is bijective on its image, any set of locations $x_1^{(i)},..,x_n^{(i)} \in i(X)$, will map uniquely onto a set of locations $x_1,..,x_n \in X$.
By definition of vector pushforward maps, $i_*$ maps tangent vectors from $T_{x_1} X \oplus .. \oplus T_{x_n} X$ into $\R^{n\times p}$ linearly.
Since Gaussianity is preserved by linearity, every set of finite-dimensional marginals in the embedded process is Gaussian, and the claim follows.
\end{proof}

This confirms that our definition of a vector-valued Gaussian process is the right one: if we embed the manifold in a Euclidean space, we obtain the correct kind of stochastic process.
\Cref{fig:gvf} illustrates a set of random samples from a Gaussian vector field, constructed using the technique presented in the sequel.

As before, the mean of such a process will simply be a given vector field.
The next step is to determine what is an appropriate notion of a \emph{kernel}.
In the Euclidean case, a \emphmarginnote{matrix-valued kernel} is a symmetric positive semi-definite function $k : X \x X \-> \R^{d \x d}$.
This notion more-or-less completely analogous to the scalar-valued case.
A priori, this notion seems completely coordinate-dependent due to the matrix appearing in the output.
To avoid this, we instead consider the map
\[
((x,\v{v}), (x',\v{v}')) \|> \v{v}^T \m{K}_{xx'} \v{v}'
\]
which describes the action of the kernel matrix on vectors.
To continue, we need an appropriate notion of bilinearity for the given setting.
We formulate this notion in general vector bundles, though we are interested in the tangent bundle $TX$ whose fibers are the vector spaces $T_x X$.

\begin{definition}[Fiberwise bilinear function]
Let $B$ be a vector bundle over $X$ with fibers $V_x$, $x\in X$.
We say that a function $k : B \x B \-> \R$ \emph{fiberwise bilinear} if for all pairs of points $x, x' \in X$ we have that
\[
k(\kappa \alpha_x + \lambda \beta_x, \gamma_{x'}) &= \kappa k(\alpha_x, \gamma_{x'}) + \lambda k(\beta_x, \gamma_{x'})
\\
k(\alpha_x, \mu \gamma_{x'} + \nu \delta_{x'}) &= \mu k(\alpha_x, \gamma_{x'}) + \nu k(\alpha_x, \delta_{x'})
\]
for any $\alpha_x, \beta_x \in V_x$, $\gamma_{x'}, \delta_{x'} \in V_{x'}$ and $\kappa, \lambda, \mu, \nu \in \R$.
\end{definition}

This leads to the following notion of a kernel.

\begin{definition}[Positive semi-definite kernel]
A symmetric fiberwise bilinear function $k : T^* X \x T^* X \-> \R$ is called a \emph{positive semi-definite kernel} if for any set of covectors $\alpha_{x_1}, \ldots, \alpha_{x_n} \in T^*X$, we have
\[
\sum_{i=1}^n\sum_{j=1}^n k(\alpha_{x_i}, \alpha_{x_j}) \geq 0
\]
\end{definition}

At first, this may seem surprising: why should the kernel be a function defined on the \emph{cotangent} bundle, rather than the tangent bundle?
A clue is given by the form of the multivariate Gaussian density, which contains a $\v{x}^T\m{K}_{\v{x}\v{x}}^{-1}\v{x}$ term---note the presence of the inverse.
This suggests that since the inverse kernel matrix acts on vectors, the kernel matrix should act on covectors.
This mirrors the duality-based view described in \Cref{ch:intro}, and is clarified further by considering the kernel of a given Gaussian vector field.

\begin{definition}[Cross-covariance kernel]
The \emph{cross-covariance kernel} of a Gaussian vector field is defined as the map
\[
\alpha_x, \beta_{x'} \|> \Cov(\dualprod{\alpha_x}{f(\.,x)}, \dualprod{\beta_{x'}}{f(\.,x')}).
\]
\end{definition}

We now verify that this is indeed the correct notion of a kernel for the given setting.
To do this, we need a general form of the Kolmogorov Extension Theorem, given by the following result.

\begin{result}[Kolmogorov Extension Theorem]
Let $(X_\alpha,\c{B}_\alpha,\c{O}_\alpha)_{\alpha\in A}$ be a family of measurable spaces, each equipped with a topology.
For each finite $B \subseteq A$, let $\mu_B$ be an inner regular probability measure on $X_B = \bigtimes_{\alpha\in B} X_\alpha$ equipped with the product $\sigma$-algebra $\c{B}_B$ and product topology $\c{O}_B$, obeying
\[
\del{\proj_C}_* \mu_B = \mu_C
\]
whenever $C \subseteq B \subseteq A$ are two nested finite subsets of $A$, where projections $\proj_C: X_B \-> X_C$ are defined by $\proj_C(\cbr{x_\alpha}_{\alpha \in B}) = \cbr{x_\alpha}_{\alpha \in C}$, and $\del{\proj_C}_*$ denotes the measure-theoretic pushforward by $\proj_{C}$.
Then there exists a unique probability measure $\mu_A$ on $\c{B}_A$ with the property that $\del{\proj_B}_* \mu_A = \mu_B$ for all finite $B \subseteq A$.
\end{result}

\begin{proof}
\textcite[Theorem 2.4.3]{tao11}.
\end{proof}

We are now ready to prove our bijective correspondence.

