Diffusion Generative Modeling on Lie Group Representations
Authors: Marco Bertolini
Tuan Le
Djork-Arné Clevert
Venue: NeurIPS 2025 spotlight ()
Keywords: generative modeling
diffusion models
lie groups
representation theory
Primary area: deep_learning
TL;DR: We introduce Diffusion-Based Generative Modeling on the (flat) Lie group representation space rather than on the (curved) Lie group

Abstract: We introduce a novel class of score-based diffusion processes that operate directly in the representation space of Lie groups.
Leveraging the framework of Generalized Score Matching, we derive a class of Langevin dynamics that decomposes as a direct sum of Lie algebra representations,
enabling the modeling of any target distribution on any (non-Abelian) Lie group.
Standard score-matching emerges as a special case of our framework when the Lie group is the translation group. 
We prove that our generalized generative processes arise as solutions to a new class of paired stochastic differential equations (SDEs), introduced here for the first time. 
We validate our approach through experiments on diverse data types, demonstrating its effectiveness in real-world applications such as $\text{SO}(3)$-guided molecular conformer generation and modeling ligand-specific global $\text{SE}(3)$ transformations for molecular docking, showing improvement in comparison to Riemannian diffusion on the group itself.
We show that an appropriate choice of Lie group enhances learning efficiency by reducing the effective dimensionality of the trajectory space and enables the modeling of transitions between complex data distributions.

Diffusion Generative Modeling on Lie Group Representations
Marco Bertolini Tuan Lef & Djork-Arné Clevert
Machine Learning Research Pfizer Worldwide Research and Development
Friedrichstraße I 110, 101 Berlin, Germany
[marco, wew.oooirr...iii. tuan. djork-arne.com
@0fizer. comm
Abstract
We introduce novel class of score-based diffusion processes that operate directly in the representation space of Lie groups. Leveraging the framework of Generalized Score Matching, we derive class of Langevin dynamics that decomposes as a direct sum of Lie algebra representations, enabling the modeling of any target distribution on any (non-Abelian) Lie group. Standard score-macching emerges as special case of our framework when the Lie group is the translation group. We prove that our generalized generative processes arise as solutions to a new class paired stochastic differential equations (SDEs), introduced here for the first time. We validate our approach through experiments on diverse data types, demonstrating its effectiveness in real-worke applications such SO (3)-dddd molecular former generation and modeling ligand-specits global SE(3) transformations for molecular docking, showing improvement in comparison to Riemannian diffusion on the group itself. We show that an appropriate choice of Lie group enhances learning efficiency by reducing the effective dimensionality of the trajectory space and enables the modeling of transitions between complex data distributions.
Introduction
Deep probabilistic generative modeling amounts creating data from known tractable prior distribution. Score-based models (Hyvärinen & Dayan 22055 SShhi-Dickttein et al.) 2015) Ho et al.
2020 [Huang al. 2021 (Song et al. 2021 2020b) achieve this by learning reverse corruption
process of the data. Most algorithms assume an Euclidean data space X, yet many scientific
applications (Brehmer & Cranmer 2020: Zhang al.) 2024 Klimovskaia
2020 Karpatne
curved manifolds M. While significant progress has been made in developing the theory diffusion curved spaces (De Bortoli et al. 2022 Huang et al. 2022), key challenges remain: parametrizing vector fields on general M is unsolved, and Langevin updates require projection to preserve the manifold structure. Even when M denoising score-matching remains a challenge for general non-Abelian groups, thus necessitating explicit trajectory simulation. Recent findings (Abramson et al.) (2024) highlight this complexity, as diffusion was performed in raw Cartesian coordinates rather than explicitly modeling the torsion space, given its representational difficulty and lack of performance gain.
et al., 2018) involve distributions
is Lie group.
An appropriate representation that leverages the symmetry property of the data should, however, enable models to better capture the underlying physical laws. The limited performance of manifold- based diffusion must thus stem from technical and computational difficulties rather than fundamental principles. This work seeks reconcile this expectation with the empirical findings by addressing the question: Given a Lie group G acting on Euclidean space X through a map (representation)
Shared first authorship.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Px G - GL(X), can we construct gen- erative process on X that models any distri- bution on G, thus retaining the advantages of flat-space diffussio while capturing non-trivial manifold structures? We address this question by constructing the diffusion process directly in the representation space, defined as the image of the group action map, [m(px) C GL(X). This yields a matrix-valued diffusion process in GL(X) which, when applied to elements of X. induces stochastic flow corresponding to infinitesimal Lie group transformations, i.e., Lie algebra elements. In this way, the process preserves the geometric inductive bias of the (curved) Lie group while remaining entirely within the flat vector space X. Our construc- tion builds on the framework of Generalized
Lie Group diffusion
representation diffusion
Figure 1: Comparison of strategies between Lie
group (top) diffusion and our proposed Lie group representation diffusion (bottom).
