Manifold Diffusion Fields
Authors: Ahmed A. A. Elhag
Yuyang Wang
Joshua M. Susskind
Miguel Ángel Bautista
Venue: ICLR 2024 poster (2024)
Keywords: Diffusion models
riemannian manifolds
graphs
Primary area: generative models
TL;DR: A diffusion model for functions on manifolds and graphs

Abstract: We present Manifold Diffusion Fields (MDF), an approach that unlocks learning of diffusion models of data in general non-euclidean geometries. Leveraging insights from spectral geometry analysis, we define an intrinsic coordinate system on the manifold via the eigen-functions of the Laplace-Beltrami Operator.  MDF represents functions using an explicit parametrization formed by a set of multiple input-output pairs. Our approach allows to sample continuous functions on manifolds and is invariant with respect to rigid and isometric transformations of the manifold. In addition, we show that MDF generalizes to the case where the training set contains functions on different manifolds. Empirical results on multiple datasets and manifolds including challenging scientific problems like weather prediction or  molecular conformation show that MDF can capture distributions of such functions with better diversity and fidelity than previous approaches.

Published as a conference paper at ICLR 2024
MANIFOLD DIFFUSION FIELDS
Ahmed A. Elhag? ; Yuyang Wang, Joshua M. Susskind, Miguel Angel Bautista
Apple {aa_e ehag, yuyang_wang4, jausskind, mbaut (haattsstamrrtinn8@ppll......
ABSTRACT
We present Manifold Diffusion Fields (MDF), an approach that unlocks learning of diffusion models data in general non-Euclidean geometries. Leveraging insights from spectral geometry analysis, define intrinsic coordinate system the manifold via the sggenffunction of the Laplace- Beltrami Operator. MDF represents functions using explicit parametrization formed by a set of multiple input-output pairs. Our approach allows to sample continuous functions on manifolds and is invariant with respect to rigid and isometric transformations of the manifold. In addition, we show that MDF generalizes to the case where the training set contains functions on different manifolds. Empirical results on multiple datasets and manifolds including challenging scientific problems like weather prediction or molecular conformation show that MDF can capture distributions of such functions with better diversity and fidelity than previous approaches.
INTRODUCTION
Approximating probability distributions from finite observational datasets is pivotal machine learning challenge, with recent strides made in areas like text (Brown et al., 2020), images (Nichol & Dhariwal, 2021), and video (Ho et al., 2022). The burgeoning interest in diffusion generative models (Ho et al.. 2020; Nichol & Dhariwal, 2021; Song et al., 2021b) can be attributed to their stable optimization goals and fewer training anomalies (Kodali et al., 2017). However, fully utilizing the potential of these models across scientific and engineering disciplines remains an open problem. While diffusion generative models excel in domains with Euclidean (i.e. flat) spaces like 2D images or 3D geometry and video, many scientific problems involve reasoning about continuous functions on curved spaces (i.e. Riemannian manifolds). Examples include climate observations on sphere (Hersbach et al., 2020; Lindgren et al., 20 or solving PDEs on curved surfaces, which is a crucial problem in areas like quantum mechanics (Bhabha, 1945) and molecular conformation (Jing et al., 2022). Recent works have tackled the problem of learning generative models of continuous functions following either adversarial formulations (Dupont et al., 2022b), latent parametrizations (Dupont al., 2022a; Du al., 2021; Bauer et al., 2023), diffusion models (Bond-Taylo & Willcocks, 2023; Zhuang et al., 2023). While these approaches have shown promise on functions within the Euclidean domain, the general case of learning generative models of functions on Riemannian manifolds remains unexplored.
In this paper, we introduce Manifold Diffusion Fields (MDF). extending generative models over functions to the Riemannian setting. We take the term function and field to have equivalent meaning throughout the paper. Note that these are not to confused with gradient vector fields typically used on manifold. These fields f: .M- - y map points from a manifold M (that might be parametrized as 3D mesh, graph or even a pointcloud, see Sect. 5.2) to corresponding values in signal space V MDF is trained collections of fields and learns a generative model that can sample different fields over manifold. Fig. we show real samples of such functions for different manifolds, as well as samples generated by MDF.
Here are our main contributions:
We borrow insights from spectral geometry analysis to define a coordinate system for points in manifolds using the eigen-functions of the Laplace- Beltrami Operator.
Laplace-Beltrami
"Work was completed while A.A.E intern with Apple.

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Published as a conference paper at ICLR 2024
S
Real samples
Samples generated by MDF
Figure MDF learns a distribution over a collection of fields f . + F where each field is defined on a manifold M. We show real samples and MDF's generations on different datasets of fields defined on different manifolds. First row: MNIST digits on the sine wave manifold. Second row Middle: ERA5 climate dataset (Hersbach et al., 2020) on the 21) sphere. Third row: GMM dataset on the bunny manifold. Fourth row: molecular conformations in GEOM-QM9 (Ruddigkeit et al., 2012) given the molecular graph.
We formulate end-to-end generative model for functions defined manifolds, allowing sampling different fields over a manifold. Focusing on practical settings, our extensive exper- imental evaluation covers graphs, meshes and pointclouds approximations of manifolds.
Our model outperforms recent approaches like (Zhuang et al., 2023; Dupont et al., 2022b), yielding diverse and high fidelity samples, while being robust to rigid and isometric manifold transformations. Results climate modeling datasets (Hersbach et al., 2020) and PDE problems show the practicality of MDF in scientific domains.
On the challenging problem of molecular conformer generation, MDF obtains state-of-the- results on GEOM-QM9 (Ruddigkeit et al., 2012).
RELATED WORK
Our approach extends recent efforts generative models for continuous functions in Euclidean space (Zhuang al., 2023; Dupont et al., 2022b;a; Du et al., 2021), shown Fig. 2(a). to functions defined over manifolds, see Fig. 2(b). The term Implicit Neural Representation (INR) is used in these works to denote parameterization of single function (e.g. single image in 2D) using neural network that maps the function's 's inputs (i.e. pixel coordinates) to its outputs (i.e. RGB values). Different approaches have been proposed to learn distributions GASP (Dupont et al., 2022b) leverages a GAN whose generator produces field data whereas point cloud discriminator operates on discretized data and aims differentiate real and generated functions. Two-stage approaches (Dupont et al., 2022a; Du et al., 2021) adopt a latent field parameterization (Park al., 2019) where functions are parameterized via a hyper-network (Ha et al., 2017) and generative model is learnt the latent INR representations. In addition. MDF also relates to recent work focusing on fitting a function (e.g. learning an INR) on manifold using an intrinsic coordinate system (Koestler et al., 2022: Grattarola & Vandergheynst, 2022), and generalizes it the problem of learning probabilistic model over multiple functions defined on a manifold.
fields in Euclidean space,
Intrinsic coordinate systems have also been recently used in the context of Graph Transform- ers(Maskey et al., 2022; Sharp et al., 2022: He et al., 2022: Dwivedi et al., 2020), where eigenvectors of the Graph Laplacian used replace standard positional embeddings (in addition to also
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f~gffi : R y po(f)f : y
[ p
(c) X x~ x(x/XX E
(a)
Figure 2: (a) Generative models of fields in Euclidean space (Zhuang et al., 2023; Dupont et al., 2022b;a; Du et al., 2021) learn a distribution pe over functions whose domain is R". We show an example where each function is the result of evaluating a Gaussian mixture with 3 random components 2D. (b) MDF learns distribution po from a collection of fields whose domain is general Riemannian manifold f q(f)lf M + V Similarly, as an illustrative example each function is the result of evaluating Gaussian mixture with 3 random components on M (i.e. the Stanford bunny). (c) Riemannian generative models (Bortoli et al., 2022; Gemici et al., 2016; Rozen et al., 2021; 1; Chen & Lipman, 2023) learn parametric distribution pe from an empirical observations (xx) x E .M of points on Riemannian manifold M, denoted by black dots on the manifold.
using edge features). In this setting, Graph Transformer architectures have been used for supervised learning problems like graph/node classification and regression, whereas we focus on generative modeling.