\begin{theorem}
The distribution of every Gaussian vector field is uniquely determined by its mean vector field and cross-covariance kernel.
Moreover, each such pair defines a Gaussian vector field.
\end{theorem}

\begin{proof}
Let $x_1,..,x_n \in X$ and for $\alpha = (\alpha_{x_1},..,\alpha_{x_n})$ and $\beta = (\beta_{x_1},..,\beta_{x_n})$ define
\[
\mu_{x_1,..,x_n} &= (\mu(x_1),..,\mu(x_n))
k_{x_1,..,x_n}(\alpha, \beta) &= \sum_{i=1}^n \sum_{j=1}^n k(\alpha_{x_i}, \beta_{x_j})
\]
Let $\pi_{x_1,..,x_n} \~[N](\mu_{x_1,..,x_n}, k_{x_1,..,x_n})$ be defined in the sense of duality.
These are our finite-dimensional marginals: by Gaussianity, it is clear these are uniquely determined by the mean and kernel.
Taking $X$ to be the index set, and $(T_x X)_{x\in X}$ with the standard topology and Borel $\sigma$-algebra to be our measurable spaces, we claim that the family of measures $(\pi_{x_1,..,x_n})_{\{x_1,..,x_n\} \subseteq X}$ satisfies the requirements of Kolmogorov's Extension Theorem.
First, for any $\{x_1,..,x_m\} \subseteq \{x_1,..,x_n\}$, by direct calculation we have
\[
(\proj_{x_1,..,x_m})_* \pi_{x_1,..,x_n} = \pi_{x_1,..,x_m}
\]
using the canonical projection induced by the direct sum.
Next, note that each $\pi_{x_1,..,x_n}$ is a probability measure on a finite-dimensional real space, it is automatically inner regular.
Thus, it follows that there exists a unique measure on the infinite product space $\bigtimes_{x\in X} T_x X$.
This is a topological measure space: if we equip it with the obvious linear structure, and define the linear operator
\[
\c{I} : \bigtimes_{x\in X} T_x X &\-> \Gamma_{\f{nns}}(TX)
(\c{I}s)(x) &= (x, \proj_x s)
\]
where $\Gamma_{\f{nns}}(TX)$ is the space of not necessarily smooth sections, equipped with the pushforward $\sigma$-algebra, we obtain by pushforward the probability distribution of our desired process.
The claim follows.
\end{proof}

\subsection{Coordinate representations}

Using the preceding recipe, we now understand what needs to be defined in order to construct vector-valued Gaussian processes on manifolds.
To make this construction concrete, we proceed to develop the calculations needed to implement such processes numerically.
The first step is to understand how to represent vector fields in coordinates---we do so using bases in each tangent space.

\label{ntn:frame}
\begin{definition}[Frame]
Define a \emph{frame} $F$ to be a collection of not necessarily smooth sections $e_i : X \-> TX$ for $i=1,..,d$ such that, for every $x \in X$, the vectors $e_i(x)$ form a basis of $T_x X$.
Given a frame, define the \emph{coframe} $F^*$ to be the collection of sections $e^i : X \-> T^* X$ defined pointwise using the dual basis with respect to $F$.
\end{definition}

\begin{figure}
\tikzset{external/export next=false}
\begin{tikzpicture}
\node at (0,2.125) {\includegraphics[scale=0.25]{figures/png/s2_a.png}};
\node at (0,0) {\includegraphics[scale=0.25]{figures/png/s2_x.png}};
\node at (2.0625,2.125) {\includegraphics[scale=0.25]{figures/png/s2_b.png}};
\node at (2.0625,0) {\includegraphics[scale=0.25]{figures/png/s2_y.png}};
\node at (3.34375,2.125) {\LARGE\ensuremath{\x}};
\node at (3.34375,0) {\LARGE\ensuremath{\x}};
\node at (4.875,2.125) {\includegraphics[scale=0.25]{figures/png/s2_vab.png}};
\node at (4.875,0) {\includegraphics[scale=0.25]{figures/png/s2_vxy.png}};
\node at (6.125,1.0625) {\Huge\ensuremath{=}};
\node at (8.875,1) {\includegraphics[scale=0.25]{figures/png/s2_xy.png}};
\node at (0.875,3.625) {Basis coefficients};
\node at (4.71875,3.625) {Frame};
\node at (8.53125,3.625) {Vector field};
\end{tikzpicture}
\caption[Representing vector fields in different frames]{A vector field can be represented by specifying its values in any frame. Here, we illustrate the same vector field on the sphere represented using two different frames.}
\label{fig:frame}
\end{figure}