Score Matching (GSM) (Lyu 2009; Lin et al. 2016), which estimates probability densities via the generalized score function L log p(x) for a suitable linear operator L. We show that the G-induced generative process satisfies a continuous time stochastic differential equation (SDE) involving this generalized score. As illustrated in Figure []] our approach differs to diffusion processes directly on the Lie group: rather than mapping data to the group and back via the representation map, we remain in X throughout, using the differential of the representation dpx TG - TX to guide the Langevin dynamics.
In short, we propose an exact SDE-based diffusion framework that enables general generative modeling on Lie group representations, thus combining the advantages of curved dynamics with the theoretical and practical effectiveness of Euclidean diffusion. Our method realizes simulation- free training of Lie group-like diffusion models, and it provides a novel approach to denoising score-matching for general non-Abelian groups. Our main contributions can be summarized as follows:
Generalized score matching via Lie algebras: We extend generalized score matching X to estimate the score of any distribution on a Lie group G acting on X. We elucidate the conditions for suitable G' (valid for any differentiable manifold X). We recover standard acore-matching as specific case of our framework, corresponding to the group G = T') of translations on X = R".
Lie group representation diffusion processes exact solution of novel class of SDEs: We introduce a new class of solvable SDEs that govern Lie group diffusion via Euclidean coordinates, significantly expanding the range of processes that can be addressed using score- based modeling techniques. We also show that our approach extends naturally to flow matching (Appendix(F)
Dimensionality reduction, bridging non-trivial distributions and trajectory disentanglement: Through extensive experiments, we demonstrate that: (1) our approach can estimate, regardless of the choice of G. any probability density (2,, distributions) and (QM9): (2) by appropriately selecting G to align with the data structure, the learning process is significantly simplified, effectively reducing its dimensionality (Section MNISTT) (3) our framework enables solutions to processes that are challenging or unfeasible with standard score matching, such as bridging between complex data-driven distributions (Section 5 (MNIST) and (CrossDocked)).
2 Diffusion dynamics through Lie algebras
We start this section by setting up notation and review the connection between vector fields and Lie algebra actions on manifolds. A Lie group G is a group that is also finite-dimensional differentiable manifold, such that the group operations of multiplication ( funnttiioo A Lie algebra g vector space equipped with an operation, the Lie bracket, : x + g. satisfying the Jacobi identity. Every Lie group gives rise to a Lie algebra as its tangent space at the identity, g = T.G. and the Lie bracket the commutator of tangent vectors,
G x G - G and inversion are
We restrict ourselves to real Lie groups It would be interesting to extend our analysis to the complex case
Le et al. 2
2

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GgG
[6]
T2X
04
FLIRRy 34/00
Figure 2: (a) Depiction of the fundamental vector field definition ① Flow coordinates for pair of commuting (b) and not-commuting ones vector fields (c).
[A AB BA. In this work, are interested in how Lie groups and Lie algebras act spaces. Given manifold X, a (left) group action of on X is an associative map Px G x + such that px (c, orbits and stabilizers. The orbit of x E X is the set of elements in X which can be reached from x through the action of G, i.c. G {px (x), to x is the set of group elements that fix x, Gx = {g E Glpx (x) = x}. The action of a Lie algebra X, 2 g Vect X) is Lie algebra homomorphism and maps elements of vector fields on X such that the map g x - TX, [A, x) 1 (x)(x)) is smooth. Given E g and group action px, the flow on X induced by px is given by {A X x IR + X, (x,
Vz E X. Fundamental concepts associated with group action are the ones of
G}. The stabilizer subgroup of G with respect
px (exp(rA)) (x), where the map exp : - G is defined by exp(A) = (I)),, where 7A: I + G is the unique one-parameter subgroup of G whose tangent vector the identity is A. The infinitesimal action of X, dpx g + (e, is defined as the differential of the map px. that is,
dpx + dr T=
px (exp(( = = (xx))
TA is called the fundamental vector field corresponding to A Given fixed point X E X, we denote т = E. (xx) (xx))=((xx) the fundamental flow coordinate. which is the parameter such that applying the flow to Xo gives Central to our discussion is the fact that any smooth vector field V x + TX on X can be interpreted as a differential operator acting on smooth functions f:X-R. R. The operator (f) represents the directional derivative of f at x E X in the direction of (x). We denote LA = II V the differential operator corresponding to II In the following will use both II, and A interchangeably, when potential confusion arises. When dim g 1 we indicate as II(x) = (Ilr Il ...) the matrix the collection of fundamental vector fields.
