Implicit Gaussian process representation of vector fields over arbitrary latent manifolds
Authors: Robert Peach
Matteo Vinao-Carl
Nir Grossman
Michael David
Emma Mallas
David J. Sharp
Paresh A. Malhotra
Pierre Vandergheynst
Adam Gosztolai
Venue: ICLR 2024 poster (2024)
Keywords: Gaussian processes
neuroscience
vector field
tangent bundle
connection Laplacian
EEG
Alzheimer's disease
Primary area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)

Abstract: Gaussian processes (GPs) are popular nonparametric statistical models for learning unknown functions and quantifying the spatiotemporal uncertainty in data. Recent works have extended GPs to model scalar and vector quantities distributed over non-Euclidean domains, including smooth manifolds, appearing in numerous fields such as computer vision, dynamical systems, and neuroscience. However, these approaches assume that the manifold underlying the data is known, limiting their practical utility. We introduce RVGP, a generalisation of GPs for learning vector signals over latent Riemannian manifolds. Our method uses positional encoding with eigenfunctions of the connection Laplacian, associated with the tangent bundle, readily derived from common graph-based approximation of data. We demonstrate that RVGP possesses global regularity over the manifold, which allows it to super-resolve and inpaint vector fields while preserving singularities. Furthermore, we use RVGP to reconstruct high-density neural dynamics derived from low-density EEG recordings in healthy individuals and Alzheimer's patients. We show that vector field singularities are important disease markers and that their reconstruction leads to a comparable classification accuracy of disease states to high-density recordings. Thus, our method overcomes a significant practical limitation in experimental and clinical applications.

Published as a conference paper at ICLR 2024
IMPLICIT GAUSSIAN PROCESS REPRESENTATION OF VECTOR FIELDS OVER ARBITRARY LATENT MANI- FOLDS
Robert Peach* University Hospital Würzzurg
Matteo Vinao-Cari*, Nir Grossman, Michael David Imperial College London (* Indicates equal contribution)
peach rêukw. de
Emma Mallas, David Sharp, Paresh A. Malhotra Imperial College London
Pierre Vandergheynst, Adam Gosztolai EPFL adam.
ABSTRACT
Gaussian processes (GPs) popular nonparametric statistical models for learn- ing unknown functions and quantifying the spatiotemporal uncertainty in data. Recent works have extended GPs model scalar and vector quantities distributed over non-Euclidean domains, including smooth manifolds appearing in numer- ous fields such computer vision, dynamical systems, and neuroscience. How- ever, these approaches assume that the manifold underlying the data is known, limiting their practical utility. We introduce RVGP, a generalisation of GPs for learning vector signals over latent Riemannian manifolds. Our method uses posi- tional encoding with eigenfunctions of the connection Laplacian, associated with the tangent bundle, readily derived from common graph-based data approxima- tion. We demonstrate that RVGP possesses global regularity over the manifold, which allows it to super-resolve and inpaint vector fields while preserving singu- larities. Furthermore, we use RVGP to reconstruct high-density neural dynamics derived from low-density EEG recordings in healthy individuals and Alzheimer's patients. We show that vector field singularities are important disease markers and that their reconstruction leads a classification accuracy of disease states comparable to high-density recordings. Thus, our method overcomes a significant practical limitation experimental and clinical applications.
INTRODUCTION
cornerstone of statistical learning theory is the manifold assumption, which posits that high- dimensional datasets often distributed over low-dimensional smooth manifolds topological spaces characterised by local Fuclidean structure. For instance, images of an object from varying camera angles or diverse renditions of written letter can all be viewed as samples from smooth manifold (Tenenbaum) (2000). Further, the common approximation of data by a proximity graph, based on a notion of atfinity or similarity between data points, induces Riemannian structure that is instrumental in geometric learning theories. For example, the analogy between the graph Laplacian matrix and the Laplace-Beltras Beltramii operator associated with a Riemannian manifold (Chung 1997) has been widely exploited in manifold learning (Belkin & Niyogi
2003
Coifman et al. (005), shape
analysis [Taubin] 1995), graph signal processing (Ortega et al.
2018).
discrete geometry (Gosztolai
& Arnaudon 2021), graph neural networks (Defferrard et al.
2016
Kipf & Welling 2017 Peach
ctt al. 2020) and Gaussian processes (Borovitskiy et al. 2020
2021).
However, many datasets contain richer structure comprising smoothly varying vector field over the manifold. Prime examples are dissipative dynamical systems where, post an initial transient phase (Fefferman al. 2016). Likewise, in neuroscience, smooth vector fields arise from the fir- ing rate trajectories of neural populations evolving over neural manifolds, which is instrumental in
neural information coding (Sussillo & Barak 2013 Khona & Fiete 2022 Gardner et al. 2022

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Smooth vector fields are also pertinent in areas like gene expression profiling during development (La Manno et al. 2018) and multireference rotational alignment in cryoelectron microscopy (Singer & Wu| 2012). These applications emphasise the need to generalise current learning paradigms to capture both the manifold structure and its associated vector field.
To address this need, promising avenue is to consider the Laplace- Beltrammm operator as hierarchy of Laplacians that act on tensor bundles of manifold with increasing order. The first member of this hierarchy is the Laplace-Beltrame operator, which acts on rank-0 tensors, i.e., scalar signals. Sim- ilarly, higher-order signals, including vector fields, have associated Laplacian operators, which can encode their spatial regularity. Among these, the connection Laplacian (Barbero et al. 2022). de- fined on vector bundles. and the related sheaf Laplacian (Knöppel et al. [013]] Bodnar et al. 2022),
which allows the vector spaces on nodes to have different dimensions, are emerging as leading oper-
ators in machine learning (Bronstein et al. 2017, [Battiloro et al.) 2023 Gosztolai et al. (023). These operators are related to heat diffusion of higher-order signals over manifolds (Singer & Wu 2012
Sharp et al. 2019) and thus intrinsically encode the signals' smoothness. The connection Laplacian is particularly appealing because it can be constructed, even when the manifold is unknown, from graph-based data descriptions (Singer & Wu 2012 Budninskiy et al., 2019). We, therefore, asked how one could use this discrete approximation to derive continuous functions that implicitly repre- sent the vector field over the manifold. Such representation could use the global regularity of the vector field to reconstruct intricate vector field structures lost in data sampling.