The key words in the above definition are \emph{not necessarily smooth}: recall again that for many manifolds, choosing a frame which varies smoothly in space is impossible---a single component of such a frame would yield a non-vanishing vector field, and such a vector field might not exist.
An example of a vector field represented in two different frames can be seen in \Cref{fig:frame}.
We can use this to calculate the coordinate expression of a cross-covariance kernel with respect to a frame.
This is given by a function $\m{K}_F : X \x X \-> \R^{d \x d}$ as
\[
\m{K}_F(x,x') = \begin{bmatrix} \label{eq:K_F}
k(e^1(x), e^1(x')) & \dots  & k(e^1(x), e^d(x')) \\
\vdots           & \ddots & \vdots \\
k(e^d(x), e^1(x')) & \dots  & k(e^d(x), e^d(x'))
\end{bmatrix}
\]
which is a matrix-valued kernel in the usual sense.
This expression is frame-dependent: to see how it transforms upon a change of frame, introduce another frame $\tl{F}$, and consider a function
\[
f(x) &= \sum_{i=1}^d f^i(x) e_i(x) = \sum_{i=1}^d \tilde{f}^i(x) \tilde{e}_i(x)
\]
expressed with respect to $F$ and $\tl{F}$, where $\tilde{f}^i$ and $g^i$ are scalar-valued functions.
We say that $\tl{F}$ is obtained from $F$ by a \emphmarginnote{change of frame} if there is a matrix-valued function $\m{A} : X \-> \R^{d \x d}$ such that
\[
\v{\tilde{f}}(x) = \m{A}(x) \v{f}(x)
\]
for all $x$.
Since $e_i$ and $\tilde{e}_i$ are bases, such a function always exists.
The range of $\m{A}$ is a subgroup of the general linear group $GL(d,\R)$, and in general depends on what frame-dependent properties one wants to preserve.
For instance, if one wants to preserve orthonormality of bases in each tangent space, one would consider a subgroup of orthonormality-preserving transformations, namely $SO(d)$.
By direct calculation, $\m{K}_F$ can be re-expressed as
\[
\m{K}_{\tl{F}}(x,x') = \m{A}(x)\m{K}_F(x,x')\m{A}(x)^T
\]
with respect to $\tl{F}$.
This condition is called \emphmarginnote{equivariance under change of frame}.
At this point, it is clear why we did not even attempt to generalize matrix-valued kernels directly by positing candidate formulas: for an arbitrary manifold, there is very little hope of simply guessing an expression that is simultaneously positive semi-definite and equivariant under change of frame.

\subsection{Projected kernels}

Now that we understand how to write kernels of Gaussian vector fields in coordinates, we are ready to consider techniques for constructing such kernels and calculating them numerically.
With our preparation complete, the construction we present is extremely simple, consisting of two steps.

\1 Embed a set of $p$ scalar-valued Riemannian Gaussian processes from $X$ into $i(X) \subseteq \R^p$, and use them to assemble a vector-valued Gaussian process defined on on $i(X)$.
\2 Project the tangent vectors onto each tangent space, obtaining a tangential vector field in the embedded space, and therefore a vector field on $X$.
\0

\label{ntn:frame-projection}
We now proceed to make this precise.
Suppose that we have an embedding
\[
i : X &\-> \R^p
i_{x,*} : T_x X &\-> \R^p
\]
where the pushforward is defined at all points $x\in X$.
Introducing a frame $F$ allows us to define the projection map, which is a matrix-valued function $\m{P}_F : X \-> \R^{p \x d}$ defined by
\[
\m{P}_F(x) =\begin{bmatrix}
i_{x,*}(e_i(x)) \\
\vdots \\
\end{bmatrix}
\]
which describes how to project arbitrary vectors onto tangent planes.
Note that if $X$ is Riemannian with metric $g$, then under the tangent-cotangent isomorphism, the linear map given by $\m{P}_F$ is a right-inverse to the map given by $\m{P}_F^T$, which describes how a vector field with respect to a frame can be re-expressed with respect to the coordinates in the embedded space.
In particular, $\m{P}_F^T \m{P}_F = \m\Gamma$, where $\Gamma_{ij} = g(e_i(x), e_j(x))$ is the coordinate representation of the metric.
It is now clear how to make our idea precise.

\begin{figure}
\tikzset{external/export next=false}
\begin{tikzpicture}
\node at (1,1) {\includegraphics[scale=0.25]{figures/png/s2_ex.png}};
\node at (0,-0.875) {\includegraphics[scale=0.25]{figures/png/s2_ey.png}};
\node at (2,-0.875) {\includegraphics[scale=0.25]{figures/png/s2_ez.png}};
\node at (5.125,0) {\includegraphics[scale=0.25]{figures/png/s2_ev.png}};
\node at (9.125,0) {\includegraphics[scale=0.25]{figures/png/s2_pr.png}};
\node at (0.84375,2.40625) {Scalar processes};
\node at (4.8125,2.40625) {Embedded process};
\node at (8.8125,2.40625) {Projected process};
\end{tikzpicture}
\caption[Projected kernels]{Here, we show how to construct Gaussian process on the sphere whose cross-covariance kernel is a projected kernel. First, we construct three scalar Gaussian processes on the sphere. Then, embed the sphere in $\R^3$, and the scalar Gaussian processes to form a vector-valued Gaussian process on the embedded sphere. Finally, we project the resulting Gaussian process onto the sphere, yielding the desired tangential vector field.}
\label{fig:proj-ker}
\end{figure}

\begin{definition}[Projected kernel]
Let $\v{f} : \Omega \x X \-> \R^{p\x p}$ be a Gaussian process defined on $i(X) \subseteq \R^p$, and define the Gaussian vector field
\[
f(\omega, x) = \m{P}_F(x) \v{f}(\omega, i(x))
\]
We call the cross-covariance kernel of a process constructed this way a \emph{projected kernel}.
\end{definition}