Let us work out the example for 2 and G SO(2), the group of rotations the plane. -a ) 0
The Lie algebra so( consists of all matrices of the form Aa
where R, and the Lie bracket is identically zero. The flow on X induced by px is given by the exponential map
cos((TT) -sin(oT)) and without loss of generality we can set 1. The Pe (exp(TA..) (x) sin(ar) cos(ar)
infinitesimal action is computed as
ApRa(A)
COS T
sin 7
x1 222
x2
dr
sin T
==
and thus the fundamental vector field defines the derivation (x = be fixed point, then the flow equation x(7) (xo, T) Pa= (exp(TT) xo) gives system of two equations, which we can solve to find the expression of the fundamental flow coordinate
0 + œ1 dzg Let x. E R² Oz
x Xo
|xo cos
Xg
T arctan
where X X x = 12 31
(2)
x
x Xo
= |xol
sin T.
x Xg
Note that
8
8z
Dr
D2 B
+ L1 Bzg
8
(x) V C
IIn the manuscript we adopt both notations px (x). derived from defining : G GL(X), and px (g. x), derived from the definition px G X. which obviously equivalent.

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2.1 Intuition behind Lie group-induced generalized score matching
Score matching aims at estimating (log) probability density p(x) by learning to match its score function, i.e., its gradient in data space. Generalized score matching replaces the gradient operator with a general linear operator L (Lyu 2009). The learning objective is given by minimizing the generalized Fisher divergence
/ p(x) |Clog ss((x)) dx,
De(pllg))=
(3)
where Sg L log qs. The requirement the choice of L is that it preserves all the information about the original density, Formally, we require C to be complete, that is, given two densities p(x) and qxx). Cp(x) Lx(x) (almost everywhere implies that p(x) ((x) (almost everywhere).
Given a Lie group G acting on X, the collection of fundamental fields II corresponding of basis A = (A1; ......) of is linear operator, thus potentially suitable for actre-mat It is then natural to set L to the derivation associated with the fundamental fields II. i.e.. L = II(x) V. then follows that C ((x)(x) computes the directional derivatives of log p(x) with respect to the fundamental flow coordinates т, and provided that II meets some consistency conditions (which we will address in the next section), we can employ log p(x) to sample from p(x) using Langevin dynamics:
choice
Xx+1 -o (xxx
X,
Σ Log (x,) At
gencralized scores A, directions
where At is the step size and we have temporarily set aside stochasticity and denoising aspects. This process mirrors the intuition depicted in Figure each infinitesimal step of the dynamics corresponds to infinitesimal transformations along the flow on X induces by the G-action, and each component of the generalized score is learned through maximum likelihood over the orbits ÇA, of the corresponding transformations.
2.2 Sufficient conditions for Lie group-induced generalized score matching
We now address the properties our setup (X, G. g. II) must satisfy meet the sufficient conditions for score-matching and Langevin dynamics. We note that these results hold for any differentiable manifold X. Proofs for these results can be found in Appendix C
Condition 1: Completeness of II. We start by establishing an algebraic-geometric condition for It's completeness:
Proposition 2.1. The linear operator IIxx) is complete if II is the local frame of a vector bundle E over whose rank is n 2 dim almost everywhere. If rank E = n everywhere, then E TX, the tangent bundle of X.
The following result specifies which Lie groups yield operators II satisfying the above proposition:
Proposition 2.2. The operator II induced by is complete if and only if the subspace U C X such C2 < n. for x E U, where = dim X, has measure zero in X.
that dim G
As an example, consider standard score-matchin on mass-centered point clouds. Here X 333-3 since the points' coordinates satisfy > X; = 0. Without loss of generality, X can be parametrized by N-. with XN determined by the center of mass condition. The group G T(3N) acts transitively on X, with a 3-dimensiona stabilizer subgroup Gx {(...... 0, a) T E RRN fixing the space. Thus, dim G G/Gx = n for all x E X, satisfying Proposition 2.2
Homogeneity of X. While the completeness of the operators is necessary for estimating the target density, it is not sufficient to ensure that the Langevin dynamics (4) will behave appropriately, as the following example illustrates. Let X R. and G RR, the multiplicative group of non-zero positive real numbers. The orbits under the action px (a, x)) = az are O = (0, oc),
Condition 2:
Almost everywhere means everywhere except for set of measure zero, where we assume the standard Lebesgue measure.

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-- 0--0->,,)), 0). and Ou = {0}. If the dynamics begins within O+, it will be never be able to reach values in O_., as G-transformations cannot move the system outside its initial orbit. We therefore ask that each pair of points of X is connected through the G action. This amounts to require that X is homogeneous for G, that is, Vx, y X there exists g G such that px (g)x = y. We note that this condition solely ensures that the generation outcome is independent of the initial sampling condition, that is, that Langevin dynamics can generate any point of the target distribution from any point of the prior. Beyond this, the formalism remains fully applicable in the non-homogeneous case, where the dynamic is restricted to orbits of the group, effectively partitioning the distribution. Though the formalism still applies within each orbit, global generation across X would not be supported without homogeneity.