The learning problem we tackle with MDF can be interpreted as lifting the Riemannian generative modeling problem (Bortoli et al., 2022: Gemici et al., 2016; Rozen et al., 2021; Chen Lipman, 2023) to function spaces. Fig. 2(b)(c) show the training setting for the two problems, which are related but not directly comparable. MDF learns generative model over functions defined on manifolds, e.g. probability density over functions f M + y that map points in the manifold M to signal space V. In contrast, the goal in Riemannian generative modeling is to learn a probability density from an observed set of points living in a Riemannian manifold M. For example, in the case of the bunny, shown in Fig. 2(c), a Riemannian generative model learns a distribution points M on the manifold.
MDF is also related to work on Neural Processes (Garnelo et al., 2018; Kim et al., 2019; Dutordoir al., 2022), which also learn distributions over functions. As opposed the formulation of Neural Processes which optimizes an ELBO (Kingma & Welling, 2014) we formulate MDF as a denoising diffusion process in function space, which results in robust training objective and powerful inference process. Moreover, our work relates to formulations of Gaussian Processes (GP) on Riemannian manifolds (Borovitskiy al., 2020; Hutchinson et al., 2021). These approaches formulations of Riemannian generative modeling (see Fig. 2). in the sense that they learn conditional distributions of points on the manifold, as opposed to distributions over functions on the manifold like MDF.
GP
3 PRELIMINARIES
3.1 DENOISING DIFFUSION PROBABILISTIC MODELS
Denoising Diffusion Probabilistic Models (Ho et al., 2020) (DDPMs) belong to the broad family of latent variable models. We refer the reader to (Everett, 2013) for an in depth review. In short, to learn a parametric data distribution Py (mr) from an empirical distribution
finite samples g( DDPMs reverse diffusion Markov Chain that generates latents 11.7 by gradually adding Gaussian noise to the data 2o ~ (xx)) for T' time-steps as follows: g(0;) xx,1)) := N (x;-1i /ā,20. (1 a)) Here, āt is the cumulative product of fixed variances with handcrafted scheduling up to time-step t. (Ho et al., 2020) introduce an efficient training recipe in which: i) The forward process adopts sampling in closed form. ii) reversing the diffusion process is equivalent to learning a sequence of denoising (or score) networks eg. with tied weights. Reparameterizing the forward process as 3; va + /1 ā,e results in the "simple" DDPM loss: Ez-(o.T) (=)))=)(())))))) [[|e- ||le-dd co (Vāaz + vi āec, 1 which makes learning of the data distribution po (20)) both efficient and scalable.
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At inference time, we compute Dg po( via ancestral sampling (Ho al., 2020). Concretely,
we start by sampling ær ~ (0.I) and iteratively apply the score network eg to denoise æy, thus reversing the diffusion Markov Chain obtain zo. Sampling 2 Po x (x,) is equivalent
VI-a. Vi-o, eo + 2, where at each inference step stochastic component -O 1) is injected, resembling sampling via Langevin dynamics (Welling & Teh, 201 1). In practice, DDPMs have obtained amazing results for signals living an Euclidean grid (Nichol & Dhariwal, 2021; Ho et al., 2022). However, the extension to functions defined on curved manifolds remains an open problem.
to computing the update: 1t-1 Va:
3.2
RIEMANNIAN MANIFOLDS
Previous work on Riemannian generative models (Bortoli et al., 2022; Gemici et al., 2016; Rozen al., 2021; Chen & Lipman, 2023) develops machinery to learn distribution from a training set of points living on Riemannian manifolds. In this work, we assume manifolds are compact and connected Riemannian manifolds .M equipped with smooth metric g: T x M + Rzo (e.g. smoothly varying inner product from which a distance can be constructed on M). A core tool in Riemannian manifolds the tangent space, this space defines the tangent hyper-plane of point x .M and is denoted by M. This tangent space T is used to define inner products (u, g) u, u E M, which in turns defines g. The tangent bundle TM is defined as the collection of tangent spaces for all points M Væ E M.
In practice we cannot assume that for general geometries (e.g. geometries for which we don't 't have access closed form and are commonly represented as graphs/meshes) one can efficiently compute While it is possible to define an analytical form for the Riemannian metric g on simple parametric manifolds (e.g. hyper-spheres, hyperbolic spaces, tori), general geometries (i.e. the Stanford bunny) are inherently discrete and irregular, which can make it expensive to even approximate To mitigate these issues MDF is formulated from the ground up without relying access to analytical form for or the tangent bundle TM and allows for learning distribution functions defined on general geometries.
3.3 BELACC- OPERATOR
The Laplace-Beltramt Operator (LBO) denoted by AM is one of the cornerstones of differential geometry and can be intuitively understood generalization of the Laplace operator to functions defined on Riemannian manifolds M. Intuitively, the LBO encodes information about the curvature of the manifold and how it bends and twists every point, reflecting the intrinsic geometry. One of the basic uses of the Laplace-Beltrami Beltrami operator is to define a functional basis on the manifold by solving the general eigenvalue problem associated with A M. which a foundational technique in spectral geometry analysis (Lévy, 2006). The degopopccopooiition of AM the non-trivial solutions to the equation AMP; = App:. The eigen- sigen-functions ;p; M - I represent an orthonormal functional basis for the space of square integrable functions (Lévy, 2006; Minakshisundaram & Pleijel, 1949). Thus, one can express a square integrable function f M + y, with f E I,2 as a (f.9:)9...
linear combination of the functional basis, as follows:
In practice, the infinite sum is truncated to the & eigen-functions with lowest eigen-values, where the ordering of the eigen-values Å λ2 ... < > enables low-pass filter of the basis. Moreover, (Lévy, 2006) shows that the cigen-functions of AM can be interpreted as Fourier-like function basis (Vallet & Lévy, 2008) on the manifold, e.g. an intrinsic coordinate system for the manifold. In particular, if M 52 this functional basis is equivalent to spherical harmonics, and in Euclidean space it becomes Fourier basis which is typically used in implicit representations (Xie et al., 2022). MDF uses the for points on M (see Fig. 3). Note that these figen-functions are only defined for points that lie on the manifold, making MDF strictly operate on the manifold.
functions of the LBO AM to define a Fourier- like positional embedding (PE)
4 METHOD
MDF is a diffusion generative model that captures distributions over fields defined on a Riemannian manifold M. We are given observations in the form of empirical distribution fo ((fo) over
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px() (om cos(2nb,, bi x), dm sin(2mb,, b
(x)x))))=)) mm
xxx) V (x)(),) (P2())))))))))))))))))))) .... (x()]]
Figure 3: Left: Fourier PE of a point æ in 2D Euclidean space. Generative models of functions in ambient space (Zhuang et al., 2023; Dupont et al., 2022b;a; Du et al., 2021) use this representation to encode a function's 's input. Right: MDF uses the feggenfunntions// ip; of the Laplace- saplace-Beltrami Operator (LBO) AM evaluated point M.
fields where a field fo .M - y maps points from manifold M signal space y. As a result, latent variables f1.T are also fields on manifolds that can be continuously evaluated.