It follows that if $\v\kappa : X \x X \-> \R^{p \x p}$ is the matrix-valued cross-covariance kernel of $\v{f}$, then the cross-covariance kernel of our Gaussian vector field is given by
\[
\m{K}_F(x,x') = \m{P}_F(x) \v\kappa(x,x') \m{P}_F(x)^T
\]
Note that here, we generally require a Riemannian structure on $X$ in order to define a useful set of scalar-valued processes in order to obtain $\v{f}$ or, equivalently, $\v\kappa$, to begin with.
The simplest approach is to make each component in the embedded space independent.
Particularly in cases where the embedding is isometric, this gives a wide class of easy-to-understand kernels.
An example can be seen in \Cref{fig:proj-ker}.

It's easy to see that all cross-covariance kernels arise this way: using Nash's Theorem, we can embed an arbitrary Gaussian vector field on a Riemannian manifold into a Euclidean space, glue together the resulting scalar components, and project back to obtain the Gaussian vector field we started with.

This completes our technical development.
Starting from first principles, we defined the notion of a Gaussian vector field, described in what sense such processes possess mean vectors and cross-covariance kernels, and defined a wide-ranging class of kernels which is completely constructive and can be implemented in software.
In particular, our constructions are automatically compatible with standard training methods such as variational inference.

\subsection{Illustrated examples}

We now explore training Gaussian vector fields on Riemannian manifolds on observed data.
Our goal is to demonstrate that the developed techniques are, in total, sufficiently explicit so as to enable Gaussian vector fields to be trained using standard methods without the need for any additional machinery.

We work with the sphere.
For training data, we consider global wind velocity, chosen because it is freely and readily available.
Specifically, we consider interpolation of wind velocity deviations from historical average, at a fixed height, using wind velocity data directly observed by a satellite.
For the prior, we use a Gaussian vector field model constructed using a projected kernel whose components are independent Matérn-3/2 scalar-valued Gaussian processes on the sphere.

We train the Gaussian process using exactly the same procedure as that used in the preceding sections: by optimizing kernel hyperparameters using gradient descent, and computing the value of the vector field at all locations using pathwise sampling.
All computations are performed with respect to a fixed frame defined by the latitude-longitude coordinate system.

Results are shown in \Cref{fig:posterior-vector-field}.
Immediately, we see that this dataset exhibits behavior not suitable for the chosen model.
Specifically: in some regions, the data exhibits large jumps, while in other regions, it is very smooth over large distances.
This means the effective length scale governing variability is spatially non-uniform, which requires a more sophisticated model to effectively represent, such as one with a spatially varying length scale.

\begin{figure}
\begin{subfigure}{0.49\textwidth}
\includegraphics[scale=0.25]{figures/png/s2_pm.png}
\caption{Posterior mean}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\includegraphics[scale=0.25]{figures/png/s2_pcc.png}
\caption{Data and posterior covariance}
\end{subfigure}
\caption[Posterior Gaussian vector field: climate data]{Posterior mean and posterior cross-covariance, with the trace of the latter denoted by color, for a projected Gaussian vector field constructed using three independent Matérn-3/2 Gaussian processes on the sphere. Training data is shown in gray. The Gaussian process is seen to oversmooth data in locations with large jumps, such as near the easterly trade wind deviations in the center. We also observe that uncertainty increases as we move away from the training locations.}
\label{fig:posterior-vector-field}
\end{figure}

The chosen Gaussian process model responds to this by overinterpolating the data and reverting to the overall mean in regions with rapidly changing wind velocity.
While this is not ideal from a modeling perspective, it is at least a relatively graceful failure mode, compared to numerical instability or incorrect results in other regions where rapid shifts do not occur.
Effects such as these are closely linked to non-linear behavior in the differential equations governing weather phenomena, and generally require special consideration.

On the other hand, we observe that the trained Gaussian process model is smooth, even though the choice of frame is non-smooth at the top of the sphere---a key motivation behind developing the theory for working with vector fields in a geometrically-sound way in the first place.
Similarly, we see that posterior uncertainty increases as we move away from locations where training data is available.
This shows that the model correctly incorporates geometry and behaves in a manner mirroring its scalar-valued analogs.

One notable characteristic of the obtained posterior vector field is that the covariance of the output vectors at any given point is close to isotropic.
This can be seen from the ellipsoids in \Cref{fig:posterior-vector-field}, which appear circular, though they are clearly elliptical from numerical examination.
This behavior appears to be a feature of the data locations chosen in this particular example, which lie approximately on a great circle, and does not necessarily occur in general.