Condition 3: Commutativity of II. The final requirement is that II forms a (locally) commuting frame of vector fields. AA, LB) [x)) VA, B and Vʃ E C (X). In this case, the coordinates T3 i are orthogonal, and their flows commute. meaning the orbits parametrized by correspond {Ty 0) jzz For non-commuting flows this is not the case, as Figure 2b-c illustrates: (b) Vi = z1 Qz + TzŪr2: V/2 = 210m, xz22, satisfy [V1 1/2 = O, and the orbits parametrized by 71 r correspond to subspaces with constant 72 0: (c) W1,2 Vi.2/|x do not commute, and the loci 0 = const no longer coincide with the r-orbits, causing 0 to vary along these, despite the fact that r, 0 are still orthogonal at each point. This last condition ensures that the updates governed by the different elements A, of in (4) remain independent of one another. Notably, this does not exclude non-Abelian groups; even A1.2 g do not commute in the Lie algebra, their flows on X can, as shown in the g s3) example in Appendix B.3
3 Lie algebra score-based generative modeling via SDEs
In this section, we formalize the framework we developed above from the point of view of SDEs. Namely, we show that there exists class of SDEs, which, when reversed, can generate data according to dynamics similar to (4) guided by the generalized score of the fundamental vector fields of the Lie algebra g. Our main result is the following.
Theorem 3.1. Let G be Lie group acting X satisfying the conditions of Section 2.2 and let g be its Lie algebra. The pair of SDEs
(()) px px() dt + +x(+)[(()))W.
dx (()))(()(((
(5)
dx [(0))((XXX(X Px (1) - -X( II(x)
72(
-11 log de (x))
(6)
R are time-dependent functions, R - Baxn the fundumentul vector fields, f IR - TR the drift, Ω = =; A is known the quadratic Casimir element of g. and
where 3,y
C = [xx] V, is such that
1. The forward-time SDE 5 is exactly solvable:
x(t)= [[ [[oo(ttt)) x(0) [ x(0),
(7)
where O (t)AA is the finite group action and T(t) is the solution to the SDE dr(x) (()) +1))W.
(8)
2. The SDE (6) is the reverse-time process of
3. The Langevin dynamic of the above SDEs decomposes direct sum of infinitesimal actions TT, defining an infinitesimal transformation along the flows E
We refer to Appendix D for the full proof the above result. Here we limit our- selves to few comments regarding the extra terms that appear in the paired SDEs.
5

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The appearance of the Casimir element (we assume the identity as bilinear form on g (Kac & Kac (983)) compensates for the deviation of the tangent vector from the orbit due to the curvature of the flow coordinates. This can be seen in the example of (2) acting on IR? (which will be discussed thoroughly below). An infinitesimal transformation along the 9 direction, represented by 1lg, moves any point x along a vector tangent to its S0(2) orbit, a circle of radius r = / + zzz Due to the orbit's non-zero curvature, this movement would shift the point to orbit of radius r' > r. The term px (S2) compensates for this displacement, ensuring the final point remains close to the original orbit. This is illustrated in Figure 3 With this result hand we can formulate our procedure for our Lie group-induced score-based generative modeling with SDEs.
0,
(x(
(x(x)
0,(0)
Figure 3: Quadratic Casimir for SO(2).
