Riemannian Score-Based Generative Modelling

Background

• SGMs add noise until data become approximately Gaussian, then generate samples by simulating a learned time-reversal diffusion whose drift depends on Stein scores (Song and Ermon, 2019; Song et al., 2021; Ho et al., 2020; Dhariwal and Nichol, 2021).The scores are estimated with neural networks using score matching.Source: §1 · p. 1

Motivation

• Because many scientific datasets lie on Riemannian rather than Euclidean manifolds, flat-geometry SGMs are unsuitable for applications including protein modelling, geoscience, robotics, and high-energy physics.The passage also identifies cell development, image recognition, graph-structured data, and hierarchical data as manifold-valued domains.Source: §1 · p. 1

Contribution

• RSGMs extend score-based generative models to Riemannian manifolds by defining geometry-aware forward diffusions and manifold-valued reverse processes.The reverse drift includes the Stein score, which can be estimated through score matching as in Euclidean models.Source: §1 · p. 1

Methodological challenges

• RSGMs address inexact transition kernels with non-standard score-estimation techniques and Geodesic Random Walks, while targeting an easy-to-sample reference distribution.On compact manifolds, transition kernels may be available only as infinite Sturm–Liouville sums.Source: §1 · pp. 1–2

Results

• RSGMs provide convergence bounds on compact manifolds and outperform recent baselines while scaling better to high-dimensional manifolds across multiple manifolds and tasks.The demonstrations include natural-disaster occurrence datasets collected by Mathieu and Nickel (2020), with baselines from Mathieu and Nickel (2020) and Rozen et al. (2021).Source: §1 · p. 2

Forward and reverse processes

• The forward process is an Ornstein–Uhlenbeck diffusion that converges geometrically to N(0, Id), while its time reversal has the data distribution p0 at the terminal time.Under mild conditions, the reversed process satisfies an SDE and would exactly recover p0 if it could be sampled.Source: §2 · p. 2

Score estimation

• SGMs replace the intractable terminal density with N(0, Id) and estimate the unknown scores using the denoising score matching identity.The identity makes ∇log pt the minimizer of a loss based on the closed-form OU transition density.Source: §2 · p. 2
• A neural network sθ is trained with a time-weighted denoising loss to approximate the score field ∇log pt across the diffusion trajectory.The loss integrates λtℓt(sθ(t, ·)) over time for a positive weighting function λt.Source: §2 · p. 2

Sampling discretization

• The reverse-time SDE is sampled numerically with an Euler–Maruyama discretization using step γ and T = γN.Later improvements include adaptive stepsizes and predictor–corrector schemes, which are reported to improve performance.Source: §2 · p. 2

3 Riemannian Score-based Generative Modelling

• RSGMs require four components: a noising process converging to an easy reference distribution, manifold time reversal, approximate SDE sampling, and efficient reverse-drift approximation.These components are combined into the RSGM algorithm, with optional predictor-corrector sampling.Source: §3 · p. 2, §3.4 · p. 5

3.1 Noising processes on manifolds

• Langevin dynamics provide generic manifold noising with invariant density proportional to e^−U, while compact manifolds can instead use Brownian motion targeting the uniform distribution.Choices include Riemannian normal and exponential-wrapped Gaussian potentials; both recover Ornstein–Uhlenbeck noising in Euclidean space.Source: §3.1 · p. 3

3.2 Time-reversal on Riemannian manifolds

• The manifold time-reversal formula shows that reversing a diffusion yields another diffusion whose backward coefficients can be computed from the forward process.The result assumes suitable smooth positive time-marginal densities and extends prior Euclidean time-reversal results to Riemannian manifolds.Source: §3.2 · p. 3

3.3 Approximate sampling of diffusions

• Geodesic Random Walks intrinsically approximate well-behaved manifold diffusions, avoiding projection errors that can accumulate in extrinsic embedded-space methods.The framework extends quantitative approximation results to time-inhomogeneous diffusions and supports sampling the backward process.Source: §3.3 · pp. 3–4