Gaussian processes- renowned family of nonparametric stochastic processes offer excellent framework for learning implicit functional descriptions of data. While GPs are traditionally defined on Euclidean spaces (Rasmussen & Williams) 2006), several studies have extended them Rieman-
nian manifolds. However, these studies have either considered scalar signals (Wilson et al. 2021
Mallasto & Feragen| 2018 Mallasto et al., 2020; Borovitskiy et al. 2020 2021] Jensen et al. 2020) or vector signals Hutchinson et al. (2021) but only in cases where the underlying manifold is known and is analytically tractable, such as spheres and tori. In this work, we generalise GPs to vector fields on arbitrary latent manifolds, which can only be approximated based on local similarities between
data points, making them applicable to real-world datasets.
Our contributions are follows. (i) We generalise GPs to vector-valued data using the connection Laplacian operator, assuming that the data originates from a stationary stochastic process. (ii) We show that the resulting Riemannian manifold vector field GP (RVGP) method encodes the manifold and vector field's smoothness as inductive biases, enabling out-of--smppe predictions from sparse or obscured data. (iii) To underscore the practical implications of RVGP, we apply it to electroen- cephalography (EEG) recordings from both healthy individuals and Alzheimer's disease patients. The global spatial regularity learnt by our method significantly outperforms the state-of-the-ar ap- proaches for reconstructing high-density electrical fields from low-density EEG arrays. This enables our method to better resolve vector field singularities and dramatically increase the classification power of disease states. In sum, our work enables differential geometric formulation of kemel- based operators and demonstrates direct relevance for fundamental and clinical neuroscience.
2 COMPARISON WITH RELATED WORKS
Let us begin by carefully comparing our method to related works in the literature.
Implicit neural representations (INRs) There has been an increasing interest in defining signals implicitly as parametrised functions from an input domain to the space of the signal. In Euclidean spaces, INRs have been a breakthrough in replacing pixel-wise description images or voxel-wise descriptions of 3D shapes by neural networks (Sitzmann et al. 2020 Lipman 2021 Gosztolai et al.) 2021 Koestler et al. 2022. Mildenhall et al. 2020). INRs have also been extended to signals over manifolds by graph Laplacian positional encoding (Grattarola & Vandergheynst, 2022). However, INRs are data-intensive due to the lack of explicit spatial regularisation. Further, they have not been extended to handle vector-valued data.
Gaussian processes specific Riemannian manifolds Several closely related works in the GP literature have provided various definitions of GPs on Riemannian manifolds. One line of works defined GPs as -aluue processes f X - M (Mallasto & Feragen 2018 Mallasto al.)
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2020) by using the exponential map of the manifold to perform regression in the tangent space. However, these works require that the manifold . be known to define the exponential map. More notable are studies which define GPs as scalar- valued functions f : X - R. For example, consider- ing data points as samples from a stationary stochastic process, the domain X can be defined based on positional encoding using eigenfunctions of the Laplace-Beltrand Beltrami operator (Solin & Särkkä 2020 Borovitskiy et al., (2000) or the graph Laplacian (Borovitskiy et al., (021). However, these works cannot be directly applied vector-valued signals by treating vector entries as scalar channels. This is because these channels are generally not independent but related through the curvature of the manifold. To address this gap, HHutchinson et al. (202T) defined GPs as functions f X + TM over the tangent bundle T.M by first mapping the manifold isometrically to Euclidean space and then using multi-input, multi-output GP learn the projected signal. However, [Hutchinson et al. (00T)) has focused on cases when the manifold is explicitly known, and its mapping to Fuclidean space can be explicitly defined. Here, we are specifically interested in the case where the manifold unknown common scenario in many scientific domains. To achieve this, generalise the works of Borovitskiy et al. 2220 2021) to vector fields on Riemannian manifolds using the connec- tion Laplacian operator and its eigenvectors positional encoding. During reviews, we have also become aware of two related works by (Robert-Nicous et al. 20023; Fichera et al. (2024).
Gaussian processes in neuroscience GPs have also been widely used in the neuroscience litera- ture, particularly combined with linear dimensionality reduction to discover latent factors underlying neural dynamics. A popular method is Gaussian Process Factor Analysis (GPFA) (Yu et al., 2009), which defines GPs in the temporal domain and does not encode spatial regularity over ensembles of trajectories as inductive bias. GPFA has been used to define time-warping functions to align neural responses across trials, i.e., individual presentation of a stimulus or task (Duncker & Sahani 2018). GPFA has been extended to Euclidean spaces by simultaneously identifying the latent dynam- ies and the manifold over which it evolves (Jensen et al., (020). However, this model is limited to manifolds built from (3) groups and requires them to be enumerated to perform Bayesian model selection. We seek a constructive framework requiring no assumption on the manifold topology.
3 BACKGROUND
Here, we introduce the function space view of GPs in Euclidean domains and their stochastic partial differential equation formulation developed for scalar-valued signals, which will then allow us to extend them to vector-valued signals.
3.1 GAUSSIAN PROCESSES EUCLIDEAN SPACES
A GP is stochastic process f : X + Rd defined over a set X such that for any finite set of samples X (.I,...,,,,, a..) E X", the random variables multivariate Gaussian, ((::;; μr, K). with mean vector μa m( and covariance matrix K &(X, X). Consequentially, GP fully characterised by its mean function m(x) := (x)())) and covariance function k(x, x²) == Cov(f f (xx)), also known as the kernel, and denoted as ~ (P(m(x), fa, x²)). It is typical to assume that m(xx) = 0, which does not reduce the expressive power of GPs (Rasmussen & Williams| 2006).