Perturbing data through the SDE. The forward-time SDE 5 defines a noising diffusion process respecting the decomposition of the Lie algebra g infinitesimal actions on X. In fact, given data sample x()) ~ Po. the solution (7) takes the form of a product of finite group element actions O₂ on x(0), where the specific order is irrelevant since the Lie algebra generators commute. For each factor, we first determine T (0) = T (x(0)), and employ these as initial conditions for the forward SDE (8) By choosing appropriately the drift terms f/s. for instance, be affine in the flow coordinates T;, can solve for (t) with standard techniques (Särkkä & Solin, (019), as this will follow a Gaussian distribution. Alternatively,
sample from [t] by first simulating (8), then performing sliced
score matching Song et al. (2020a) Pang et al. 2020) to sample from pr [x(TTT (())
Generating samples through the reverse SDE. The time-reverse SDE (6) guides the generation of samples x0) (x(x) starting from samples xT) ~ (x), provided we can estimate the
generalized score C log (x) of each marginal distribution. To sample from pr, we use the fact that the distribution in the flow coordinates T is tractable (with an appropriate choice of the drift terms and time-dependent functions B, y in (5)) and that (since Pe (x(xdx = (c())d
(x()) = PT(
= (((T)
a (x)
0x
where the extra term corresponds to the determinant of the Jacobian of the coordinate transformation induced by the fundamental flow coordinates. In particular, when f(T) affine in t. follows that Pr (7) [TT 0, E), where Σ = d of. oo.......,,, o.). Thus, can sample (TT) ~ Pr(T) simply as a collection of independent Gaussian random variables, and use the flow map to obtain x(T) & (T(T), xo), which will follow the distribution (9) for T. We describe training and sampling procedures in Algorithms 1 [nd 22 Appendix B
Estimating the generalized score. Analogously to standard score matching, we train a time- dependent neural network so (x)),,))) R. x - R to estimate the generalized score L Clog (xo))(((())xx((((( at any time point, that is, we minimize the objective
(10) where w : [0, T] + R, tiighweighting function. Now, from Condition above and the property that LA, computes the direction derivative along the flow of IT (x), it follows that Under the above assumptions, Pt(T) ((T))(((()) 1), ({)), where the form of the mean and the variance depends on the explicit form of Using the parametrization (()) = pxx(0), 1) + E()), where 77 (0, I), we obtain
E, (x)))) (x())) (x)(0)) [s)(((()),) - C (x)))((
C Llog (x (x)))((0)) = Vi(c) log pt (x())()))(())())))
C Clog - (T-((((()) (x(()),
Et)
n:
(11)
3.1 Examples
Standard Score Matching. Standard score matching can be recovered as a special case of our formalism by choosing X = R" and G = T(n). As we show explicitly in Appendix B.I we have L V and the Lie algebra action II(x) = I, the identity on X. Since II is x-independent, its divergence vanishes, well as the quadratic Casimir (T(N) ) is Abelian), that the SDEs (5) take the known form
dx (()) (t)dW, dx [[(1)((x) x(t)² log (xx)) dt + 7t)dW (12)

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[0]
10)
Figure 5: (a) 2d mixture of Gaussians (top: ground truth, bottom: generated); (b) generating process using single scores for the subgroups S0(2), R with the corresponding scores (c); (d,e) one- dimensional learning for a symmetric distributions; 3dddistributions: torus (f) and Möbius strip (g) (top: ground truth, bottom: generated); (h) 4d mixture of Gaussian for the group SO(4) x R+.
G = (2) x R4+ A simple but non-trivial case in given by G = SO(2) x R, describing rotations and dilations acting on X RR2 A basis for = so (2) @ R is given by A, and
Ag -1) yielding II(x)
-y which satisfies all the conditions of section 2.2 x
Following our discussion above and in Appendix B.1 we have (since p( A A dx = {t)) (f,()),,x+ + fo(0) A5x) dt + 7(t) (dW,A,x + d
0)
(13)
and we see that the SDE splits into contributions from the two Lie algebra summands. To find an explicit solution, let (t) /B(t) and fr corresponds, in the flow coordinates system, to 2d Ornstein-Uhlenbeck system (Gardiner) 1985)
log(x²2 y²),
arctan V This choice
(((0) 0(0) e lö ((s)de and variance (1 e fü 8())da
T. Let us define (t) = VI - e J p(s)ds such that r()) = r(0) + A(t) = r(0) - r()) + (f)n, and similarly (()) (0) + sp(1) 0(0) ((0) ( where T/r, E (0, 1), then it is an
which has Gaussian solution with mean
easy calculation to show that
x (t)
= x01) cos p(t)
/cos>(()
sin ((t)
x 21
(t
p(t
cos p(t)
x2(
(14)
We can look at the asymptotic behavior of the solution. Assuming that (t) is monotonous increasing function, that is, B(t+ > B(t) for e >> O, then lim44400 o(t) and hence
lim x(t) -o 1-00
cos Bu sin 0u)
(
coss7 179 sin no sin
(e"0 cos Op
e cos 110
-in du
cos 00
cos m7
e sin 0u
e sin
where 0g = 0(0), ro = r(0). Note that, even if (15) is not Gaussian, we can still easily draw samples from it by sampling the two Gaussian variables 77,9-
Dihedral and bond angles. The above for- malism can be applied to obtain transformations of physically meaningful quantities, as bond and torsion angles for molccules' conforma- tions. Let % be the dihedral angle between the planes identified by the points X;, X and {x,, X;+1, X,+2). respectively (Figure 4a). The Lie algebra element corresponding finitesimal change % is given by a 3N x 3.- dimensional 3-block diagonal matrix, whose
in-
Figure 4: Lie algebra (2) C so(3 dynamics for torsion (a.b) and bond angles (c,d) in molecular conformers.
j == = I,.., N block is given by H(j - (i + 1)) A), where A- (Ap, Ay. A.) is the
vector of the Lie algebra basis for (3), X (x - xx)/((((44 - x,) and H()) = 1 if > 0 and 0 otherwise the Heaviside step function. For bond angles 3, (Figure 4 we construct the corresponding so(2) E s0(3) algebra element blocks as Híj - (x x;-1,5) A. Examples of the dynamics generated by these operators are presented in Figure (bd).