3.4 Score approximation on Riemannian manifolds

• Riemannian score approximation uses denoising or implicit score matching and parametrizes tangent-vector scores through smooth vector fields spanning the tangent bundle.The resulting algorithm combines score learning, manifold time reversal, and Geodesic Random Walk sampling to define RSGM.Source: §3.4 · pp. 4–5

Compact-manifold construction

• Compact manifolds support a uniform distribution and heat-kernel approximations, so RSGMs use Brownian motion as the noising SDE with that uniform stationary distribution.Table 1 summarizes differences between compact-manifold RSGMs, generic-manifold RSGMs, and Euclidean score-based models, but the supplied passage gives no cell values.Source: §4 · p. 5, Table 1 · p. 6

Heat-kernel approximations

• When spectral data are available, truncating the Sturm–Liouville heat-kernel expansion gives a score approximation converging to ∇xt log pt|0 as J → +∞.The eigenvalues and eigenvectors are computable for the d-dimensional torus and sphere, enabling approximation for any t > 0 (Chavel, 1984; Saloff-Coste, 1994, Section 2; Jones et al., 2008, Lemma 1).Source: §4 · p. 6
• When spectral quantities are unavailable or intractable, Varadhan asymptotics approximate the heat kernel for small t away from the cut-locus.The approximation applies for x, y ∈ M with y /∈ Cut(x) (Bismut, 1984, Theorem 3.8; Chen et al., 2021, Theorem 2.1; Lee, 2018, Chapter 10).Source: §4 · p. 6

Convergence results in the compact setting

• The analysis assumes manifolds M whose heat-kernel density satisfies pt|0(x|x) ≤ Ct−α/2 for C, α > 0, t ∈ (0, 1], including spheres, tori, compact matrix groups, and products.Source: §4 · p. 7
• For any ε > 0, choosing T large, score error M small, and step size γ small yields W1(L(YN), p0) ≤ ε under the compact-manifold assumptions.The result assumes smooth positive p0, T > 1/2, and a uniform bound on the score approximation error; M includes neural-network and heat-kernel approximation bias.Source: §4 · p. 7

• Prior work generally prescribes the manifold structure, unlike methods that jointly learn manifold geometry and density, such as Brehmer and Cranmer (2020) and Caterini et al. (2021).Source: §5 · p. 7

Push-forward of Euclidean normalizing flows

• Push-forward normalizing flows transform Euclidean flows through invertible maps, but global representation requires M to be topologically equivalent to R^n; exponential-map variants target Lie groups and hyperbolic space.This approach appears in Falorsi et al. (2019) and Bose et al. (2020).Source: §5 · p. 7

Neural ODE on manifolds

• Riemannian neural ODE methods define continuous normalizing flows directly on manifolds, avoiding embedding-specific artifacts and numerical instabilities; Moser flows avoid solving an ODE during training.These methods include Falorsi and Forré (2020), Mathieu and Nickel (2020), Falorsi (2021), and Rozen et al. (2021).Source: §5 · p. 7

Optimal transport on manifolds

• Optimal-transport approaches construct discrete-time manifold flows from exponential maps of gradients of c-convex potentials, trading expressivity against scalability through the chosen parametric family.This line of work includes Sei (2013), Rezende and Racanière (2021), and Cohen et al. (2021).Source: §5 · p. 7

6 Experiments

• The experiments benchmark RSGMs against manifold-valued methods across spherical, toroidal, rotational, and hyperbolic manifolds, assessing modelling capacity, dimensional scalability, regularity, and time complexity.The study includes a stereographic score-based model and reports implementation details and code availability separately.Source: §6 · p. 8

6.1 Earth and climate science datasets on the sphere

• On spherical earth and climate datasets, all methods have comparable log-likelihoods, with RSGMs marginally better on most datasets.The evaluation covers volcanic eruptions, earthquakes, floods, and wildfires; Moser flows train slowly, while Moser flows and stereographic SGMs are costly to evaluate.Source: §6.1 · p. 8, Figure 2 · p. 9

6.2 Synthetic data on tori

• RSGMs fit wrapped Gaussian targets on high-dimensional tori with linear or constant computational cost, whereas Moser flows cannot be trained for d ≥10.Moser flows scale poorly because their complexity grows with d and Monte Carlo samples K, while accurate estimation requires K = O(e^d); their likelihood gap also widens with dimension.Source: §6.2 · pp. 8–9, Figure 3 · p. 9