To tackle the problem of learning a diffusion generative model over fields we employ a similar recipe to (Zhuang al., 2023), generalizing from fields defined on Euclidean domains to functions on Riemannian manifolds. In order to this we use the first l eigen-functions ffi=1:k of AM to define Fourier-like representation parametrization of the LBO, e.g. cotangent, point cloud (Sharp & Crane, 2020) or graph laplacians be used depending on the available manifold parametrization (see Sect. 5.2 for experimental
M. Note that our model is independent of the particular
results). We use the term ()x:) = vn (2()),
(x)] E TR& to denote the normalized
eigen-function representation of a point æ E M. In Fig. 3 we show visual comparison of standard Fourier PE on Euclidean space and the eigen-functions of the LBO on a manifold.
We adopt an explicit field parametrization (Zhuang al., 2023), where field is characterized by pairs {(((x)) Mc..))).. I E M, V(c.0) E V, which is denoted as context set. We row-wise stack the context set and refer to the resulting matrix via C = (x(),), ((,)))) Here, (xx)) denotes the eigen-function representation of the coordinate portion and (e.0)) denotes the signal portion of the context set at time t = O. We define the forward process for the context set by diffusing the signal and keeping the eigen-functions fixed:
set of coordinate-signe
C₁ c= Y ((e.)) = V Y (c,00 + V1 aefe),
where 6, ~ No, I) is noise vector of the appropriate size. We now turn to the task of formulating score network for fields. Following (Zhuang et al., 2023), the score network needs to take as input the context set (i.e. the field parametrization), and needs to accept being evaluated continuously in M. We do this by employing a query set {æg, V(g,0) J. Equivalently to the context set, we row-wise stack query pairs and denote the resulting matrix as Qo = (x(())), Y ((,,))- Note that the forward diffusion process equivalently defined for both context and query sets:
Q: = (P(()) (a.1) = V (9.0) + VV-
where e, ~ -O, I) is a noise vector of the appropriate size. The underlying field is solely defined by the context set, and the query set are the function evaluations to be de-noised. The resulting score field model is formulated as follows, Eg = Eo (C,, Q.).
Using the explicit field characterization and the score field network, we obtain the training and inference procedures in Alg. 1 and Alg. 2, respectively. which are accompanied by illustrative examples of sampling a field encoding a Gaussian mixture model over the manifold (i.e. the bunny). For training, we uniformly sample context and query sets from fo ~ (niform(q(fo)) and only corrupt their signal using the forward process in Eq. equation I and Eq. equation 2. We train the score field network eg to denoise the signal portion of the query set, given the context set. During sampling. to generate field fo ~ (fo) we first define query set QT = [x(())), (g.)) (0, I)] of random values to be de-noised. Similar to (Zhuang et al., 2023) we set the context set to be a random subset of the query set. We use the context set to denoise the query set and follow ancestral sampling as in the vanilla DDPM (Ho et al., 2020). Note that during inference the digen-functio representation (x)) of the context and query sets does not change, only their corresponding signal value.
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Algorithm 1 Training
Context aet: Co
1: AMPi = ypi // LBO eigen-dccommositiee
R
2: repeat
Query set: Q
3: (Co, Qo) Uniform(g(fo))
4: t> Uniform((1......
Ee ~ (O. 6g (0,,
xxx)
6: C. = (x(X.), Vū, Y (c.v) /1 [cec]
Diffused context act: C,
Diffused query act: Q:
Q. [x(X,) ;) vā Y (a.0) + /1 dleg
8: Take gradient descent step on
Ve el co,C,, 1, Q.)1
co(Ct,te t, Qc)
until converged
Figure 4: Left: MDF training algorithm. Right: Visual depiction of training iteration for field on the bunny manifold M. See Sect. for definitions.
Algorithm 2 Sampling
Contect
Query oet Or
I: AMP = (,A, // LBO digen-decomposition
t T
2: Qr = (x()), ((.) (0, (0,,1)) 1n)
3: CT C QQ
b Random subset
4: for T T..... do
1
x(x)
5: ~~ (O ift 1, clss 2 =0
6: Yint-1) ~ेzz ((()))
1-as - colo., Q.)) tos
Contect act: Cpe
Q M, (9.0-11)]
Query set: Qu
Same subset step R
t=t-1
C.-1 Q
9: end for
10: return fo evaluated coordinates xxx,))
(PX)
Figure 5: Left: MDF sampling algorithm. Right: Visual depiction of the sampling process for a field on the bunny manifold.
EXPERIMENTS
We validate the practicality of MDF via extensive experiments including synthetic and real-world problems. In Sect. 5.1 we provide results for learning distributions of functions fixed manifold (e.g. climate science), where functions change but manifolds are fixed across all functions. In addition, in Sect. 5.2 show that MDF is robust different manifold parametrizations. Finally, in Sect. 5.3 we also provide results on generalized setting where manifolds are different for each function (e.g. molecule conformer generation). As opposed generative models over images, cannot rely on FID (Heusel et al., 2017) type metrics for evaluation since functions are defined on curved geometries. We borrow metrics from generative modeling of point cloud data (Achlioptas et al., 2018), namely Coverage (COV) and Minimum Matching Distance (MMD). We compute COV and MMD metries based on the la distance in signal space for corresponding vertices in the manifolds.
5.1 DISTRIBUTIONS OF FUNCTIONS ON A FIXED MANIFOLD
We evaluate MDF on 3 different manifolds that are fixed across functions: a sine wave, the Stanford bunny and a human mesh. These manifolds have an increasing average mean curvature K (averaged over vertices), which serves as measure for how distant they are from being globally Euclidean. On each manifold we define 3 function datasets: a Gaussian Mixture (GMM) with 3 components (where in each field the components are randomly placed on the manifold), MNIST (LeCun et al., 1998) and CelebA-HQ (Karras et al., 2018) images. We use off-the-shelff texture mapping approach (Sullivan Kaszynski, 2019) to map images to manifolds, see Fig. 1. We compare MDF with Diffusion Probabilistic Fields (DPF) (Zhuang al., 2023) generative model for fields in ambient space, where points in the manifold are parametrized by the Fourier PE of its coordinates in 3D space. To provide fair comparison equate all the pyrameeas in both MDF and DPF (Zhuang

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GMM
MNIST
Co
GMM
MNIST
CelebA-HQ
COVT
MMD
COVT
MMDJ
COVT
MMD
COV1
MMD J.
COVT
MMDJ
COVT
MMD 4
MDF
0.444
0.01405
0.564
0.0954
0.354
0.11601
MDF
0.551
0,00100
0.529
0.08895
0.346
0.14162
DPF
0.352
0,01339
0.552
0.09633
0.361
0.12288
DPF
0.479
0,00112
0.422
0.09537
0.318
0.14502
Table 1: COV and MMD metries for different Table 3: Human manifold (mean curvature| datasets the wave manifold (mean curvature 25.966). At high mean curvatures MDF consis- | = 0.004).
tently outperforms DPF (Zhuang et al., 2023).