4 Related Work
Representation theory applied to neural networks has been studied both theoretically (Esteves) 2020
Chughtai et al. 2023 Puny et al. 202 Smidt 202 and applied to variety of groups, architectures
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q
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Figure 6: (a) Original and rotated MNIST samples with generated samples from our model and BBDM. (b) Reverse diffusion trajectories of our model against BBDM. Intermediate samples from
BBDM resemble interpolation of mixed digits. For the first BBDM case, the 4-digit transitions into a 6-digit.
and data type: CNNs (Cohen & Welling 2016 Romero et al. 2020: Liao & Liu 2023 Finzi et al. 2020; Weiler & Cesal 2019; Weiler et al. 2018). Graph Neural Networks (Satorras et al. 202
Transformers, (Geiger & Smidt 2022 Romero & Cordonnier] 2020 Hutchinson et al. 2021 ). point clouds (Thomas et al.) 2018), chemistry (Schütt et al. 2021: Le et al., 2022a). On the topic of disentanglement of group action and symmetry learning. Pfau et al. (2020) factorize a Lie group from the orbits in data space, while Winter et al. (2022) learn through
autoencoder architecture invariant and cquivariant representations of any group acting on the data, Fumero et al. (202T) learns disentangled representations solely from data pairs. Dehmamy et al. (222)) propose architecture based on Lie algebras that can automatically discover symmetries from data. [uu et al.] (2022) predict molecular conformations from molecular graphs in aot--translation invariant fashion with equivariant Markov kernels.
Related to our study is the field of diffusion on Riemannian manifolds. De Bortoli et al. (2022) propose diffusion in product space, a condition which is not necessary our framework, defined by the flow coordinates in the respective Riemannian sub-manifolds. When the Riemannian manifold a Lie group, their method yields dynamics similar to ours, as illustrated in an example in Scction 3.1 In fact, our formalism could be integrated with their approach to create unified framework for diffusion processes on the broader class of Riemannian manifolds admitting a Lie group action.
These techniques has been applied in variety of use cases (Corso et al. 2023 Ketata al. 2023
Yim et al.) 2023] Jing et al.] 2022) for protein docking, ligand and protein generation. The works
Zhu al. 2024): Kong & Tao (2024) leverage trivialized momentum perform diffusion on the Lie
algebra (isomorphic to R²") instead of the Lie group, thereby eliminating curvature terms, although their approach is to date only feasible for Abelian groups. An interesting connection with our work is the work of Kim et al. (2022): the authors propose a bijection to map a non-linear problem to a linear one, to approximate a bridge between two non-trivial distributions. Our case can be seen as a bijection between the (curved) Lie group manifold and the (flat) Euclidean data space.
Experiments
2d, 3d and 4d distributions. In Figure 5 we illustrate the framework for a variety of d = 2, dimensional distributions. In all cases take G SO(d) x R4. Figure/ja.b.o b..b displays a mixture of Gaussians: in (a) (bottom) we see that our generalized score-matching can learn any distribution, regardless of its inherent symmetry; (b) shows the output of the generation process using only one score (top g = 52),,, bottom g = r+), while (c) shows the vector fields corresponding to the scores, where we color-coded the field directions. Figures(3] (j,d,) depicts radial and angular distributions, where the score is learned using the respective Lie algebra elements. This reflects the ability to leverage the symmetry properties of the data and perform diffusion in lower-dimensional space. We also show in Figure (G = (4) x IR.,) that our method can be applied to higher dimensional Lie groups. We list quantitative comparisons in terms of W2-distances for our generalized score model against standard diffusion model in Appendix E.
In this experiment we show that our framework can be applied to effectively learn bridge between two non-trivial distributions, adopting however only techniques from score- matching and DDPM. Let (x(x) (x) be the rotated MNIST dataset and po (x) the original (non-rotated) MNIST dataset. We can learn to sample from po starting from element pr by simply modeling (O( dynamic. Some examples of our results are shown in Figure 6 Notice that our formalism
Rotated MNIST.
8

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allows us to reduce the learning to a 1- --dimensional score Co = z102, - -xgdry, which reflects the true dimensionality of the problem. We trained the model with T = 100 time-steps, but for sampling it suffices to set T 10. As it can be seen in the example trajectories G. the model starts converging already at 1/T ~ 0.5. We employ a CNN which processes input images x(t), and the resulting feature map is flattened and concatenated with a scalar input t, then passed through fully connected layers to produce the final output.