6.3 Synthetic data on the Special Orthogonal group

• RSGMs fit SO3(R) mixtures consistently across component counts, while Exp-wrapped SGMs and Moser flows perform well only over limited ranges of M.Moser flows also require K = 10^4 Monte Carlo samples for regularization, making them much slower to train; RSGMs use significantly fewer NFEs, whereas Moser flows require ≫10^3.Source: §6.3 · pp. 9–10, Table 5 · p. 10

6.4 Synthetic data on hyperbolic space

• On hyperbolic space H2, both score-based models qualitatively fit a wrapped-Gaussian mixture target using Langevin dynamics for noising.This experiment extends RSGM evaluation to a non-compact manifold with constant negative curvature.Source: §6.4 · p. 10

Contributions

• RSGMs represent target densities on manifolds as time reversals of Langevin dynamics, with scalability, broad manifold applicability, robustness, complex-data capacity, and convergence guarantees.The method's manifold coverage follows from diverse available loss functions.Source: §7 · p. 10

Future work

• Future work includes extending RSGMs to manifolds with boundaries, alternative noising processes, and efficient stochastic-control algorithms for Schrödinger bridges on manifolds.The Schrödinger-bridge direction follows the spirit of De Bortoli et al. (2021) on Euclidean state spaces and cites Thornton et al. (2022).Source: §7 · p. 10

• The checklist identifies the main contribution as extending diffusion models to Riemannian manifolds and directs readers to Section 7 for limitations.Source: Checklist · p. 17
• The paper reports no immediate societal impact and marks human-subject, consent, personally identifiable information, and offensive-content considerations as not applicable.Source: Checklist · p. 17
• Theoretical assumptions are stated, and complete proofs are provided in the supplementary material.Source: Checklist · p. 17
• Experimental details, training specifications, reproducibility instructions, compute resources, and error bars are reported in the appendices or supplementary materials.Source: Checklist · p. 17

A Organization of the supplementary

• The supplementary material collects the proof of Theorem 3.1, convergence proof for RSGM, likelihood extensions, algorithmic variants, comparisons, and experimental details.It also covers density estimation, conditional SGMs, Schrödinger Bridges, noncompact settings, stereographic SGMs, and related score-matching results.Source: A Organization of the supplementary · p. 18
• It organizes stochastic Riemannian geometry, likelihood computation, parametric vector fields, spherical and toroidal Laplace–Beltrami facts, predictor-corrector schemes, and time-reversal theory across appendices.The organization also includes a comparison with Rozen et al. (2021) and extensions to density estimation, conditional SGMs, and Schrödinger Bridges.Source: A Organization of the supplementary · p. 18

B Notation

• The notation section defers detailed introductions to Riemannian geometry and manifold stochastic processes to Appendix C, including stochastic processes specifically in Appendix C.2.The section itself only introduces notation used in the work.Source: B Notation · p. 18
• The framework assumes a smooth, connected, complete Riemannian manifold M equipped with a smooth metric g defining distances and inner products.The gradient satisfies ⟨∇f, v⟩_g = df(v), while distance is defined by infimizing curve lengths.Source: B Notation · p. 18
• The notation defines geodesics, exponential maps, and the metric-induced natural volume form used on the manifold.The exponential map sends an initial tangent vector v to the endpoint at time one of its geodesic, and the volume form is ω(x) = |g(x)|^1/2 dx1 ∧ ... dxd.Source: B Notation · p. 18

C Preliminaries on stochastic Riemannian geometry

• The stochastic Riemannian geometry preliminaries set M as a d-dimensional smooth manifold with tangent bundle TM, cotangent bundle T⋆M, smooth functions C∞(M), and vector fields X(M).They recall foundational concepts following Hsu (2002), Lee (2018), and Lee (2006), with broader manifold references to Lee (2010) and Lee (2013).Source: C Preliminaries on stochastic… · p. 18