((x))) E Rexd are jointly
One may obtain the fest-fit GP to a set of training data points (X. 3) = {(x,, y:)li = L...., nn by Bayesian linear regression, assuming that the observations y: differ from the predictions of f by some Gaussian measurement noise, i.e., y; = f(x:) + 6,, where f ~ GP k) and 6, ~ NN(, a2) for some standard deviation an. Then, the distribution of training outputs y and model outputs f. := fxx at test points x., is jointly Gaussian, namely
N ((,
[(X,, x)+ + + X)
A(X,x.) x(X.,X.))
y f.
(1)
To generate predictions for test set X, = (Ø).... x,,') ). one can derive the posterior predictive distribution conditioned on the training set (Rasmussen & Williams| 2006), namely fx| A/ (H Ky whose mean vector and covariance matrix are given by the expressions
(xy = K (2x, k(x,, xx) kb X)
+ - y,
(2)
(3)
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The advantage of GPs is that the smoothness of the training regularises their behaviour, which is controlled by the kernel function. We focus on kernels from the Matérn family, stationary kernels of the form:
(a,, x²) la x') = 0² 21~ [(D)
V2 || - K. (V2 -
x')
(4)
K
K
for < OC, where [(2) is the Gamma function and K2 the modified Bessel function of the second kind. Matérn family kernels are favoured due to their interpretable behaviour with respect to their hyperparameters. Specifically, o, K. V control the GP's variability, smoothness and mean-squared differentiability. Moreover, the well-known squared exponential kernel, also known radial basis function 6 (x - =x) = x exp (-11x x' x'12²/2 /1²/2x²) obtained the limit as v + oo.
3.2 SCALAR-VALUED GPs ON RIEMANNIAN MANIFOLDS
In addition to their interpretable hyperparameters, Matérn GPs lend themselves to generalisation over Euclidean domains. A formulation that will allow extension to the vector-valued case is the one by Whittle (1963) who has shown that in Euclidean domains X = RRd Matérn GPs can be viewed as a stationary stochastic process satisfying the stochastic partial differential equation
+
f=W,
(5)
R²
where is the Laplacian and W is the Gaussian white noise. Likewise, for v -> oc, the limiting GP satisfies exp(-x²A/4)) A/4)f W, where the left-hand side has the form of the heat kernel.
As shown by Boroovitkkiy et al. (. Eq. 5 readily allows generalising scalar-valued (d 1) Matérn GPs to compact Riemannian manifolds X MM by replacing A by the Caplace-Beltrang operator AM- The corresponding GPs are defined by SM ~ GP(O, kM), with kernel
kM (1)xxx) = C₁
0²
Σ
2v
+
f((x)f((z'),
(6)
R2
1=0
where (Ai, are the eigenvalue- pairs of AM and C₁ is a normalisation factor.
3.3 SEALAR-VALURE GPs ON GRAPHS
Analogously, scalar-valued Matérn GPs can be defined graph domains, X G (Borovitskiy et al. 2021) by using the graph Laplacian in place of A M as its discrete approximation. Specifically, the graph Laplacian is L = D W with weighted adjacency matrix W E Raxx of and diagonal node degree matrix D = diag(W1 The graph Laplacian admits a spectral decomposition.
(7) where U is the matrix of eigenvectors and A is the diagonal matrix eigenvalues and, by the spectral theorem, ((L) U U for some function & R + R. Therefore, choosing (())
L = UAUT.
(2v/n² + x) obtains the operator on the left-hand side of Eq. up to a scaling factor) Using this, one may analogously write
L
W.
(8)
for vector R". Thus, analogously to Eq. fG ~ GP(O,, kc), with kernel
the scalar-valued GP on graph becomes
q) = =c ()) Pu(g)" (u(()"
(9)
where u(i), ulf) are the columns of corresponding to nodes i, E V. respectively.
INTRINSIC REPRESENTATION OF VECTOR FIELDS OVER ARBITRARY LATENT MANIFOLDS
We may now construct GPs on unknown manifolds and associated tangent bundles.
Note the different signs due the opposite sign convention of the Laplacians.
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A
Vector field on unknown manifold
Approximate manifold
C
Approxmate tangent bundle
Positional encoding
Veclor field predictin
T.M
TM
Riemamiian Veder Feld G (RVGP) JTM
M
Iransport
Cornector on Lap 00 (kghTsSicier
Figure 1: Construction of vector-valued Gaussian processes on unknown manifolds. A Input consists of samples from a vector field over latent manifold M. B The manifold is approximated by a proximity graph. Black circles mark two sample points, and and their graph neighbourhood. The tangent bundle is a collection of locally Euclidean vector spaces over M. It is approximated by parallel transport maps between local tangent space approximations. D The eigenvectors of the connection Laplacian
used positional encoding to define the GP that Iearns the vector field. E The GP is evaluated as unseen points to predict the smoothest vector field that consistent with the training data. We this GP to accurately predict singularities, where sampling is typically sparse.
4.1 VECTOR-VALURE GPs ON UNKNOWN MANIFOLDS
We consider training data consisting of pairs { u²) L. .z). where I; are samples from a manifold M C Rdd and e are sampled from the tangent bundle TM U,T.M. If the dimension of the manifold is m S d the tangent spaces T:M := Ma anchored to are isomorphic as a vector space to Rm Importantly, we assume that both .M and T.M are unknown and seek a GP to provide an implicit description of the vector field over M that agrees with the training set and provides continuous interpolation at out-of-sample test points with controllable smoothness properties.
Approximating the manifold and the tangent bundle We first fit a proximity graph G = (V, to X, defined based on some notion of similarity (spatial or otherwise) in the data. While G approx- imates M, it will not restrict the domain to V in Boooviitkki al. (202T). Then, to approximate TM, note that the tangent spaces do not have preferred coordinates. However, being isomorphic to R.. T.M can be parametrised by m orthogonal vectors the ambient space R to form local frame, or gauge, T,, To obtain this frame, we take vectors eiy E F from i to N nearest nodes j, assuming that they span T.M and form matrix by stacking them column-wise. The left singular vectors corresponding to the m largest singular values yield the desired frame
T, = (t(" (!!))