The model used is also not realistic in other ways, beyond its use of a single global length scale: in particular, it does not incorporate time or any kind of physical information into its design.
Nonetheless, we believe that it serves as a useful demonstration that constructing vector-valued Gaussian processes using the ideas developed is practical.
We hope this motivates further development to apply such models and extensions thereof more broadly, both for the sphere and for other manifolds.

\section{Geometry-aware Bayesian optimization}

We now study the role that geometry plays when working with decision systems built using Gaussian processes.
To do so, we perform Bayesian optimization of standard benchmark functions on Riemannian manifolds using the Riemannian Matérn and squared exponential Gaussian processes with the numerical techniques developed herein.
Following \textcite{jaquier20}, we call this setting \emph{geometry-aware Bayesian optimization}.

\begin{figure}
\begin{subfigure}{0.32\textwidth}
\includegraphics[scale=0.25]{figures/png/fn_a.png}
\caption{Ackley function}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\includegraphics[scale=0.25]{figures/png/fn_l.png}
\caption{Levy function}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\includegraphics[scale=0.25]{figures/png/fn_r.png}
\caption{Rosenbrock function}
\end{subfigure}
\caption[Global optimization benchmark functions]{Global optimization benchmark functions used in the geometry-aware Bayesian optimization experiments. All three are defined and shown on an ambient Euclidean space of dimension two.}
\label{fig:benchmark-functions}
\end{figure}

We consider three manifolds: the spheres $\bb{S}^3$ and $\bb{S}^5$, and the special orthogonal group $SO(3)$, each embedded within a Euclidean space in the natural manner.
We use three target functions commonly used as benchmarks in global optimization: the \emph{Ackley} \cite{ackley87}, \emph{Levy} \cite{levy82}, and \emph{Rosenbrock} \cite{rosenbrock60} functions.\footnote{The Ackley, Levy, and Rosenbrock benchmark functions, respectively, are given as
{\setlength{\tabcolsep}{0.25ex}\begin{tabular}{r c l}
$f_{\f{A}}(x_1,..,x_d)$ & $=$ & $20-20\exp(-0.2 (\frac{1}d \sum_{i=1}^d x_i^2)^{1/2}) - \exp(\frac{1}d \sum_{i=1}^d \cos(2\pi x_i)) + e$
\\
$f_{\f{L}}(x_1,..,x_d)$ & $=$ & \makebox[10.375cm][s]{$\sin(\pi w_i)^2 + \sum_{i=1}^{d-1} (w_i - 1)^2 (1 + 10\sin(\pi w_i + 1)^2) + (w_d - 1)^2 (1 + \sin(2\pi w_d)^2)$}
\\
$f_{\f{R}}(x_1,..,x_d)$ & $=$ & $\sum_{i=1}^{d-1} 100(x_{i+1} - x_i^2)^2 + (x_i - 1)^2$
\end{tabular}\newline}
where $w_i = 1 + \frac{x_i - 1}{4}$ and $d$ is the dimension of the space on which the functions are defined.}
These are defined in the ambient spaces: we optimize their restrictions onto the manifold of interest, centered to ensure the minima is on the manifold.
\Cref{fig:benchmark-functions} plots these benchmark functions for a Euclidean space.

\begin{figure*}[p!]
\includegraphics{figures/tex/gabo}
\caption[Geometry-aware Bayesian optimization benchmark results]{Geometry-aware Bayesian optimization benchmark results.}
\label{fig:gabo}
\end{figure*}

For each manifold, we use two priors: Matérn with smoothness $\nu = 5/2$ and squared exponential, both with a Gaussian likelihood.
The remaining kernel hyperparameters and likelihood error variance are learned from observed data via gradient descent.
We use expected improvement for our acquisition function, computed directly from the kernel.
We run for a total of $n=200$ iterations, and repeat each experiment $30$ times to assess variability.

To establish a performance baseline, we compare against Bayesian optimization in ambient Euclidean space.
Here, the acquisition point is selected by performing constrained optimization on the acquisition function, ensuring that points selected for evaluation are located on that manifold.

Results can be seen in \Cref{fig:gabo}.
We plot median regret curves, along with first and third quartiles.
From this, we see that the performance of geometry-aware Bayesian optimization depends on interplay between the manifold and the overall shape of the function being optimized.

For the $3$-dimensional sphere, all techniques produce comparable regret curves.
This shows working with geometry-aware models and acquisition optimization, compared to constrained acquisition optimization on Euclidean models defined in ambient spaces, offers less benefit in cases where the ambient space is not too different from the manifold.
The performance of Matérn and squared exponential models, in this case, is similar.

For the $5$-dimensional sphere, the role of geometry becomes more pronounced.
For the Ackley and Levy functions, we see a clear improvement from using geometry-aware methods.
For the Rosenbrock function, which possesses less local variability, performance of all methods remains similar.
For all three target functions, Matérn and squared exponential models perform similarly.

On the $3$-dimensional special orthogonal group, geometry-aware models perform similarly to models defined on the ambient Euclidean spaces for all three benchmark functions.
On the other hand, here, Matérn models outperform squared exponential models.
Therefore, we see that the effect of kernel smoothness on performance of Bayesian optimization may depend on the geometry of the manifold, as well as the target function.