C.1 Tensor field, metric, connection and transport

• A Riemannian metric is a smooth positive-definite tensor field, while the uniquely defined Levi–Civita connection is torsion-free and metric-compatible.The connection extends covariant differentiation to functions, one-forms, and general tensor fields.Source: §C.1 · p. 19
• Parallel transport gives linear isomorphisms between tangent spaces, and geodesics define exponential mappings and locally unique transport between nearby points.On connected compact manifolds, geodesic completeness ensures a geodesic exists between any two points.Source: §C.1 · pp. 19–20

C.2 Stochastic Differential Equations on manifolds

• Manifold SDEs are defined through their action on smooth functions using Stratanovitch integrals, preserving the ordinary chain rule and diffeomorphism covariance.Under the stated smoothness conditions, the corresponding manifold SDE has a unique solution up to the relevant stopping time.Source: §C.2 · pp. 20–21

C.3 Brownian motion on manifolds

• Brownian motion on a connected compact Riemannian manifold is characterized by the Laplace–Beltrami operator and can be sampled intrinsically, extrinsically, or through local-coordinate SDEs.The Laplace–Bochner representation on the orthonormal bundle and the Nash-embedding projection representation both express the relevant generator through sums of squared operators.Source: §C.3 · pp. 21–22
• Geodesic random walks are introduced as a limiting construction whose small-step behavior is intended to support analysis of the Brownian time-reversal process.The construction assumes tangent-space increment distributions with finite third moments and prescribed first- and second-moment fields.Source: §C.3 · p. 23
• Geodesic random walks converge to a manifold diffusion whose generator is determined by the increment distribution, recovering Brownian motion when the mean is zero and covariance is identity.On compact manifolds, Brownian motion admits a density and preserves the normalized Hausdorff measure as an invariant distribution.Source: §C.3 · pp. 23–24

D.1 ODE likelihood computation

• The probability-flow ODE transports densities through the divergence of its vector field, enabling approximate likelihood evaluation with a score network and manifold ODE solver.For sufficiently large terminal time, the reversed noised distribution is close to the invariant reference measure used for initialization.Source: §D.1 · p. 24

D.2 Difference between ODE and SDE likelihood computations

• ODE and SDE likelihood computations generally do not coincide, agreeing only when the SDE sampler’s density matches the time-reversed forward density.The SDE likelihood can instead be approximated using a second network for the reverse-process score.Source: §D.2 · p. 25

E Parametric family of vector fields

• RSGM represents scores with projected ambient fields, divergence-free Lie-group frames, or local-coordinate fields, while accounting for non-parallelizable manifolds such as the sphere.Projected fields preserve tangent-space membership and have divergence equal to their Euclidean divergence under the stated construction.Source: E Parametric family of vector fields · p. 25

F Eigensystems of the Laplace–Beltrami operator and heat kernels

• Laplace–Beltrami eigensystems provide heat-kernel representations on tori and spheres, whose logarithmic gradients can be approximated by spectral truncation or Varadhan’s small-time approximation.The sphere uses eigenvalues −k(k+d−1) and spherical harmonics, while compact-manifold heat kernels admit Sturm–Liouville expansions.Source: F Eigensystems of the Laplace–Beltrami… · p. 26

G Predictor-corrector schemes

• The predictor-corrector scheme alternates reverse-time prediction with correction steps that project the evolving distribution back toward the corresponding forward noising marginal.The corrector is designed so that the reverse process has pT−t as its invariant measure at time t.Source: G Predictor-corrector schemes · p. 27, Figure 2 · p. 27

H Time-reversal formula: extension to Riemannian manifolds

• The time-reversal proof extends diffusion-process arguments to Riemannian manifolds, drawing on Cattiaux et al. (2021) and related manifold stochastic-process theory.The appendix also provides an informal Haussmann and Pardoux (1986) derivation and notes García-Zelada and Huguet (2021)’s manifold Girsanov theory.Source: H Time-reversal formula: extension to… · pp. 27–28