We compare our approach to the Brownian Bridge Diffusion Model (BBDM) al. 2023). Unlike our method, BBDM operates unconstrained in the full MNIST pixel space (IR28x25), where intermediate states represent latent digits. As shown in Figure 64, this can result in incorrect transitions. such as adding extraneous pix- els or altering the original digit, even generating entirely different digits (Figure 6p).
We further evaluate both methods on the classification accuracy as well as FID scores of generated MNIST digits. Since the task is to correctly rotate a MNIST digit into the correct orientation aligning with the ground-truth data distribution, we observe that our GSM model achieves superior classification accuracies (0.96 vs 0.80) and FID scores (85.77 vs 133.4) as shown in Table [
Table I: FID and Accuracy scores com- paring GSM against BBDM.
Avg
Avg FID (4)
Model
Acc (t)
GSM BBDM
0.96 + 0.02
85.8 + 15.7
0.80 0.10
133.4 +: 19.0
Further details can be found in Appendix
framework train a generative model po M) for conformer sampling of small molecules M from the QM9 dataset (Ramakrishnan et al.] 2014). We only keep the lowest energy conformer as provided in the original dataset, that is, for each molecule only one 3D conformer is maintained. Here X = R&N and we choose G = (SO(3) x ) where each factor acts on the space R spanned by the Cartesian coordinates of the molecule's atoms, respectively. As Figure 7 shows, our generative process yields conformers that are energet- ically very similar to the ground truth conformers, while showing some vari- ability, as it can be seen in the last example where the torsion angle is differently optimized. We train an- other model p,y (XIM) via standard Fisher denoising score-matching, i.c., choos-
QM9. We
Figure 7: (a) Generated 3D conformer from the QM9 validation set (top) and ground truth conformer (bottom). (b) Energy difference distribution between diffusion models (pe, P.y) and ground-trute energy. Both models generate a similar A energy distribution.
ing G = T(3) as in Sec. 3.1 and generate 5 conformers per molecule for both models pa: Py- We then compute the UFF energy (Rappe al. 1992) implemented in the RDKit for all generated conformers and extract the lowest energy geometry as generated sample. To compare against the reference geometry,
compute the energy difference A = Unue Ugen for both models. Figure 7p shows that both diffusion models tend to generate conformers that have lower energies than the ground true conformer according the UFF parametrization, while the diffusion model that implements the dynamics according to G = (SO(3) x RR,,) (colored in blue) achieves slightly lower energy conformers, mean Ag 0.2119 against mean A., -0.2144 for the standard diffusion model (colored in orange).
DroksDocke22020: Global E(3) and Protein-Ligand Complexes. In this final experiment, we train a generative model for global SE(3) transformations acting on small molecules. Specifically, given a pair consisting of compound and protein pocket, our goal is to generate the trajectory by which the ligand best fits into the pocket. Importantly, the internal structure of the compound remains fixed, which presents challenge with standard diffusion processes. Thus, while the SE(3) transformations are global with respect to the ligand, they do not represent global symmetries of the overall system. We derive in appendix B.4 the relevant operators that guide the dynamics (6). Figure &h shows examples of docked molecules using SF((3-guidde) score-matchie cotee-ntthhing diffusion. The true and generated molecules at different generation steps are visualized as point clouds, showing

---

(A)
(b)
Figure (a) SE(3) trajectories for molecular docking; (b) Comparison with RSGM.
good agreement. Figure shows that our model achieves lower RMSD (2.9 + 1.0 Á vs 5.6 + 1.2 A) for the docked ligands than the RSGM method (De Bortoli et al. 2022 Corso et al. 2023) (for details, we refer to Appendix E...T. We also compare our method against the Brownian Bridge Diffusion Model (BBDM) which operates the T(3)* N group, baseline with the constraint to start and end at valid rigidly transformed molecules during training. We use the same network architecture in the GSM and RSGM experiments to learn the correct SO(3) rotation. Unlike existing experiments (our method and RSGM), the Euclidean BBDM in this setting attempts learn only global SO(3) rotation, neglecting translation. Since the problem is implicitly 3-dimensional but the equivariant score network predicts all 3N ligand atom coordinates, final samples with implausible coordinate trajectories tend to have higher energies due unphysical poses including bond stretching, non-planar aromatic rings, and deformed rings. In terms of mean/std RMSD on the Docke2220) test set, our method (Lie algebra: 2.91 + 1.0 Å) is comparable with BBDM (2.92 = 1.57 Å). However, since BBDM models all atomic coordinates, the overall dynamics do not follow global SO(3) rotation, achieving MAE (# zo), D fr)) 0.43 + 0.21), while RSGM and our method achieve 0.0 by design. This indicates that Lie algebra induced diffusion offers clear advantage over standard diffusion models in this particular bridging problem.