Overall, the results show that Riemannian models perform comparably to Euclidean ones in some cases, and outperform them in others.
The effect varies according to the interplay between the target function being optimized and the manifold's geometry.
We also see that slightly less smooth Matérn Gaussian processes perform comparably to squared exponential Gaussian processes in some cases, and outperform them in others.

Bayesian optimization is often used in settings where function evaluations are extremely expensive, and therefore even the relatively modest performance gains seen here can be consequential.
In such cases, taking advantage of constraints, symmetries, and other geometric properties is a promising route towards improved performance.
Geometry-aware Bayesian optimization offers a principled framework for doing so.

\section{Conclusion}

In the preceding section, we have developed three key classes of \emph{non-Euclidean Gaussian process} models: the \emph{Riemannian Matérn} and \emph{graph Matérn} classes of scalar-valued Gaussian processes, and the \emph{projected kernel} class of vector-valued Gaussian processes on Riemannian manifolds.
Each of these provides a unifying view of previously-proposed methods, and expands Gaussian processes models' scope of applicability.

The key idea for defining Matérn Gaussian processes on Riemannian manifolds was to not attempt to define kernels using geodesic distances---such constructions, while intuitively appealing, turn out to not lead to an effective mathematical formalism.
Instead, we relied on the idea of \emph{functional calculus} built using spectral theory of the Laplace--Beltrami operator to introduce operators through which we constructed the desired Gaussian processes as transformations of Gaussian white noise processes.

Adopting this viewpoint enabled us to reinterpret models previously-proposed within a cumbersome finite element framework using a more abstract approach built via the theory of stochastic partial differential equations and reproducing kernel Hilbert spaces.
From this approach, we obtained formulas for pointwise evaluation of the \emph{Riemannian Matérn kernel} in terms of eigenvalues and eigenfunctions of the Laplace--Beltrami operator.
The formalism also enabled us to define and compute the \emph{Riemannian squared exponential kernel}.

While our original original formalism was introduced for working with manifolds, it is clear that the constructions are more general.
Inspired by the discrete computations used in implementing the Riemannian Gaussian processes we defined, we explored purely-discrete analogs of these models, obtaining the \emph{graph Matérn kernel} and \emph{graph squared exponential kernel}.
These enabled us to apply Matérn Gaussian processes to model phenomena in purely discrete spaces whose geometry departs significantly from manifold-like settings.

Finally, we returned to the manifold setting, this time studying formalism for defining \emph{Gaussian vector fields}.
This setting is non-trivial from the point of conceptualization: a vector-valued Gaussian is only \emph{locally} a vector-valued random function, and instead should be formulated using the differential-geometric formalism of \emph{random sections}.
We therefore began by clarifying what the appropriate notion of a \emph{Gaussian vector field}  and \emph{cross-covariance kernel} should be, and how to work with this notion numerically.

Having done so, we introduced a wide class of cross-covariance kernels for defining such processes called \emph{projected kernels}.
These kernels can be built from the Riemannian Gaussian process models already defined, together with an embedding of the manifold into an ambient Euclidean space, and produce covariances that reflect the manifold's geometry.
This construction enabled us to implement Bayesian learning of vector-valued data on the sphere.

Having explored these constructions from a purely model-building point of view, we concluded by considering them within a decision-making setting, by studying \emph{geometry-aware Bayesian optimization}.
There, saw that using geometry-aware models also resulted in a modest performance improvement in cases where topology and geometry play a non-trivial role, particularly for higher-dimensional spaces.

It is clear that our ideas apply within a wider scope than what we have explored.
For example, our derivations for the Matérn class apply essentially-unmodified to closely-related settings such as \emph{manifolds with boundary}, provided one introduces boundary conditions which ensure the operators of interest behave the same way as in our setting.
Extensions such as this make excellent candidates for future work.

Our differential-geometric framework also provides a promising way to study \emph{symmetry} within Gaussian processes and Bayesian optimization.
Since symmetry plays a fundamental role in geometry and in physics, understanding how to handle symmetries is a promising avenue for building richer classes of Gaussian process models, and applying techniques such as Bayesian optimization to wider classes of scientific problems.

We hope that techniques such as the ones presented here provide a foundation upon which these and other developments can be built.
We therefore conclude our presentation of contributions, and move to discuss the state and overall picture of Gaussian processes and decision systems that we have studied.

\chapter{Discussion}
\label{ch:discussion}

\lettrine{B}{ayesian learning} with Gaussian process models offers a clear and comprehensive framework for making decisions that assess and propagate uncertainty in order to resolve explore-exploit tradeoffs inherent in decision-making settings.
In this dissertation, we explored two avenues for broadening applicability of Gaussian process models in such settings.

In \Cref{ch:pathwise}, we presented pathwise conditioning methods.
These methods allow a posterior Gaussian process, as an actual random function, to be expressed as a sum of a prior Gaussian process and a dependent update term.
Pathwise conditioning gives rise to Gaussian process approximations for which all stochasticity can be evaluated in advance, which makes it substantially easier for upstream algorithms to interface with the Gaussian process.