((1)) E Rdxm
(10)
Then, ú; = T u. acts as a projection of the signal to the tangent space in the & sense. Note that m does not need to be known ahead of time but is estimated as the average of the dominant single values across all estimated tangent spaces, c.g, based on a cutoff.
Constraining the vector field over the manifold Armed with the approximation of M and TMM, by G and (T.). we may define the connection Laplacian operator L, that will regularise the GP's behaviour by using the global smoothness of the vector field.
The notion of vector field smoothness is formalised by the parallel transport map P Pj-vi that aligns TjM with T.M allow the comparison of vectors in common space. While parallel transport is generally path dependent, we assume that i, are close enough such that P is the unique smallest rotation. Indeed, constructing the nearest neighbour proximity graph limits pairs i, to be close in space. This is known as the Lévy-Civita connection and is computed as matrix Oj E O(m) in the orthogonal group (rotation and reflection)
Oj: = arg min
|[T - -T,O|L.
(11)
(m)
where || || is the Frobenius norm and is uniquely computable in ((m)-time (Kabsch 1976).
'n practice, we pick N closest nodes to i on the proximity graph where N is a hyperparameter. Larger N increases the overlaps hetween the nearby tangent spaces. We find that N = 2D is often good compromise between locality and robustness noise of the tangent space approximation.
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Using the parallel transport maps, define the connection Laplacian (Singer & Wu 2012). block matrix Le E Rmmxxm, whose (i block entry is given by
= {W
{ DaaTmxm for ij for i adjacent.
Lo(i.5)
(12)
O
Let us remark that Le prescribes the smoothness of the vector field over an underlying continuous manifold that agrees with the available training data. In fact, as n + co, the eigenvectors of Le converge to the eigenfunctions of the connection Laplacian over the tangent bundle (Singer & Wu 2017), such that the corresponding continuous signal satisfies the associated vector diffusion process (Berline N. 1996). The solution of this diffusion process minimises the vector Dirichlet energy LIJEE Wig 01- Oit which quantifies the smoothness of the vector field (Knöppel et al. 2015).
Vector-field GP on arbitrary latent manifolds We can now define GP to regress the vector field over M. To this end. we consider a positional encoding of points on the tangent bundle TM based on the spectrum of the connection Laplacian, L. U,A.U where Ac, U, E R Compared with Eq. 7 where each node corresponds to a point, each node now represents a vector space of dimension m. Thus, the positional encoding of some vector at node i is given by an lmmx xk matrix, rather than an 1x vector, whose columns are the eigenvectors corresponding to the k smallest eigenvalues in A. and rows are the coordinates of T.M:
Wim,1
Uim,k kk
(Ū.): = Vnm
E ERmx
(13)
((++1)m.1
((++1)m,k/
The matrix (Ü.); can also be thought of as the i-th slice of an Raxmxk tensor. This allows us to define the positional encoding of by mapping the eigencoordinates defined in the respective tangent spaces using T; back into the ambient space
P. TŪ.)) E gd
(14)
Then, by analogy to Eqs. 5 6 we define the vector-valued Matérn GP frM : TM + R represent- ing the vector field over the manifold with -gduud kernel (Fig. 2 top)
KTM(U, (,
(15)
where v, v E TM and A. = (Ac)1:H..:k Note that we recover fM for scalar signals (m 1), where Caplace-Beltrane
operator equals the connection Laplacian for a trivial line bundle on M. Therefore, the tangent spaces become trivial (scalar), recovering the well-known Laplacian eigenmaps Pa
n muu......... mi.k). Likewise, ( w²) reduces to the ecalar-valued kernel (M( for æ, x E M in Eq. 6 However, kM x²) and ( v') are not linearly related because the underlying Laplace- Laplace-Beltrami and connection Laplacian operators are linked by the cur- vature of the manifold as given by the Weitzenböck identity. Note that the kernel in Eq. [5] differs from that defined in Hutchinson et al. (021), which considers an isometric projection of the tangent bundle into Euclidean space and constructs a valued kernel therein (Fig. 2A, bottom). Instead of approximating this isometric projection, which may be challenging for unknown manifolds, our constructive approach uses an intrinsic parametrisation of the manifold using only local similarities. This yields matrix-valued kernel (Fig, ZA top). which accounts for alignment of the vector field to principal curvature directions over M. This is made explicit using the connection Laplacian po- sitional encoding (Eq. [3) and tangent spaces (Eq. TT). In Sect. 5, we show that this construction leads to specific performance advantages in recovering vector field singularities.
4.2 SCALABLE TRAINING VIA INDUCING POINT METHODS
drawback of GPs is their computational inefficiency, driven by the need to compute an n X n. covariance matrix. Therefore, inducing point methods, which reduce the effective number of data points to a set of n'' < n inducing points, have become mainstay of GPs in practice (Titsias) 2009). Hutchinson al. (202 showed that inducing point methods, e.g., Titsias 2209),, are extendible to vector fields over Riemannian manifolds, provided the covariance matrix of inducing points is represented in tangent space coordinates. Since the kernel of RVGP is constructed from positional encoding using connection Laplacian eigenveetors, which are expressed in local coordinates, our method is readily compatible with inducing point methods, which we provide implementation of.

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Superresclution
Inpainting (RBF kemel)
Inpainting (RVGP)
RVGP kernel
algnment
0.-
traniing tost
lest
0.4
02-
R0NPP
diam.)
Scalaar kernel
Meen aignment
Figure 2: Superresolution and inpainting. Matrix-valued kernel (RVGP) against a scalar-valued kernel (e.g., in Hutchinson et al. (202T)) B Uniformly distributed samples over the Stanford bunny and torus are interpolated to higher resolution (k 50). C Ablation studies for the Stanford bunny, showing the dependence of aligmment of the superresolved vector fields in the test set against data density quantified by the average distance between manifold points a (for & = 50 fixed) and the number of eigenvectors k. The vectorial RVGP representation is compared against channel- representation using an RBF kernel with Laplacian eigenvectors as positional encoding. D Predic- tion of singularity in masked area. RBF kernel predicts discontinuities along the masked boundary (triangle), vectors that protrude the mesh surface (star) and do not converge to zero magnitude at the singularity. RVGP predicts smoothly varying inpainting (I: 50).