H.2.1 Diffusion processes and integration by part formula

• The appendix formalizes diffusion processes on Polish spaces through path measures, filtrations, local martingales, and extended-generator domains to support integration-by-parts arguments.This section states a simplified version of Cattiaux et al. (2021, Theorem 3.17).Source: §H.2.1 · p. 30
• The integration-by-parts theorem is formulated for Markov path measures using extended generators and carré du champ operators, which reduce on Riemannian manifolds to gradient inner products.For u,v ∈ C2(M), the carré du champ is ⟨∇u, ∇v⟩; the theorem is then applied under stated continuity, convergence, and finite-measure conditions.Source: §H.2.1 · pp. 30–31
• Finite entropy ensures the integrability conditions needed to apply the integration-by-parts theorem to diffusion processes on compact manifolds.The appendix states that these domain and integrability properties are obtained using Girsanov theory.Source: §H.2.1 · p. 31, §H.2.2 · p. 31

H.2.2 Girsanov theory on compact Riemannian manifolds

• Under finite entropy relative to Brownian motion, Girsanov theory provides tangent drifts and martingale problems for both a manifold diffusion and its time reversal.Proposition H.4 applies when KL(P|Q) < +∞ and constructs βP and βR(P) taking values in TxM, with generators AP and AR(P).Source: §H.2.2 · p. 33
• Applying the framework gives an expectation identity involving βP(t,Xt), βR(P)(T−t,Xt), the gradient of u, the Laplacian, and the gradient pairing.The displayed relation is E[v(Xt)(⟨βP(t,Xt) + βR(P)(T −t,Xt), ∇u(Xt)⟩+ ∆Mu(Xt)) + ⟨∇u(Xt), ∇v(Xt)⟩] = 0.Source: §H.2.2 · p. 34

H.1 Informal derivation

• The derivation introduces forward and time-dependent infinitesimal generators and uses martingale identities to express the reversed process dynamics.The generator formulation and change-of-variable argument yield the desired time-reversal relation.Source: §H.1 · pp. 29–30
• The informal derivation proves Theorem 3.1 by reversing the diffusion, applying backward Kolmogorov equations, and identifying the reversed generator under smooth positive density assumptions.The argument assumes sufficient regularity of the conditional-expectation function v; the authors note that a fully rigorous treatment requires establishing this regularity.Source: §H.1 · pp. 28–30

H.2 Proof of Theorem 3.1

• The rigorous proof of Theorem 3.1 adapts integration-by-parts and Girsanov arguments to Riemannian manifolds by embedding the processes into Euclidean space via the Nash embedding theorem.The proof is organized around infinitesimal-generator properties, an adapted integration-by-parts formula, manifold Girsanov theory, and a concluding argument.Source: §H.2 · p. 30

H.2.3 Concluding the proof

• The appendix concludes that the reversed-process drift is characterized through its inner product with test-function gradients, completing Theorem 3.1.The identification β^P = b is used together with the density-based divergence calculation in the final step.Source: §H.2.3 · p. 34
• In the concluding proof, smooth positive transition densities and the divergence theorem identify the reversed drift contribution as −b(x) + ∇log p_t(x).This identity is stated for all t ∈ [0,T] and x ∈ M and completes the proof of the time-reversal formula.Source: §H.2.3 · p. 34

I Convergence of RSGM

• The convergence section studies convergence of RSGM, states its main results, and provides discretization bounds following Cheng et al. (2022).The main convergence results are stated in Appendix I.1, while discretization bounds are developed separately.Source: I Convergence of RSGM · p. 34

I.1 Main results

• Theorem I.1 assumes A1, smooth positive p0, uniformly bounded score error M, and T > 1/2, then states existence of C independent of T; W1 is order-one Wasserstein distance.The proof controls initialization, score approximation, and GRW discretization terms using geometric ergodicity, Girsanov theory, and GRW convergence.Source: §I.1 · pp. 34–35
• Theorem I.2 gives the corresponding continuous-time guarantee under the same assumptions, without a discretization error term.The result controls an L1 density bound and is compared with the L∞ density bound of Rozen et al. (2021, Theorem 3).Source: §I.1 · p. 36