In \Cref{ch:noneuclidean}, we presented a collection of techniques for working with Gaussian process models defined on non-Euclidean spaces, including Riemannian manifolds and graphs.
We derived constructive and practical expressions which enable such models to be trained using standard methods.
For the Riemannian setting, we did so for both scalar-valued and vector-valued processes.

For both pathwise conditioning and non-Euclidean Gaussian process models, the end result of our constructions was the ability to perform Bayesian optimization in a flexible and effective manner tailored to the setting at hand.
This allowed us to benchmark our ideas in one of the simplest, but most important, classes of decision systems.

\section{Future work}

One end goal of Gaussian process research is the design of decision-making systems which simply work in the settings where they apply, without the usual tuning or numerical difficulties, and fail gracefully in settings where they do not apply.
The ideas presented naturally lead to a number of further research directions towards this goal, which we briefly comment on now.

\subsection*{Decision systems beyond optimization}

Perhaps the most promising research direction made possible by pathwise conditioning methods is the design of data-efficient decision systems for tasks beyond optimization.
\textcite{neiswanger21} propose \emph{Bayesian algorithm execution}, a framework which generalizes Bayesian optimization by allowing for the task to be defined by using a general \emph{underlying algorithm} in place of optimization.

For this, \textcite{neiswanger21} introduce \emph{InfoBAX}, an information-theoretic acquisition function constructed directly from the algorithm being considered.
This enables the introduction of \emph{Bayesian Dijkstra's algorithm} for finding shortest paths in a graph in a data-efficient manner, and other novel decision systems.

Pathwise conditioning methods play a key role in computing the InfoBAX acquisition function.
Specifically, one runs the underlying algorithm on posterior random functions, producing a set of \emph{execution paths} of points queried by the algorithm.
InfoBAX is then constructed from the execution trace and output via a set of information-theoretic calculations.
Approximate pathwise conditioning makes it possible to compute the required execution traces in an efficient, accurate, and convenient manner.

Gaussian processes have not yet been widely considered for tasks not conveniently expressed as maximization of rewards.
Bayesian algorithm execution enables such methods to be considered for settings beyond Bayesian optimization and model-based reinforcement learning, which are well-established and whose limitations are largely known.
Developing this framework to the same degree of detail afforded to those cases is therefore a particularly promising research direction.

From a theoretical point of view, one of the first steps to be taken in understanding Bayesian algorithm execution is to understand what is the appropriate notion of \emph{regret} for this setting, among a number of possible variations, and to develop techniques for studying it.
The analysis of closely-related state-of-the-art bandit algorithms, such as \emph{information-directed sampling} \cite{russo14}, offers hints on how to proceed.

In parallel, it seems fruitful to look for applications where more general data-efficient decision systems beyond optimization are potentially useful, and yet where Gaussian process models are viable.
\emph{Multi-objective Bayesian optimization} \cite{emmerich05,campigotto14,hernandezlobato16,suzuki20}, where the goal is finding a Pareto frontier, rather than obtaining a global minima, gives a concrete algorithm class to consider studying from this point of view.

From the Bayesian algorithm execution framework alone, we see that pathwise conditioning methods enable potential new avenues for constructing decision systems and therefore growing the applicability and scope of Gaussian process methods within machine learning.
We hope that, in the coming years, more of these avenues are explored, enabling Gaussian processes to have larger role in the machine learning toolbox.

\subsection*{Connections to numerical analysis}

Another promising research direction is exploring the connections between Gaussian processes and numerical analysis.
Interest in these connections has grown rapidly in the Gaussian process community within machine learning in recent years, as a way to improve performance and usability of software implementations \cite{matthews17,gardner18,vanderwilk20}.
Ideas based on pathwise conditioning and stochastic partial differential equations offers new avenues for taking advantage of these connections.

Firstly, one can study use of iterative solvers, such as the conjugate gradient method, in combination with pathwise conditioning and inducing points.
Iterative solvers are promising because they can in principle tackle larger linear systems for which Cholesky factorization simply fails, provided they are set up correctly.
A large set of techniques in this class have recently been proposed for Gaussian process models \cite{dong17,gardner18,pleiss18,meanti20,pleiss20}.

The computational costs of an iterative solver depend on how many iterations are required for convergence, which vary according to the condition number of the linear system being solved \cite{saad03}.
Variational approximations such as the inducing point construction presented in \Cref{ch:pathwise} make it possible to work with linear systems that are substantially better-conditioned than those in the true posterior, due to additional diagonal terms added to the matrix.
Exploring such combinations is a promising route for improving performance.

Pathwise conditioning makes it possible to avoid computation of matrix square roots when working with iterative solvers.
Matrix square roots generally require further considerations than simply solving linear systems \cite{pleiss20} and can be less convenient.
With pathwise conditioning, the only additional quantities needed, beyond solution of the linear system itself, are generally log-determinant terms.
Techniques for computing these terms have been proposed \cite{saibaba17,dong17}, and are key part in making iterative methods effective.