5
EXPERIMENTS
5.1
MANIFOLD-CONSISTENN INTERPOLATION OF VECTOR FIELDS
We expected that RVGP fitted to sparse vector samples over an unknown manifold would leverage the global regularity of the vector field provide accurate out-of-sample predictions. Thus, we ducted two experiments on the Stanford bunny and toroidal surface mesh to test the RVGP's ability super-resolve sparse samples and inpaint missing regions containing singularities on diverse man- ifold topologies. Given n uniformly sampled anchor points {æ;} on the surface mesh, we generated a smooth ground truth vector signal over these points by sampling vectors {w;) from a uniform distribution on the sphere S², projecting them onto their respective tangent spaces ú; TT v; and using the vector heat method (Sharp et al.) 2019) to find the smoothest vector field. Specifically, concatenating signals as 6 0= IR | the vector heat method obtains e + b(r)/u(T), where iT) = 0 exp (- ) is the solution of the vector heat equation and u(r) = | exp ( LT) is the solution of the scalar heat equation. We ran the process until diffusion time 100. On surfaces of genus this process will lead to at least one singularity.
Superresolution First, we asked if RVGP can smoothly interpolate (also known as super-resolve) the vector field from sparse samples. To this end, we fitted a graph the bunny mesh and trained RVGP using vectors over 50% the nodes, holding out the rest for testing. For benchmarking. also trained a radial basis function (RBF) kernel that treats vector entries as independent scalar channels. We found that RVGP predictions were in excellent visual alignment with the training vector field for dense (Fig. 2B) and sparse data (Fig. S2). The predictions of our model remain accurate when the points do not lie on the manifold surface but are drawn from a distribution centred on the manifold. Indeed, when we added Gaussian geometric noise of increasing standard deviation to manifold points, we found that the prediction accuracy was largely unaffected (Fig. (3))
We conducted two ablation studies to further investigate the robustness of RVGP predictions. First, we subsampled the surface mesh using the furthest point sampling algorithm (Qi et al.) (017)) to simultaneously reduce the resolution and the amount of data used for training. Here, a parameter a controls the average pairwise distance of points relative to the manifold's diameter. As quantified by the inner product between predicted and test vectors, RVGP produced accurate alignment with only 10 cigenvectors over broad range of sampling densities (Fig. 2C). By contrast, the benchmark method suffered a drop in performance at lower densities (higher a).. Next, we found that on a nigh-resoluttin surface (a 1.5%), RVGP yields high accuracy already with few eigenvectors
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(k:) which progressively increased with k. By contrast, the benchmark achieved good representation only from k = 50 eigenvectors, thus achieving inferior dimensionality reduction.
Inpainting In the second experiment, we tested RVGP's ability to inpaint whole vector field re- gions containing singularities. This experiment is more challenging than classical inpainting because it requires our method to infer the smoothest topologically consistent vector field. To this end, we masked off the vortex singularity and used the remaining points to train RVGP. We found that vec- tors predicted by RVGP closely followed the mesh surface, aligned with the training set on the mask boundary and smoothly resolved the singularity by gradually reducing the vector amplitudes to zero (Fig. (2)). By contrast, the RBF kernel did not produce vectors with smooth transitions on the mask boundary and often protruded from the mesh surface, showing that vectors treated as independent scalar fields do not capture the geometry of the tangent spaces. Thus, the connection Laplacian positional encoding provides sufficient regularity to learn vector fields over complex shapes.
5.2 SUPERRESOLUTION OF EEG DATA
Finally, as a biologically and clinically relevant use case, we applied RVGP to superresolve elec- troencephalography (EEG) recordings from humans. EEG recordings measure spatiotemporal wave patterns associated with neural dynamics, which play a fundamental role in human behaviour (Sato) jtt al. 2012; Xu et al. 2023). Accurately resolving these dynamics requires high-density EEG se- tups in excess of 200 channels (Robinson et al., 2017, [Seeber et al. 2019: (Siclari et al., 2018).
However, due to long setup times during which signal quality can rapidly degrade, experimentalists and clinicians commonly resort to low-density recordings with 32 or 64 channels (Chu 2015).
Thus, we asked whether superresolving low-density 64-channel recordings (Fig. AA using RVGP (Fig. BC) can facilitate biological discovery and clinical diagnostics. As ground truth, we collected 256-channel EEG recordings of resting-state brain activity from 33 Alzheimer's 's patients (AD) and 28 age- metchad healthy controls (sce Appendix I..I) We focused on www-frequency alpha waves spanning 8-15 Hz, which represent the dominant rhythm at rest (Berger 1934) and exhibits impaired dynamics in AD (Moretti et al. 2004 [Besthorn et al.) 1994 [auwell et al. 2010). After prepro- cessing of EEG time series (see Appendix [I. we used a triangulated mesh of the scalp, with 256 known electrode locations as vertices, and finite differencing to compute the corresponding vectors (Fig. BBC (ground truth), Appendix(A.2) Then, we constructed RVGP kernels using the connection Laplacian eigenvectors derived from K-nearest neighbour graph (K 5) fit to vertices. Finally, we trained RVGP using vectors at 64 training vertices (Fig. 3) and used it to infer the vectors at the remaining 192 test vertices (Fig. BC). As benchmark, we used a channel-wise interpolation of the vector field using linear, spline and RBF kernel methods.