I.2 Discretization bounds for GRW

• The GRW analysis extends Cheng et al. (2022) to time-inhomogeneous drift terms and establishes discretization bounds on progressively finer grids.The construction uses minimizing geodesics, parallel transport, exponential-map updates, and a discrete Grönwall argument.Source: §I.2 · pp. 37, 41
• Under Assumption A1, Proposition I.4 states the section’s main result with some C ≥ 0; another bound uses C1 ≥ 0 independent of k or ℓ for t ∈ [kγℓ, (k + 1)γℓ].Source: §I.2 · pp. 37–39
• Proposition I.6 supplies a further GRW discretization estimate under A1, completing the section’s convergence analysis.Its proof combines the preceding GRW lemma with Lemma I.5.Source: §I.2 · p. 41
• The discretization argument uses independent zero-mean identity-covariance Gaussian variables and a frame for TX1.Source: §I.2 · p. 41
• Under assumption A1, Proposition I.7 establishes the existence of a constant C ≥ 0 for every k ∈ N.Source: §I.2 · p. 41
• The proof applies Lemma I.3 and the inequality √(1 + t) ≤ 1 + t/2 for t ≥ 0 to derive the stated bound.Source: §I.2 · pp. 41–42

J Proof of proposition 3.3

• Proposition 3.3 holds when p_t|s(x_t|x_s)s_t(x_t) is a C1 vector field for every x_s, with conditions controlled through the score network and noising process.These choices provide the required regularity assumptions for the proposition.Source: J Proof of proposition 3.3 · p. 42
• The proof applies a divergence theorem for non-compact manifolds to transform the relevant divergence integral and conclude the proposition.The argument takes t ∈ (0, T] and s_t ∈ C∞(M).Source: J Proof of proposition 3.3 · p. 42

K Comparison with Moser flows

• Moser flows and RSGM both interpolate between a reference distribution p_ref and a target distribution p_0, assuming p_ref is known and sampleable.The reference distribution is often uniform on M.Source: K Comparison with Moser flows · p. 43
• Moser flows use the density interpolation p̂_t = (1 − t)p̂_0 + t p̂_1 and a velocity field derived from u with div(u) = p̂_0 − p̂_1.The resulting process satisfies X_t ∼ p̂_t through the Fokker–Planck equation.Source: K Comparison with Moser flows · p. 43
• Moser flows also enable density estimation through p̂_1 = p̂_0 − div(u_θ), whereas density estimation is not directly accessible with RSGM.The paper proposes Fisher-score-based estimation for RSGM in Appendix L, following a manner akin to Choi et al. (2021).Source: K Comparison with Moser flows · p. 43
• Moser flows interpolate in density space, causing spontaneous density creation, while RSGM interpolates in sample space, corresponding to density displacement.The paper characterizes these as vertical displacement for Moser flows and horizontal displacement for RSGM, following Santambrogio (2017).Source: K Comparison with Moser flows · p. 43
• When reference and target distributions overlap little, Moser-flow trajectories can require extremely high Stein-score amplitudes for sampling or likelihood evaluation.This issue is attributed to spontaneous density creation during density-space interpolation.Source: K Comparison with Moser flows · p. 43

L Density estimation with Fisher score

• The noising density converges exponentially toward the uniform reference, with |p_T(x) − 1| ≤ C exp[−λ_1T/2] for sufficiently large T.This convergence supports the Fisher-score density approximation on compact manifolds.Source: L Density estimation with Fisher score · p. 43
• Fisher-score estimation decomposes density evaluation through the time derivative of log p_t, enabling joint estimation of the spatial Stein score and Fisher score.The Fisher score is characterized as the minimizer of a loss, while approximation error is quantified under bounded score discrepancy.Source: L Density estimation with Fisher score · pp. 45–46

M.1 Schrödinger bridge.

• Schrödinger bridges address initialization mismatch and enable interpolation between dataset-defined distributions, beyond conventional SGM reference-distribution sampling.Standard methods may require many backward steps to keep T large and the GRW stepsize γ small.Source: §M.1 · p. 46

M.2 Conditional RSGM.