GMM
MNIST
CelehA-HQ
M + M .M + Ma Map Mio
COVT MMD J. COVT
MMDJ
COVT
MMD J
COVT MMDJ covt MMD J. COVT MMD 4
MDF
0.575
0,00108
0.581
0,07205
0.346
0.11101
MDF 0,595 0.00177
0.595 0,00177
0.582 0.00191
DPF
0.00120 0.454 0.11525 0.313
0.11530
DPF 0.547 0.00189
0,003 0.08813
0.306 0,00742
Table 2: Results on the bunny manifold (mean Table 4: Training MDF on manifold M and curvature |K| 7.388). As the mean curvature evaluating it on an isometric transformation Mise increases the boost of MDF over DPF (Zhuang does not impact performance, while being on par al., 2023) becomes larger across all datasets. with training directly on the transformed manifold.
et al., 2023). Tab. 1-2-3 show results for the different approaches and tasks. We observe that MDF tends to outperform DPF (Zhuang al., 2023), both in terms of covering the empirical distribution, resulting in higher COV, but also in the fidelity of the generated fields, obtaining a lower MMD. In particular, MDF outperforms DPF (Zhuang al., 2023) across the board for manifolds of large mean curvature J We attribute this behaviour to our choice of using intrinsic functional basis (e.g. eigen-functions of the LBO) to represent a coordinate system for points in the manifold. Fig. I shows a side to side comparison of real and generated functions on different manifolds obtained from MDF.
We also compare MDF with GASP (Dupont et al., 2022b), generative model for continuous functions using an adversarial formulation. We compare MDF and GASP performance on the CelebA-HO dataset (Karras et al., 2018) mapped on the bunny manifold. Additionally, we report results on the ERA5 climate dataset (Dupont et al., 2022b), which is composed of functions defined on the sphere f + R. (see Fig. 1). For the ERA5 dataset we use spherical harmonies to compute yp, which are equivalent the analytical figen-function GASP we use their pre-trained models to generate samples. In the case of CelebA-HQ, we use GASP to generate 2D images and map them to the bunny manifold using (Sullivan & Kaszynski, 2019). Experimental results in Tab. show that MDF outperforms GASP in both ERAS(Hersbach et al., 2020) and CelebA-HQ datasets, obtaining both higher coverage but also higher fidelity in generated functions. This can be observed in Fig. 6 where the samples generated by MDF are visually crisper than those generated by GASP.
of the LBO the sphere (Lévy, 2006). To compare with
ERAS
CelebA-HQ on M
COVT
COVt
MMD4
MDF
MDF
0.347
0.00498
0.346
0.11101
GASP
0,114
0.00964
0.309
0.38979
Table 5: MDF outperforms GASP on ERAS(Hersbach et al., 2020) and CelebA- HQ both in terms of fidelity and distribution coverage. For GASP, we generate CelebA-HQ images and texture map them to the bunny manifold using (Sullivan & Kaszynski, 2019).
GASP
Figure 6: CelebA-HQ samples generated by MDF and GASP (Dupont et al., 2022b) on the bunny.
Furthermore, we ablate the performance of MDF as the number of eigen-functions used to compute the coordinate representation ip increases (e.g. the zigen-decomposition of the LBO). For this task we use the bunny and the GMM dataset. Results in Fig. 7 show that performance initially increases with the number of not needed to faithfully encode the distribution of functions.
up to a point where high frequency eigen-function of the LBO
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0.6
Figure 7: Performance of MDF as a function of the num- ber of sigen-functions of the LBO, measured by COV and MMD metries. As expected, performance increases ini- tially as more eigen- sigen-functions are used, followed by a pllateau phase for more than k = 32 functionn.
0.5
0.8
MMD x10
0.5
0.4
0.2
0.1
0.0
0.0
20
100
120
40
80 100
120
Number of eigenfunctions
Number of eigenfunctions
5.2
MANIFOLD PARAMETRIZATION
MDF uses the figen-function: of the LBO as positional embeddings. In practice, different real-world problems parametrize manifolds in different ways, and thus, have different ways of computing the LBO. For example, in computer graphies the usage of 3D meshes and cotangent Laplacians (Rustamov et al., 2007) is widespread. In computer vision, 3D geometry can also be represented as pointclouds which enjoy sparsity benefits and for which Laplacians can also be computed (Sharp & Crane, 2020). Finally, in computational chemistry problems, molecules are represented as undirected graphs of atoms connected by bonds, in this case graph Laplacians are commonly used (Maskey et al., 2022). In Fig. 8 we show the top-2 cigenvectors of these different Laplacians on the bunny manifold. In Tab. 6 we show the performance of MDF on the bunny mesh on different datasets using different manifold parametrizations and their respective Laplacian computation. These results show that MDF relatively robust to different Laplacians and can be readily applied to any of these different settings by simply computing eigenvectors of the appropriate Laplacian.
GMM
MNIST
CelebA-HQ
COVT
MMD 1 COV-
MMD4
COVT
MMD
Graph
0.575
0.00108
0.551
0,07205
0.346
0.11101
(x))
Cotangeut
0.581
0,00384
0,568
0.06890
0.374
0.12440
Pointcloud
0.588
0,00417
0.571
0,06909
0,337
0.12297
(P(())
Table 6: Performance of MDF using different Laplacians for different datastets on the bunny manifold, where we see that MDF is relatively robust and can be readily deployed on different settings depending on the manifold parametriza- tion.
Graph
Cotangeut
Pointcloud
Figure 8: Visualizing top-2 eigenvectors on the bunny manifold for Graph, Cotangent and Point- cloud (Sharp & Crane, 2020) Laplacians.
5.3 GENERALIZING ACROSS MANIFOLDS
We now generalize the problem setting learning distributions over functions where each function is defined on a different manifold. In this setting, the training set is defined as {fi]i=o:N with functions Ji M; + y/ mapping elements from different manifolds M, shared signal space Y. This is a generalization of the setting in Sect. 5.1 where functions are defined as Sa : M + Y. with the manifold M being fixed across f. 's. This generalized setting is far more complex than the fixed setting since the model not only has to figure out the distribution of functions but also it needs to represent different manifolds in consistent manner. To evaluate the performance of MDF this setting we tackle the challenging problem of molecule conformer generation (Xu et al., 2021; 2022; Ganea et al., 2021; Jing et al., 2022) which is a fundamental task in computational chemistry and requires models to handle multiple manifolds. In this problem, manifolds M, are parametrized as graphs that encode the connectivity structure between atoms of different types. From MDF's perspective conformer is then function f. : M, - R that maps elements in the graph (e.g. atoms) to a point in 3D space. Note that graphs are just one of different the manifold representations that are amenable for MDF show Sect. 5.2.
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Following the standard setting for molecule conformer prediction use the GEOM-QM9 dataset (Ruddigkeit et al., 2012; Ramakrishnan et al., 2014) which contains ~ 130K molecules ranging from ~ 10 to ~ 40 atoms. We report our results in Tab. 7 and compare with CGCF (Xu et al., 2021), GeoDiff (Xu et al., 2022), GeoMol (Ganea et al., 2021) and Torsional Diffusion (Jing et al., 2022). Note that both GeoMol (Ganea al., 2021) and Torsional Diffusion (Jing al., 2022) make strong assumptions about the geometric structure of molecules and model domain-specific characteristics like torsional angles of bonds. In contraposition, MDF simply models the distribution of 3D coordinates of atoms without making any assumptions about the underlying structure. We use the same train/val/test splits as Torsional Diffusion (Jing et al., 2022) and use the same metries to compare the generated and ground truth conformer ensembles: Average Minimum RMSD (AMR) and Coverage. These metrics are reported both for precision, measuring the accuracy the generated conformers, and recall, which measures how well the generated ensemble covers the ground-truth ensemble. We generate 2K conformers for molccule with K ground truth conformers. Note that in this setting, models are evaluated unseen molecules (e.g. unseen manifolds M,).
We report results on Tab. where we see how MDF outperforms previous approaches. is important to note that MDF is general approach for learning functions manifolds that does not make any assumptions about the intrinsic geometric factors important in conformers like torsional angles in Torsional Diffusion (Jing al., 2022). This makes MDF simpler implement and applicable to other settings in which intrinsic geometric factors are not known.