To assess the quality of reconstructions, we computed the divergence and curl of the predicted EEG vector field and computed the mean absolute error (MAE) relative to the ground truth 256-channel EEG. Divergence and curl have previously been used to identify singularities in neural wave patterns in human neuroimaging and have been linked to cognitive function and behaviour (Roberts et al.) 2019 Xu et al., 2023). We found that the RVGP reconstruction closely approximated the diver- gence and curl of the high-density EEG (Fig. 3C) For visual comparison, we show a snapshot with characteristic vector field singularities such as sources, sinks and vortices, which were significantly better preserved by RVGP than benchmarks. Linear interpolation underfitted the vector field, while spline interpolation introduced spurious local structures. RBF kernel interpolation performed most similarly to RVGP due the homogeneous curvature of the human head. This observation is cor- roborated by significantly lower angular eиror and curl-divergence MAE for all subjects for RVGP compared with benchmarks (n 61. Fig. 3P)). To test the variability against data density variation, we repeated the experiment for 32-channel EEG signal and found that the errors were too large for all methods to be of practical relevance (Fig. ST). This shows that 64-channel EEG represents an empirical limit for resolving small-scale singularities.
Given the superior reconstruction accuracy of RVGP, we asked if it could enhance the classification accuracy for patients with Alzheimer's 's Disease (AD). Contemporary diagnostics for AD are costly, invasive, and laborious (Zetterberg & Bendlin 2021). We instead employed a linear support vector machine to classify AD patients versus age- matched healthy controls based on the reconstructed divergence and curl fields derived from a brief, one- minute resting-state low-density EEG recording procedure that can be feasibly integrated into clinical settings due to its non-invasive nature and
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Lovidensity EEG channels)
High-density ground-truth channels)
RVGP prediction
Singulsrity prediction
0.2
and avdrgeageecc
MAE (cmm6)
0.0
Bnkk
vartex
RVEP
R3F
Spline
Lincar
Low-density vector tield
Linar preciction
Spine prodiction
0.0
True poslive
0.
Ground buh
RVGF
RSF
Spline
0.2
0.4
0.6
Palse postive rato
Figure 3: Reconstruction of spatiotemporal wave patterns in human FEG. Snapshot of an alpha wave pattern (8-15 Hz) recorded low density (64 channel) EEG from healthy subject projected in two dimensions. B Phase field of an alpha wave. Vector field denotes the the spatial gradient of the voltage signal. Ground-truth and reconstructed high-density phase field (256 chan- nel) using RVGP, linear and spline interpolation. Streamlines, computed based on the vector field, highlight features of the phase field. RVGP significantly better preserves singularities, i.e., sources, sinks and vortices. D Reconstruction accuracy, measured by the preservation of singularities. E Receiver operating characteristic (ROC) for binary classification of patients with Alzheimer's 's dis- case against healthy controls using linear support vector machine trained on the divergence and vorticity fields. Shaded areas indicate a 95% confidence interval.
cost-efficieney. Our results show significantly higher than state-of-the-ar classification accuracy, approaching that derived from the ground truth high-density EEG signal (Fig. 3E).
6 DISCUSSION
We introduced RVGP, a novel extension of Gaussian processes designed to model vector fields on latent Riemannian manifolds. Utilising the spectrum of the connection Laplacian operator, RVGP intrinsically captures the manifold's geometry and topology and the vector field's smoothness. This enables the method to learn global patterns while preserving singularities, filling a significant gap in existing approaches limited to known, analytically tractable manifolds. A key strength of RVGP is its data-driven, intrinsic construction via a proximity graph, which enhances its practical utility by making it highly applicable to real-world datasets where explicit manifold parametrisation, e.g., for spheres and tori, often unavailable. Demonstrated across diverse scientific domains such as neuroscience and geometric data analysis, RVGP advances the field of gcometrically-infore probabilistic modelling and offers a statistical tool for various high-impact applications, including clinical neuroscience, that is robust to sampling density and noisy data.
As RVGP uses proximity graph to approximate the manifold, one needs a sampling density that high enough, combined with suitable similarity metric and graph algorithm to build an intrinsic approximation of the latent manifold, especially in high dimensions. Otherwise, RVGP will use the closest manifold approximated by thresholding the spectral decomposition of the connection Laplacian. Further, although we find that RVGP gives higher reconstruction accuracy than modelling the signal channel-wise on complex manifolds, as expected, this difference diminishes when the manifold curvature is homogeneous. In addition, bundle the manifold. However, one may also consider non-tangent but smooth vector fields by finding (non-unique) SO(d) (d) rotation that maps the vectors into the tangent space, applying RVGP, and mapping back the predicted vectors by inverting the rotation. Future work may also study when the tangent spaces not of uniform dimension by considering the sheaf Laplacian operator.
considered vector fields that lie in the tangent

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A
EEG ANALYSIS
The following analyses were scripted in MATLAB R2023a:
A.1 EXPERIMENTAL DATA, AND PRE-PROCESSING
Experimental data: We analysed 10-minute resting state EEG recordings from a cohort of 61 subjects containing patients with a clinical diagnosis of AD and age-matched healthy control subjects. Data was acquired from the clinical imaging facility at Hammersmith Hospital in London. The AD group consisted of 15 Females and 18 Males with mean age of 76 years (std = 7.8 years), healthy control group contained 15 Females and 13 Males with mean age of 77 (std 4.9 years). 10 par- ticipants were excluded from the original full cohort of 71 subjects due to poor data quality, During pre- pe-pprocessing the EEG bandpass filtered from 2-40 Hz with second-order Butterworth filter and downsampled to 250Hz. Channels exceeding a kurtosis threshold of were rejected before av- erage re-referencing. We performed independent component analysis with the Picard algorithm and applied an automatic artefact rejection algorithm in eeglab (Delorme & Makeig 2004 with a maximum of of components removed from each dataset. The pre-processed EEG was then filtered in the alpha band (8- (8-15Hz). The low-density EEG consisted of 64 channels sampled evenly across the scalp. These channels were then used to reconstruct the full 256-channel high-density recording.