In the partial differential equation and mathematical physics communities, iterative solvers for linear systems constructed using finite element methods achieve state-of-the-art performance for many problems \cite{evans10,lord14}.
Connections with stochastic partial differential equations, such as the ones explored here, offer a fruitful avenue for seeking inspiration for improving Gaussian process performance further.

A key lesson from numerical analysis is that, in order to solve a linear system, the most important step is often the construction of a \emph{preconditioner} to partially solve the problem before an iterative method is used \cite{saad03,e11,xu17}.
There, the degree of sophistication used in constructing preconditioners is often at least as large as that used in constructing iterative solvers themselves.

Multi-grid methods provide a broad framework for constructing preconditioners which achieve state-of-the-art performance for many partial differential equations \cite{e11,xu17}.
In essence, such methods provide a framework for resolving long-range effects at a coarse resolution, and short-range effects at a fine resolution, improving computational efficiency in the process.
Adapting such ideas to the Gaussian process setting offers possibilities for improvement.

Numerical analysis can also be a source of inspiration for improving variational inference.
Inducing points often struggle on datasets where the overall data size is large, but the amount of data points within any given length scale ball is small.
Multi-scale methods suggest that right kind of approximation for this regime should involve replacing long-range effects with averaged, sparsified versions thereof \cite{e11,xu17}.
Understanding how construct such a variational approximation is therefore a promising avenue for future work.

In total, ideas from numerical analysis provide a fruitful source of ideas for improving Gaussian processes further.
We hope that in the coming years, these ideas come to fruition, and using Gaussian processes in most practical settings become just as easy and seamless as training neural networks on well-understood problem classes has become.

\subsection*{Improved understanding of kernels on geometric spaces}

The constructions presented in \Cref{ch:noneuclidean} for building non-Euclidean Gaussian process models give general formulas applicable for a wide variety of spaces.
Many manifolds encountered in practice, particularly those used in physics and engineering, possess additional mathematical structure, which can be used to both construct new kernels and provide avenues for more efficiently computing the Matérn and squared exponential kernels studied previously.

For example, the eigenfunctions of the Laplace--Beltrami operator on the sphere are the spherical harmonics \cite{chavel84,canzani13}.
The spherical harmonics, in turn, can be re-expressed in terms of the \emph{Gegenbauer polynomials}, which are a set of polynomials on $[-1,1]$ which are orthogonal with respect to a certain weight function \cite{gradshteyn14}.
The Riemannian Matérn and squared exponential kernels can therefore be written in terms of these polynomials, and turns out to only depend on the geodesic distance between two points.

This is a substantial simplification compared to the general case, and one can use it to pre-compute the kernel for various values of the geodesic distance, and interpolate the resulting values, rather than evaluate an infinite series every time the kernel needs to be computed.
Note that the spherical squared exponential kernel, when expressed using this simplification, is not the squared exponential of the geodesic distance: instead, it is a different, more complicated function of the geodesic distance.

Similar simplifications also occur on the special orthogonal group, suggesting that there may be a general theory that describes them.
Finding such a theory is a promising direction for future work.
To do so, the first step would be to understand how symmetries of a manifold affect kernels defined over it via spectral methods similar to the ones we have used.

Simplifications may also occur due to other structures beyond symmetries.
Lie groups are particularly important in many areas, including robotics, computer vision, physics, and many engineering disciplines.
Due to their prominence, it therefore seems fruitful to study potential methods for computing kernels on Lie groups, provided they are equipped with a Riemannian metric compatible with the group structure.

In particular, studying kernels on non-compact Lie groups, and related spaces such as certain matrix manifolds, is a promising direction towards a general theory of Riemannian kernels which does not require compactness.
Here, one would need more general forms of spectral theory than Sturm--Liouville decompositions.
These are mathematically sophisticated, requiring additional machinery such as integration over projection-valued measures \cite{lang12}.
Exploring such generalizations is a promising avenue for further work.

Finally, one can consider extending our constructions to more general spaces than manifolds, such as for example manifolds with boundary, which possess similar spectral properties.
Spaces like this are often important in applications areas \cite{solin18,krainski18,solin19,coveney20}.
Studying heat kernels, which have been considered in settings as general as metric measure spaces \cite{grigoryan03,grigoryan14}, gives hints about the total scope which may be possible.

Summarizing, the constructions we have studied only begin to describe the full scope of geometry-aware kernels such as the Matérn and squared exponential class constructed using Sturm--Liouville theory and described here.
We hope this first step leads to further substantive understanding of this class of methods within machine learning.

\section{Conclusion}

The contributions presented in this thesis expand the set of settings where Gaussian processes can be used.
In particular, pathwise conditioning methods reduce the barriers needed to deploy Gaussian process models once they have been trained, and make it easier to obtain posterior quantities of interest for the setting at hand.
Simultaneously, the obtained kernel expressions for non-Euclidean Gaussian processes enable such models to be trained using off-the-shelf methods in a standard manner.

Overall, these ideas reduce the barriers that need to be overcome in order for practitioners to use Gaussian processes in day-to-day work.
We hope the ideas presented enable more people to consider using Gaussian process models as part of their workflow, and bring decision-making systems such as Bayesian optimization to new areas of science, technology, and engineering.

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