Recall
Precision
Coverage 1
AMR
Coverage
AMR +
mean
median mean median mean median mean
median
CGCF
69.47
96.15
0,425
0.374
38.20
33.33
0.711
0.695
GeaDiff
76.50
100.00
0.297
0.229
50.00
33.50
0.524
0.510
GeoMol
91.50
100.00
0.225
0.193
87.60
100.00
0.270
0:241
Torsional Diff.
92.80
100.00
0.178
0.147
92.70
100.00
0.221
0.195
MDF (ours)
95.30
100.00
0.124
0.074
91.50
100.00
0.169
0.101
MDF (k 16)
94.87
100.00
0.139
0.093
87.54
100.00
0.220
0.151
MDF (6: 8)
94.28
100.00
0.162
0.109
84.27
100.00
0.261
0.208
MDF (A 4)
94.57
100.00
0.145
0.093
86.83
100.00
0.225
0.151
MDF (A: 2)
93.15
100.00
0.152
0.088
86.97
100.00
0.211
0.138
Table 7: Molecule conformer generation results for GEOM-QM9 dataset. MDF obtains comparable or better results than the state-of-the-art Torsional Diffusion (Jing et al., 2022), without making any explicit assumptions about the geometric structure of molecules (i.e. without modeling torsional angles). In addition, we show how performance of MDF changes as a function of the number of eigen-functions h. Interestingly, with as few k eigen-functions MDF is able to generate consistent accurate conformations.
Finally, In the appendix we present additional results that carefully ablate different architectures for the score network in A.7.1. As well as an extensive study of the robustness of MDF to both rigid and isometric transformations of the manifold M A.7.2. Finally, we also show conditional inference results on the challenging problem of PDEs on manifolds A.7.3.
CONCLUSIONS
In this paper introduced MDF diffusion probabilistic model that is capable of capturing distri- butions of functions defined on general Riemannian manifolds. We leveraged tools from spectral geometry analysis and use the figen-functions of the manifold Laplace-Bal Laplace-Beltrami Beltrami Operator to define an intrinsic coordinate system on which functions are defined. This allows us to design an efficient recipe for training diffusion probabilistic model of functions whose domain are arbitrary geometries. Our results show that we can capture distributions of functions on manifolds of increasing complexity outperforming previous approaches. while also enabling the applications of powerful generative priors to fundamental scientific problems like forward and inverse solutions to PDEs, climate modeling, and molecular chemistry.

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A APPENDIX
A.1 BROADER IMPACT STATEMENT
When examining the societal implications of generative models, certain critical elements warrant close attention. These include the potential misuse of generative models to fabricate deceptive data, such as "DeepFakes" (Mirsky & Lee, 2021), the risk of training data leakage and associated privacy concerns (Tinsley et al., 2021), and the potential amplify existing biases in the training data (Jain et al., 2020). For a comprehensive discussion on ethical aspects in the context of generative modeling, readers directed to (Rostamzadeh et al., 2021).
A.2 LIMITATIONS AND FUTURE WORK
As MDF advances in learning function distributions over Riemannian manifolds, it does encounter certain constraints and potential areas of future enhancement. One primary challenge is the computa- tional demand of the transformer- based score network in its basic form, even at lower resolutions. This stems from the quadratic cost of calculating attention this, the PerceiverlO architecture, which scales in a linear manner with the number of query and context points, is utilized (Jaegle et al., 2022) in our experiments. Further exploration of other efficient transformer architectures could be promising direction for future work (Zhai et al., 2022; Dao al., 2022). Furthermore, MDF, much like DDPM (Ho al., 2020), iterates over all time steps during sampling to generate a field during inference, a process slower than that of GANs. Current studies have accelerated sampling (Song et al., 2021a), but at the expense of sample quality and diversity. However, it's worth noting that improved inference methods such as (Song et al., 2021a) can be seamlessly incorporated into MDF.
context and query pairs. To mitigate
Since MDF has the capability to learn distributions over fields defined on various Riemannian mani- folds within single model, in future work we are poised enhance its capacity for comprehending and adapting to a broader range of geometrical contexts. This adaptability will further pave the way towards the development of general foundation models to scientific and engineering challenges, which can better account for the intricate geometric intricacies inherent in real-world scenarios.
For example, we aim to extend the application of MDF to inverse problems PDEs. A noteworthy attribute of our model is its inherent capability model PDEs Riemannian manifolds trivially. The intrinsic structure of MDF facilitates not only the understanding and solving of forward problems, where PDEs are known and solutions the forward problem are needed, but also inverse problems, where certain outcome or boundary conditions are known and the task is to determine the underlying PDE. Expanding our application handle inverse problems in PDEs have profound implications for complex systems modeling, as it enhances our understanding of the manifold structures and the way systems governed by PDEs interact with them.
Riemannian manifolds can
A.3 DISCUSSION ON COMPUTING EMBEDDINGS
When considering how to compute embeddings for points in a manifold M there are several options explore. The simplest one adopt the ambient space in which the manifold is embedded as coordinate system to represent points (eg. a plain coordinate approach). For example, in the case of 3D meshes one can assign a coordinate in IR to every point in the mesh. As shown in Tab. 1-2-3-4 this approach (used by DPF) is outperformed by MDF. In addition, in Sect. A.6.2 we show that this approach is not robust wrt rigid or isometric transformations of the manifold. Note that manifolds are not always embedded in an ambient space. For example, in molecular conformation, molecular graphs only represent connectivity structure between atoms but are not necessarily embedded in a higher dimensional space.
Another method that one can consider is to use a local chart approach. Local charts are interesting because they provide a way assigning a set of coordinates to points in a local region of the manifold. While the manifold may have arbitrary curvature, local charts are always Euclidean spaces. Each point in the manifold can be described by unique set of coordinates in the chart, but different charts may overlap. However, this requires computing transformations (often complex to implement) to convert coordinates from one chart to another.
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Finally, the eigen-funn sreen-ffnction of the LBO not only provide way of assigning coordinate to points manifold but also do this by defining an intrinsic coordinate system. This intrinsic coordinate system is global, and does not require transformations like local charts do. In addition, this intrinsic coordinate system is robust wrt rigid or isometric transformations of the manifold (ref A.6.2). Summarizing. this intrinsic coordinate system is fundamental way of describing the manifold, based its inherent properties, without reference to an external ambient space.
A.4 IMPLEMENTATION DETAILS
In this section we describe implementation details for all our experiments. These include details about the data: manifolds and functions, well details for computing the eigen-functions ip. We also provide hyper-parameter and settings for the implementation of the score field network and compute used for each experiment in the paper.
A.4.1 DATA
Unless explicitly described in the main paper, we report experiments on different manifolds which we show in Fig. 9. This manifolds are: parametric sine wave Fig. 9(a) computed using (Sullivan Kaszynski, 2019) containing 1024 vertices. The Stanford bunny with 5299 vertices Fig. 9(b). A human body mesh from the Tosca dataset (Bronstein et al., 2008) containing 4823 vertices, show in Fig. 9(c). A cat mesh and its reposed version from (Sumner & Popovic, 2004), show in Fig. 9(d) and Fig. 9(c), respectively containing 7207 vertices. To compute the mean curvature values [K] for each mesh reported in the main paper we compute the absolute value of the average mean curvature, which we obtain using (Sullivan & Kaszynski, 2019).