A.2 WAVE VELOCITIES
We calculated the velocity vector field at each time point using a method similar to Roberts et al. 2019). We compute the instantancous phase each channel using the Hilbert transform and estimate the wave velocity v from the spatial and temporal derivatives of the unwrapped phase Φ(x, U, 2, t), (CNEM) (Illoul & Lorong, 2011). CNEM is a mesh-free calculus method for solving partial dif- ferential equations that avoids artefacts that can arise from mesh-based interpolation of nodes posi- tioned on the outer boundary of the brain's 's convex hull.
- |tt0ll// Vøll2 implemented using the constrained natural element method
A.3 LINEAR AND SPHERICAL SPLINE INTERPOLATION
Spherical spline interpolation was carried out on the raw EEG for each subject using 64 electrodes sampled evenly across the scalp and implemented in eeglab (Delorme & Makeig (004). Each channel was mapped onto a unit sphere, and the electrical potential at the missing channel locations was interpolated using the spline function, which weights the contributions of the neighbouring electrode based on its distance to the interpolation point (Perrin et al. 1989).
Linear interpolation was performed the 64-node 3-dimensional phase flow field. The gradient vector for each node was projected onto the tangent plane prior to linear interpolation.
A.4 FEATURE EXTRACTION
To determine the behaviourally relevant regions of the cortex, we computed the time-resolved diver- gence and vorticity fields for each subject. Next, estimated the probability density of sources, sinks and spirals by computing the mean positive/negative vorticity and mean positive/negative di- vergence fields thresholded
and < -1 averaged over time for each node. This produced four cortical probability maps per subject; mean source probability (positive divergence, D > 1), mean sink probability (negative divergence D -1), clockwise spiral probability (positive vorticity, C 1), anticlockwise spiral probability (negative vorticity, C < -1). We extracted two predictors from these four probability maps. The first predictor the probability ratio of sources sinks a given node. The second predictor was the probability ratio of clockwise to anti-clockwise spirals given node:
(A(A)
R, = -logi0
(16)
P.((B)
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where (A) represents either the probability of a source clockwise spiral node and (B) represents the probability of sink or an anti-clockwise at node n. We ran a group-level cluster permutation analysis to identify nodes where the source-to-sink. ratio or spiral ratio differed between the patients with AD and healthy controls the reference EEG (alpha 0.05, 50000 permutation). The three clusters with the lowest p-value were then used as seed regions where we extracted the source-to-sink probability ratio and clockwise to anti-clockwise spiral probability ratio in the interpolated EEG for each reconstruction method (RVGP, spline and linear interpolation).
A.5 BINARY CLASSIFICATION
To evaluate whether the RVGP reconstruction contained relevant information about cognitive func- tion, we tested whether healthy controls and Alzheimer's 's disease patients could be accurately clas- sified by the ratio source to sinks and the ratio of clockwise to anti-clockwise spirals. We tested the classification accuracy using three seed brain regions and compared the accuracy from each constructed approach (RVGP, spline, linear interpolation) against the high-density reference EEG. For binary classification, used linear support vector machine and computed the accuracy and ROC curves for each approach separately using 10-fold cross-validation classifiers categorise pooled samples from all 3 brain regions and across subjects into each class (AD or healthy controls). For each training epoch, we fit the posterior distribution of the scores to 90% of the data and used this determine the probability of each sample belonging to each class in the test set (10%). To handle class imbalance, we repeated the classification 500 times, with each iteration trained on a random subset 100 AD and 100 healthy control samples. The AUC and ROC reported in the results were calculated from the distribution estimated over 500 iterations.
We trained two one-vs-all
A.6 VISUALISATION
For visualisation, the y, coordinates for each channel location were flattened onto a 2D grid us- ing multi-dimensional scaling to most accurately preserve the local distances between nodes across the scalp. For plotting, we use the 2D tangent vectors of the phase field and streamlines computed using the 'streamslice' function in MATLAB.
A.7 DATA COLLECTION
Data was collected in the UKDRI Care Research & Technology Centre at the Micheal Uren Hub in London from 2022-2023. AD exclusion criteria were a previous history of head injury or any other neurological condition. All patients had clinical diagnosis of AD, with subject retrospectively excluded after a change in diagnosis to frontal-temporal lobe dementia (FTLD). Subjects on AD medication were asked to abstain the day of testing and to avoid caffeine. 11 subjects were excluded from EEG analysis due to low signal quality, leaving 61 subjects total (33 AD, 28 controls).
Singularity prediction
Curl and diverge rgence
MAE (cm/s)
1.5
Divergencs
RVGP
RBF
Linear
Figure S1: Reconstruction of spatiotemporal wave patterns in 32-channel human EEG. Recon- struction accuracy, measured by the preservation of singularities. The accuracy is 10-fold lower than using 64-channel BEG, meaning that predictive power is lost for all methods at this resolution.
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training test
Figure S2: Superresolution for sparse training data. Same as in Fig. 2A, but for points placed at average distance of 5% of manifold diameter.
Standard deviation: 0.5% diam.
Standard deviation: 5% diam.
Insiiing test
Mean nmmnt
points
make larning led
02-
2
Standard deviation (% diam.)
Figure S3: Training and prediction for off-manifold points. To test the regularity in our method, took distribution of on-manifold points, spaced approximately 3% of manifold diameter, and added additive Gaussian noise to push them off the manifold. We increased the noise until the two dominant dimensions of the tangent space approximations explained at least 80% of the variance, amounting to a standard deviation of approximately 5% of the manifold diameter. We repeated the experiment twice, once with noise affecting only test data points and once adding noise all data points. We trained the model as described in the main text in Section 5.1. Parameters used: k = 50.