• Conditional RSGM amortizes the score network with an observation y, allowing approximate posterior sampling when the posterior is supported on a manifold.The network receives the additional input s_θ(t, x; y).Source: §M.2 · p. 46

M.3 Invariant distributions

• For distributions invariant under a group action, an equivariant probability-flow map preserves invariance when the reference density is invariant and the score network is equivariant.The construction assumes the representation and drift commute, as in the linear-drift case.Source: §M.3 · pp. 46–47

N Stereographic baseline details

• Stereographic baselines project manifold data through an invertible map, train a Euclidean score model, and push its density back to the manifold, but they generally underperform intrinsic RSGM.Seam behavior requires points to map toward Euclidean infinity, causing numerical challenges and artifacts; results also depend on the chosen bijection.Source: N Stereographic baseline details · p. 47

O Experimental details

• Experiments were implemented in JAX using a modified Geomstats Riemannian-geometry library.Anonymized code and the modified Geomstats version are provided in the supplementary materials, with full repositories planned after publication.Source: O Experimental details · p. 47
• The score network uses a sinusoidal multilayer perceptron with five hidden layers for Earth and SO(3), three for high-dimensional experiments, and 512 units per layer.Its output is decomposed over divergence-free vector fields.Source: O Experimental details · p. 47
• Unless otherwise specified, models use sliced score matching with a Hutchinson divergence estimator, while denoising score matching produced similar results.For DSM, the weighting uses conditional variance with a Euclidean closed-form standard deviation as a compact-manifold proxy.Source: O Experimental details · p. 48
• Likelihood ODEs use an adaptive Dormand–Prince Runge–Kutta 4(5) solver with 1e−5 absolute and relative tolerances, while SDE rollouts use Euler–Maruyama prediction without correction for 100 steps.The stereographic sphere projection can omit one point because it is not necessarily bijective.Source: O Experimental details · pp. 47–48

O.1 Sphere

• Sphere datasets use random 0.8/0.1/0.1 training, validation, and test splits, with Earth approximated as a perfect sphere.This subsection reports the sphere-specific data protocol.Source: §O.1 · p. 48
• Kent-mixture baselines select components by validation likelihood from a specified grid, yielding 25 for Volcano, 50 for Earthquake, and 100 each for Flood and Fire.The mixtures are optimized with EM for 250 iterations.Source: §O.1 · p. 48
• Sphere score models train for 600k iterations except Flood, where 300k performs best, and the stereographic SGM uses βmin = 0.001 and βmax = 2.These settings define the sphere-specific training duration and noise schedule.Source: §O.1 · p. 48

Additional experimental results

• From N = 5, geodesic-random-walk samples closely match exact forward samples by MMD, and log-likelihood remains robust except at very small N.Figure 4a measures distribution discrepancy against N = 1000 exact samples; Figure 4b evaluates log-likelihood across discretization steps.Source: Additional experimental results · p. 48, Figure 4 · p. 49
• The Varadhan approximation alone yields decent DSM performance, while combining it with heat-kernel truncation can improve results when τ and J are sufficiently high.Test log-likelihood is evaluated on Flood across a grid of τ and J values.Source: Additional experimental results · p. 48, Figure 5 · p. 49, §O.2 · p. 49

O.2 Torus

• On Tn, the synthetic data are wrapped Gaussians with uniformly chosen random means and standard deviation 0.2.The distribution is formed by mapping tangent-space normal densities through the exponential map at each mean.Source: §O.2 · p. 49
• For the Torus experiments, RSGMs use one consistent unit-norm vector field per product dimension, a shared three-layer 512-unit MLP, and 50k optimization iterations.Training uses both implicit and sliced score-matching losses.Source: §O.2 · p. 49

O.3 Special Orthogonal group

• The SO3(Rd)11 dataset is a 32-component mixture of wrapped normal distributions with uniformly sampled component means and inverse-Gaussian variance parameters.The model also demonstrates conditional sampling for individual mixture components using the DSM loss with the Varadhan approximation.Source: §O.3 · pp. 49–50
• On SOd(R), the score vector field is parameterized using the Lie-algebra basis QEij through the divergence-free vector-field approach.The construction uses skew-symmetric basis matrices Eij induced on tangent spaces.Source: §O.3 · p. 50
• The SO3(Rd) comparison includes RSGM, Moser flows, and a wrapped-exponential baseline, with models trained for 100k iterations and tuned by specified parameter grids.The wrapped-exponential transformation uses a radial operator whose constant 2π equals the group’s injectivity radius, making it injective except at the measure-zero cut locus.Source: §O.3 · p. 50