(a)
(b)
(c)
(d)
(e)
Figure 9: Manifolds used in the different experiments throughout the paper. (a) Wave. (b) Bunny. (c) Human (Bronstein et al., 2008). (d) Cat (Sumner & Popovic, 2004). (e) Cat (re-posed) (Sumner & Popovic, 2004).
terms of datasets of functions on these manifolds we use the following:
A Gaussian Mixture Model (GMM) dataset with 3 components, where each field the components randomly placed the specific manifold. We define held out test set containing 10k samples.
MNIST (LeCun et al., 1998) and CelebA-HQ (Karras et al., 2018) datasets, where images are texture mapped into the meshes using (Sullivan & Kaszynski, 2019), models are evaluated the standard tests sets for these datasets.
The ERA5 (Hersbach al., 2020) dataset used compare with GASP (Dupont et al., 2022b) is available at https://gites neural- functtunn-oaae stributionns ions. This dataset contains train of 8510 samples and test set of 2420 samples, which are the settings used in GASP (Dupont al., 2022b). To compare with GASP used their pretrained model available available at
com/ coom/mmil iienDupott/ /
'https://gii com ienhuuentt/eerrr funct di i rt but ons
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C2 = (e(x,), Yocl
R
Context C
Latent amy
Encodd
Query set:
40 2
R
R R1 (o(X), Y Yonl
Figure 10: Interaction between context and query pairs in the PerceiverlO architecture. Context pairs C₁ attend latent array of learnable parameters via cross attention. The latent array then goes through several self attention blocks. Finally, the query pairs Q: cross-attend to the latent array to produce the final noise prediction cq.
A.5 COMPUTING THE LAPLACIAN AND &P
In practice, for general geometries (e.g- general 3D meshes with n vertices) we compute eigenvectors of the symmetric normalized graph Laplacian L. We define L as follows:
L L= D- D -D-AJD- A
where A E {0, 1]"xx is the discrete adjacency matrix and D is the diagonal degree matrix of the mesh graph. Note that eigenvectors of L converge to the figen-function of the LBO A as n + co (Belkin & Niyogi. 2001; Bengio et al., 2003b;a). The degopoecconooiiiion of L can be computed solvers (Hernandez et al., 2009) and only needs to be computed
efficiently using sparse eigen- once during training. Note that eigen-vectors
are only defined for the mesh vertices. In MDF, we sample random points the mesh during training and interpolate the eigenvector representation of the vertices in the corresponding triangle using barycentric interpolation.
A.5.1 SCORE FIELD NETWORK
In MDF, the score field's design space covers all architectures that can process irregularly sampled data, such Transformers (Vaswani et al., 2017) and MLPs (Tolstikhin et al., 2021). The model is primarily implemented using PerceiverlO (Jaegle et al., 2022), an effective transformer architecture that encodes and decodes. The Perceiver1O was chosen due its efficiency in managing large numbers of elements in the context and query sets, as well as its natural ability to encode interactions between these sets using attention. Figure 10 demonstrates how these sets are used within the PerceiverIO architecture. To elaborate, the encoder maps the context set into latent arrays (i.e., group of learnable vectors) through cross- attentint layer, while the decoder does the same for query set. For more detailed analysis of the PerceiverlO architecture refer to (Jaegle et al., 2022).
The time-step is incorporated into the score computation by concatenating a positional embedding representation of / to the context and query sets. The specific PerceiverlO settings used in all quantitatively evaluated experiments are presented in Tab.8. Practically, the MDF network consists of 12 transformer blocks, each containing 1 cross-attention layer and 2 self-attention layers, except for GEOM-QNE we use smaller model with 6 blocks. Each of these layers has attention heads. Fourier position embedding is used to represent time-steps t with 64 frequencies. An Adam (Kingma Ba, 2015) optimizer is employed during training with a learning rate le 4. We use EMA with decay of 0.9999. A modified version of the publicly available repository used for Perceiver10 2
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Wave
Bunny
Human
GEOM-QM9
train
1024
5299
4823
variable
#context set
1024
5299
4823
variable
#query set
1024
5299
4823
variable
#eigenfuncs (k)
64
64
64
28
#freq pos. embed t
64
64
64
64
#latents
1024
1024
1024
512
#dim latents
512
512
512
256
#blocks
12
12
12
#dec blocks
#self attends per block
2
2
2
#self attention heads
4
4
4
#cross attention heads
4
4
4
batch size
96
96
96
96
Ir epochs
1e -
1e
1c
1c
4
1000
1000
1000
250
Table 8: Hyperparameters and settings for MDF on different manifolds.
A.5.2 COMPUTE
Each model was trained on an machine with 8 Nvidia A100 GPUs, we trained models for 3 days.
A.6 METRICS
Instead using FID type metrics commonly used for generative models over images (Heusel et al., 2017), must take different approach for evaluating functions curved geometries. Our suggestion to use metrics from the field generative modeling of point cloud data (Achlioptas al., 2018), specifically Coverage (COV) and Minimum Matching Distance (MMD).
Coverage (COV) refers how well the generated data represents the test set. We first identify the closest neighbour in the generated set for each field in the test set. COV is then calculated as the proportion of fields the generated that have corresponding fields in the test set. The distance between fields is determined using the average l2 distance in signal space on the vertices the mesh, usually either R. or RR3 space in our experiments. high COV score implies that the generated samples adequately represent the real samples.
Minimum Matching Distance (MMD), on the other hand, provides measure of how accurately the fields are represented in the set. This measure required because in the COV metric matches don't 't necessarily have to be close. To gauge the fidelity of the generated fields against the real ones, we pair each field in the generated set with its closes neigbour in the test set (MMD), averaging these distances for our final result. This process also utilizes the l2 distance in signal space on the mesh vertices. MMD provides a good correlation with the authenticity of the generated set, it directly depends the matching distances.
As a summary, COV and MMD metrics are complementary to each other. A model captures the distribution of real fields with good fidelity when MMD is small and COV is large. In particular, equivalent levels of MMD higher COV is desired (Achlioptas et al., 2018), and vice-versa. This observation correlates well with results shown in Tab. 1-2-3 the main paper, where MDF obtains comparable or better MMD score that DPF (Zhuang et al., 2023) while greatly improving COV.
A.7 ADDITIONAL EXPERIMENTS
In this section provide additional empirical results using different network architectures to implement the score field network eg. Furthermore, we provide additional experiments on robustness to discretization.
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A.7.1 ARCHITECTURE ABLATION
The construction of MDF does not rely on a specific implementation of the score network eg. The score model's design space encompasses a broad range of options, including all architectures capable of handling irregular data like transformers or MLPs. To substantiate this, we conducted an evaluation on the GMM dataset and the Stanford bunny at a resolution of 602 vertices, comparing three distinct architectures: a PerceiverIO (Jaegle et al., 2022), a standard Transformer Umncder-Deecoder (Vaswani et al., 2017), and an MI.--mieer (Tolstikhin et al., 2021). For a fair comparison, we approximated the same number of parameters (around 55M) and settings (such as the number of blocks, parameters per block, etc.) for each model and trained them over 500 epochs. Note that because of these reasons the numbers reported in this section are not directly comparable to those shown in the main paper. We simplified the evaluation by using equal number of context and query pairs. Both the Transformer Encoder and MLP-mixer process context pairs using their respective architectures; the resulting latents then merged with corresponding query pairs and fed into a linear projection layer for final prediction.