16


Reviews:
[{"review_id": "cyEVkRalU0", "reviewer": "Reviewer_3KnT", "summary": "Gaussian processes is a popular Bayesian method that easily incorporates prior knowledge and provides good uncertainty quantification. GP is originally defined in Euclidean space which limits its application in certain domains. In this paper, the author proposes the Riemannian manifold vector field GP (RVGP) which extends GP to learn vector signals over latent Riemannian manifolds with the use of connection Laplacian operator. Experiment results show RVGP can encode the manifold and vector field's smoothness as inductive biases and have good performances on electroencephalography recordings in the biological domain.", "questions": "See weakness above.", "limitations": "", "rating": 8, "confidence": 2, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "- Extending GP to vector-valued signals is novel. The proposed method also removes the common assumption that the manifold is known in non-Eucleadian GP, which expands GP's applicability.\n- The paper is well written.", "weaknesses": "- One advantage of GP is data efficiency. In section 5.1's superresolution experiment RVGP is trained using vectors over 50% of the nodes, I would be interested to see the results when trained with less data.\n- In section 5.2 the authors compare RVGP with interpolation methods, which might be a too simple baseline comparison. The RVGP's performance is also close to linear prediction. I would be interested to see the comparison against a stronger baseline method. \n- A section discussing the limitations of the proposed method would be good.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ELEQaRURRx", "reviewer": "Reviewer_TuVU", "summary": "This paper looks at performing Gaussian Process regression over vector fields on unknown manifolds.\n\nThe underlying manifold and tangent space is estimated by combining a proximity graph approach to modelling the underlying manifold, and approximating the tangents space by taking the highest singular values of the matrix of directions to neighbours.\n\nThe discretised connection Laplacian on these tangent spaces is then used to construct a kernel by projecting the spectral decomposition of the connection laplacian onto the estimated tangent spaces.\n\nThis kernel is then used in a number of experiments, from some simple inpainting and super-resolution tasks to superresolution of real EEG data, and show improved diagnostic capabilities using this method.", "questions": "- Can you explain the procedure of constructing the kernel in \"Vector-field GP on arbitrary latent manifolds\"? \n- How does this kernel differ from using the method of Hutchinson et. al. more directly? I.e. Using the scalar kernel defined by the graph laplacian, $k(i,j)$, from this creating a diagonal kernel $ K(i,j) = k(i,j) * I_{d\\times d}$, and then restricting this to the estimated tangent spaces, $\\mathbb{K}(i, j) = \\mathbb{T}_i K(i,j)  \\mathbb{T}_j$\n- Presumably one needs to know the dimension of the unknown manifold ahead of time?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The method is clean and simple\n- The method demonstrably works in the single task presented\n- Most of the paper is easy to follow", "weaknesses": "- I found the section \"Vector-field GP on arbitrary latent manifolds\" difficult to follow. For example it is not clear to me what $(U_c)_i$ is. To me this would denote the $i'th$ row, but it clearly is not as it is the wrong shape. Also in equation 15, $\\Phi(\\Lambda_C)^{-2}$ is a $\\mathbb{R}^{mn \\times mn}$ matrix, but is being producted with $P_v$, a $\\mathbb{R}^{d\\times k}$ matrix?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "A8YyzCpqVV", "reviewer": "Reviewer_bAaw", "summary": "This paper presents a Riemannian manifold vector field Gaussian process (RVGP), a generalization of Gaussian processes (GPs) for learning vector signals over latent Riemannian manifolds. The core of the idea is to use positional encoding derived from the connection Laplacian. The authors demonstrated the effectiveness of the proposed method via super resolution and inpainting for the vector field on a 3D mesh and EEG analysis.", "questions": "1. The reviewer has a question about the statement \"vector field on latent Riemannian manifolds\". I suppose two interpretations; each of the vector itself should lie on the manifold, or the domain of the field is enclosed on the manifold but the vector can be out of the manifold. In the experimental result (Fig. 2C), the authors point out the \"vectors that protrude the mesh surface\". For the former, this should be an undesirable result, but for the latter, it is okay. An example of the latter case is the normal vector of the Stanford bunny for Figure 2. It is also possible to consider such a problem, but I'm curious if the proposed method can model it. At least I expect the proposed method to work for $m=n-1$ (2D manifold in 3D space), but I doubt it for $m<n-1$ because the Levy-Civita connection is insufficient to address the complementary subspace of the tangent space.\n\n2. The reviewer has some doubts about the term \"unknown manifold\". For 5.1, the training data is dense enough to approximate the manifold well. For 5.2, the authors \"constructed RVGP kernels using the connection Laplacian eigenvectors derived from a k-nearest neighbour graph (k = 5) fit to mesh vertices\". It seems that the Laplacian was computed from 256 vertices, not n=61. If I understand correctly, this violates the prerequisite of \"unknown manifold\". Is there no value in analyzing the relationship between the coverage of the data for the manifold and the accuracy of the vector field prediction? I'm also curious about the analysis of the predicted vector field for out of the manifold to check for regularity.", "limitations": "", "rating": 6, "confidence": 5, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "- The theoretical part is mostly well written and technically sound.\n- Related work is well addressed.\n- Practicality is well demonstrated on real data.", "weaknesses": "Lack of precision in some statements.\n- V is defined multiply: for vectors and for the nodes of the graph. One of the important statements \"While G approximates M it will not restrict the domain to V\" becomes unclear.\n- The shape of $O_{ij}$ is unclear. It seems to be $m \\times m$ for eq. 11, but it seems to be $d \\times d$ for eq.12. (The rank of $O_{ij}$ will be m.) The shape of $L_{c}$ is also unclear. It is clearly defined as $nd \\times nd$ in eq. 12, but it is inconsistent with $\\Lambda_{c},U_{c} \\in R^{nm\\times nm}$. Maybe something is wrong.\n- A quantitative evaluation and analysis is missing for 5.1. I could not judge whether \"smoothly resolved the singularity by gradually reducing the vector amplitudes to zero\" is okay or not. Does this mean that the result is different from the ground truth? A quantitative evaluation is also appreciated.\n\nDetailed comments:\n- Something wrong: {($x_i$}), $\\hat{v} = ∥_{i=0}^{n}\\hat{v}_{i}$\n- I could not understand what the authors were doing: \"We then sampled corresponding vectors {$v_{i}$} from a uniform distribution on the sphere\".\n- $S^{3}$ should be $S^{2}$", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (poster)