In Tab. 9 we show that the MDF formulation is compatible with different architectural implementa- tions of the score field model. We observe relatively uniform performance across various architectures, ranging from transformer-base based MLPs. Similar patterns are noted when examining qualitative samples displayed in Fig. 11, corroborating our assertion that MDF's advantages stem from its formulation rather than the specific implementation of the score field model. Each architecture brings its own strengths forgghhhhhrhoo for instance, MLP- M-iiers enable high throughput, transformer encoders are easy to implement, and PerceiverlO facilitates the handling of large and variable numbers of context and query pairs. We posit that marrying the strengths of these diverse architectures promises substantial advancement for MDF. Please note, these empirical results aren't directly comparable to those reported elsewhere in the paper, these models generally possess around 50% of the parameters of the models used in other sections.
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Table Quantitative evaluation of image generation on the GMM + Stanford Bunny dataset for different implementations of the score field eg.
Finally, to measure the effect of random training seed for weight initialization we ran the exact same model fixing all hyper-parametent and training settings. For this experiment we used the PerceiverIO architecture and the GMM dataset on the Stanford bunny geometry with 602 vertices. We ran the same experiment times and measured performance using COV and MMD metrics. Our results show that across the different training runs MDF obtained COV=0.569 0.007 and MMD=0.00843 + 0.00372.
A.7.2 ROBUSTNESS OF MDF
We evaluate MDF's robustness to rigid and isometric transformations of the training manifold M. We use the cal category geometries from (Sumner & Popovic, 2004) and build a dataset of different fields on the manifold by generating 2 gaussians around the area of the tail and the right paw of the cat. Note that every field is different since the gaussians are centered at different points in the tail and right paw, see Fig. 13(a). During training, the model only has access to fields defined on fixed manifold M (see Fig. 13)).. We then evaluate the model on either a rigid Magia (shown Fig. 13(b)) or isometric Mise (Fig. 13(c)) transformation of M. Qualitatively comparing the transfer results of MDF with DPF (Zhuang et al., 2023) in Fig. 13(d)-(e), we see a marked difference in fidelity and coverage of the distribution.
In Fig. 13 we show how performance changes as the magnitude of a rigid transformation (e.g. a rotation about the =- axis) of M increases. As expected, the performance of DPF (Zhuang et al., 2023) sharply declines as we move away from the training setting, denoted by a rotation However, MDF obtains stable performance across transformations, this is due to the eigen- funntffnntionn basis being intrinsic to M, and thus, invariant to rigid transformations. In addition, in Tab. 4 we show results under an isometric transformation of the manifold (e.g. changing the pose of the cat, see Fig.
radians.
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(a) Real samples.
(b) Transf. Enc-Dec (Vaswani et al., 2017).
(c) MI.P-mixer (Tolstikhin et al., 2021).
(d) PerceiverIO (Jaegle al., 2022).
Figure 11: Qualitative comparison of different architectures to implement the score field model
13(c)). As in the rigid setting, the performance of DPF (Zhuang et al., 2023) sharply declines under an isometric transformation while MDF keeps performance constant. In addition, transferring to an isometric transformation (M + Miaa) performs comparably with directly training on the isometric transformation (Mio + Miso) up to small differences due to random weight initialization.
Robustness of MDF and DPF (Zhuang et al.. 2023) with respect to rigid transformations of M. The distribution of fields learned by MDF invariant with re- spect to rigid transformations, while DPF (Zhuang et al., 2023) collapses due to learn- ing distributions in ambient space.
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We also provide transfer results to different discretizations of M. To do so, we train MDF on a low resolution discretization of manifold and evaluate transfer to high resolution discretization. We the GMM dataset and the bunny manifold at 2 different resolutions: 1394 and 5570 vertices, which we get by applying loop subdivision (Loop, 1987) the lowest resolution mesh. Theoretically, the Laplacian cigenvectors are only unique up to sign, which can result in ambiguity when transferring a pre-trained model to a different discretization. Empirically we did not find this be an issue in our experiments. We hypothesize that transferring MDF from low to high resolution discretizations largely a function of the number of digen-functions used to compute ş. This is because eigen- functions of small eigenvalue represent low- low-frequency components of the manifold which are more stable across different discretizations. In Fig. 14 we report transfer performance as a function of the number of -igen-funttions used compute sp. We observe an initial regime where more eigen- functions aid in transferring (up to 64 eigen- (eggetffnntions) followed by stage where high-frequency eigen-functions negatively impact transfer performance.
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(b) MDF M - Miled
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Figure 13: (a) Training set composed of different fields f M + R where gaussians randomly placed in the tail and the right paw of the cat. Fields generated by transferring the MDF pre-trained on M (b) rigid and (c) isometric transformation of M. Fields generated by transferring the DPF (Zhuang et al., 2023) pre-trained on M to (d) rigid and (e) an isometric transformation of M.
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Figure 14: Transferring MDF from low high resolu- tion discretizations as func- tion of the number of eigen- functions. We observe that eigen-functions small eigen- value transfer better since they encode coarse (i.e. low- frequency) information of the manifold.
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We additionally run a transfer experiment between low resolution and high resolution discretizations of a different manifold (e.g. mesh of the letter '', show in Fig. 15(b)). In this setting the low resolution mesh contains 1000 vertices and the high resolution mesh contains 4000 vertices. As show in Fig. 16 the results are consistent across manifolds, and a similar trend as in Fig. 14 can be observed. This trend further reinforces our hypothesis that low frequency eigen- functions transfer better across discretization than high frequency ones.
aa (b)
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Figure 15: (a) Low and high resolution discretizations of the Stanford bunny manifold used for the transfer experiments in the main paper (Fig. 14). (b) Low and high resolution discretizations of the letter ''' manifold, used for the experiments in Fig. 16.
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A.7.3 CONDITIONAL INFERENCE ON PDES
In this section we evaluate MDF on conditional inference tasks. In particular, we create dataset of different simulations the heat diffusion PDE on manifold. As a result, every sample in our training distribution fo ~ ((Jo) is temporal field f M x IR + R. We create training set of 10k samples where each sample a rollout of PDE for 10 steps given initial conditions. We generate initial conditions by uniformly sampling 3 gaussian heat sources of equivalent magnitude on the manifold and use FEM (Reddy, 2019) to compute the rollout over time. For this experiment we use a version of the bunny mesh with 602 vertices manifold M and set the diffusivity term of the PDE to = 0.78. We then train MDF on this training set of temporal fields f M practice simply means concatenating Fourier PE of the time step the eigen- irgnttnnnttion of the LBO.
: x TR + R. which in
We tackle the forward problem where we are given the initial conditions of the PDE and the model is tasked to predict the forward dynamies on test set of 60 held samples. To perform conditional inference with MDF we follow the recipe in (Lugmayr et al., 2022) which has been successful the image domain. We show the forward dynamics predicted by FEM (Reddy. 2019) Fig. 17(a) and MDF Fig. 17(b) for the same initial conditions in the held out set. We see how MDF successfully captures temporal dynamics, generating temporal field consistent with observed initial conditions. Evaluating the full test set MDF obtains mean squared error MSE 4.77e10 - 3. In addition, MDF can directly be used for inverse problems (Isakov, 2006). Here we focus on inverting the full dynamics of the PDE, conditioned the FEM rollout, amounting to observing 10% of the field. Fig. 17(d) shows probabilistic solution to the inverse PDE problem generated by MDF which is consistent with the FEM dynamics in Fig. 17(a).
sparse observations. Fig. 17(c) shows sparse observations of
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Figure 17: (a) Forward prediction the heat diffusion PDE computed with FEM (Reddy, 2019). (b) Conditionally sampled field generated by MDF. (c) Sparse observations of the FEM solution for inverse prediction. (d) Conditionally sampled inverse solution generated by MDF.
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A.8 ADDITIONAL VISUALIZATIONS
In this section we provide additional visualizations of experiments in the main paper. We show real and generated fields for the manifold (Fig. 18), ERA5 dataset (Hersbach al., 2020) (Fig. 19) and GMM dataset on the bunny (Fig. 20) and human (Bronstein et al., 2008) manifolds (see Fig. 21). In summary. MDF captures the distribution of real fields for different datasets and manifolds, with high fidelity and coverage.
Finally, under /videos we include video visualizations:
A visualization of training data well the sampling process for the MNIST dataset the wave manifold.
A visualization of GT and temporal fields generated by MDF for the PDE dataset introduced in Sect. A.7.3.
A visualization of the sampling process for QM9 molecules for experiments Sect. 5.3.
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Figure 18: Real and generated samples for MNIST (LeCun al., 1998) digits on the wave manifold.
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(a) Real samples
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Figure 19; Real and generated samples for the ERA5 (Hersbach et al., 2020) climate dataset.
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(a) Real samples
(b) Generated samples
Figure 20: Real and generated samples for the GMM dataset on the Stanford bunny manifold.
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(a) Real samples
(b) Generated samples
Figure 21: Real and generated samples for the GMM dataset on the human manifold (Bronstein et al., 2008).
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Reviews:
[{"review_id": "ZeNIOQJTaB", "reviewer": "Reviewer_WPEM", "summary": "The authors propose a methodology for learning fields over manifolds with denoising diffusion probabilistic models. It uses an approximation of eigen-functions of the Laplace-Beltrami Operator for the manifold at hand to create an intrinsic functional basis for the manifold. This approximation is the eigendecomposition of the graph Laplacian of the mesh approximating the manifold. This formulation theoretically allows the model to learn from data that live in manifolds, even in the absence of analytical forms for the metric of the manifold and related operators.", "questions": "1. Is the model's robustness to rigid transformations on the manifold a sufficient explanation for its performance for molecule generation? Is it possible that it is also robust to other types of transformations which are useful in solving this task?\n\n*Minor comments*\n- Sec. 3.3: were $<$ and $>$ the intended symbols when writing $f$ as a linear combination of the basis?\n- Sec. 3.3: \"making MDF strictly operate the manifold\" => \"on the\"?\n- p. 7: \"available at [URL]\" I think a footnote here is less disruptive for reading", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "The paper is excellently written, with very clear explanations, illustrations, and algorithms. Experimentally, the paper covers a lot of \"ground\" in terms of exploring the capabilities of their model. This includes comparisons against the closest-related model in multiple scenarios and comparisons against task-specific models (molecule generation).\n\nI believe the paper does a good job at showing how this relatively direct tweak to how DPF works is a very appropriate one. This is supported by a sufficient amount of experimental evidence, which explores most practical questions about such a change in representation.\n\nInterestingly, the paper also shows the robustness of the model to rigid transformations on the manifold, having a much more appropriate behavior in this case than the model it is mostly based on.", "weaknesses": "I believe the model's biggest weakness is its novelty. Theoretically, there is only a small contribution compared to DPF (Zhuang et al., 2023).\n\nWhile the authors theoretically discuss the possibility to use Laplace-Beltrami Operators as the source for a functional basis on the manifolds, this is only explored as graph Laplacians, which are a discrete approximation of manifolds. Understandably, in many scenarios it is not possible to analytically solve this problem, but it is unclear how challenging it is to adapt the ideas presented here to scenarios where it is possible (or how beneficial that is). A simple fix to this issue, in my opinion, is to claim what is being extensively evaluated experimentally: graphs/meshes as an approximation of a manifold.\n\n**Update after rebuttal**\n\nThe authors have updated their paper to clarify their claims and point out the equivalence, at least in one experiment, of their use of graph Laplacians to the analytical LBO (Table 5 shows results of this case). After reading the comments of other reviewers and the rebuttals, I am increasing my score to 8.", "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": "BJwmBDi273", "reviewer": "Reviewer_YGhV", "summary": "The paper proposed the extension of diffusion probabilistic fields onto Riemannian manifolds. The proposed method utilizes the Laplace-Beltrami operator to build an efficient representation of the points on Riemannian manifolds, which makes the proposed method invariant with respect to rigid and isometric transformations of the manifold. Numerical experiments are provided to show the efficacy of the proposed method.", "questions": "1. The main concern, as stated in the Weakness section, is the comparison of this work with [1]. To me, the difference of this work with [1] is only the different ways of representing points on the Riemannian manifold, where [1] directly use some coordinate since the manifolds we consider are always embedded, and this work uses the leading eigen-function of the LBO to approximate the points. It remains not very clear why and how LBO sign-functions outperforms a plain coordinate approach, or a local chart approach (since for every point on the manifold we have a local coordinate chart). Calculating eigen-functions of LBO potentially also consumes time. A more elaborative discussion of different approaches to do the embedding seems necessary.\n\n2. The authors choose the top k LBO eigen-functions to do the embedding. is there a guidance on how to determine k?\n\n3. Regarding Algorithm 2: First, what’s $M_q$, which seems to exist in [1] but not here? Second why do we need a random subset $C_{t}$? I thought we just need the same context and query sets as in Algorithm 1.\n\n4. Regarding the numerical experiments, I’m particularly interested in the ERA5 dataset, which seems to be the only natural instance of manifold-supported distributions. It would be interesting to see how the proposed algorithm performs in terms of COV and MMD for this dataset, comparing to other algorithms as DPF and GASP.\n\nReferences: \n[1] Zhuang, Peiye, et al. \"Diffusion probabilistic fields.\" The Eleventh International Conference on Learning Representations. 2022.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "The proposed method is the first work that can handle the diffusion process where the support of the distribution is on Riemannian manifolds. The proposed usage of the eigen-functions of Laplace-Beltrami operator makes the proposed method invariant w.r.t. affine and isometric transformations. The proposed method outperforms existing diffusion models on manifolds.", "weaknesses": "(Please respond to the questions section directly) The proposed method seems to be a plain extension of a previous work of diffusion probabilistic field. This extension shows a great potential for non-Euclidean domain but the work seems not to focusing on the discussion of how this new representation improves the work quantitatively.", "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": "F29g6jWCnf", "reviewer": "Reviewer_wDdD", "summary": "The authors extend diffusion fields onto Riemannian manifolds. They leverage the Laplace Beltrami operator as a way to get local coordinates. The authors then present experimental results in many situations comparing their approach to SOTA techniques.\n\nMinor comment: in 3.3 end of the first paragraph should use langle rangle rather than less than greater than signs,", "questions": "I understand the usefulness of using function and field interchangeably but I am curious as to why this choice was made. I ask this because over Riemannian manifolds it's common to use gradient vector fields which are defined over tangent spaces and the use of fields can cause confusion here.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "The overall idea the paper presents is relatively simple to understand so it is easy to follow what the authors are attempting to accomplish.", "weaknesses": "There are some errors when introducing Reimannian manifolds, for instance the authors state $g:\\mathcal{M}\\times\\mathcal{M}\\rightarrow\\mathbb{R}_+$ but this is not true, the metric tensor takes in inputs from the tangent spaces and further can be 0.\n\nThe authors have a specific goal in mind for the manifolds they wish to consider but the theory seems to be more general in a sense. That is, typical Riemannian manifolds such as the cone of SPD matrices or a Grassman manifold go largely un mentioned. The authors focus on specific manifolds which could be interesting such as the Stanford bunny, but I lack to see the importance/applicability of it. That being said, it is a nice application but a little mysterious as to its usefulness.", "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)