standard (Euclidean) diffusion
6 Conclusions and Outlook
We presented a method for generative modeling on any Lie group G representation on space X through generalized score matching. Our framework generates curved Lie group diffusion dynamics in flat Euclidean space, without the need to transform the data and of performing group projections. Specifically, we introduced a new class of exactly-solvabll SDEs that guide the corruption and generation processes. Thus, our framework does not merely complement existing methods, but expands the space of exactly solvable diffusion processes. Our framework is particularly relevant given recent findings (Abramson et al.) 2024) showing that unconstrained models outperform equivariant ones: with our framework there is no need of tradeoff, as we retain the expressivity of unconstrained models raw Cartesian coordinates with the benefits of group inductive bias. flow matching (Lipman al., 2022).
Moreover, our techniques descend quite straightforwardly We spell out the connection in appendix FF and we plan to expand on this in future work.
Code Availability
Our source code will made available on httt:://hhhiicooompprrrpopooooer score-matching
10

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A
Appendix A: Summary of Notation and Intuition
Symbol
Name
Definition
Intuition
G
Lie group
A continuous symmetry group, e-g., ro- tations (SO(3)), translations, scalings. Encodes the structure of transforma- tions acting on the data.
e
Identity element G
e e-g g
gVg EG
The identity transformation leaving ev- erything unchanged.
g
Lie algebra of
T&G
Tangent space at the identity; represents infinitesimal group transformations.
X
Data manifold
The space where the data lives, often RR, but can be more general or discrete (e.g., graph for molecules, grid for images, etc.).
Group action
Px : G.Gxx -
Specifies how each abstract group ele- ment g E transforms data points X via matrices.
G X
Orbit of under
(xx(0)(x), g EEG)
The set of all points reachable from x via group actions. Captures the "sym- metry class" æ.
Gx
Stabilizer subgroup at
tg E Glpx(o)(x) x}
Subgroup of G' that leaves unchanged. Describes residual symmetries at that point.
dpx
Infinitesimal action
dpx + Vect(X)
Maps infinitesimal transformations to vector fields X; captures how tiny "step" in G moves point in X.
exp
Exponential map
exp(A) (A(1), where
Geodesic path on G determined by the direction given by the vector A E Path on X corresponding to a geodesic path on G determined by A.
7A: R - G
Flow on X induced by px
px (exp(TA)) (x)
(xxx)
Fundamental
vector
field
d =x
vector field on X generated by direc- tion A in the Lie algebra; describes how x moves under an infinitesimal group transformation.
from A
Table 2: Summary of Lie group/Lie algebra related quantities with their notation, definition and intuitive meaning.
B Examples of Lie groups and Lie algebra actions
In this appendix we list some important Lie groups and Lie algebra actions, their corresponding fundamental vector fields as well as the fundamental flow coordinates. These will be useful in the main text.
B.1 T(N)
Let X = N and G T(N), the group of translations in N-dimensional space. Element of T(N) are represented by vector (v), 12,,.... uN) E RRN, where are the translation components along the x; axes for 1 .,.... N, thus T(N) R.V. Explicitly, for x X its action given by PR (v,x) = x
15

---

The corresponding Lie algebra ((N) is also isomorphic to R", and it consists of vectors
(01, dg...... aN. E RRN. The Lie bracket of any two elements in t(N) vanishes, as T(N) is Abelian.
To derive the infinitesimal action, we first note that the exponential map is trivial exp(rA) =A. Hence, we have
d
d
(x)
PEN (TA,
(x+ A.
(15)
dr 7=
dr ==0
Thus, the fundamental vector field II corresponding to A E t(N) is the constant vector field:
A
0/ Orz
ON
A V.
Or1
DEN
B.2 X=R R4 (group of dilations)
Let us consider X = R. N and G = R the group of dilations in N-dimensional space. The group F consists of all positive scaling factors. Each element of G E can be represented by scalar >> that scales all vectors in IR. N by this factor.
The action of G = 1 on RR is a dilation, meaning that every vector x 02, IN) E RR scaled by the factor A. Explicitly, the group action is given by
PRN (A, x) = Ax.
(16)
The Lie algebra = R corresponding to the dilation group G R consists of real numbers representing the logarithm of the scaling factor. Specifically, element A g corresponds to generator of the dilation, and the exponential map exp - G is given by:exp(TA) = A where is real parameter.
The infinitesimal action corresponds to taking the derivative at t 0. For a vector E R- and A g. the fundamental vector field IIA is computed
d
d (ETAx) Ax
dpp> (A)
PaN (e xx)
(17)
dr r=0
dr r=0
and
(x(x) = Ax V.
Now, solving the equation
x Axxx
(18)
terms of 7 we obtain
2
2
1
x
x 2
log
log
log
log
(19)
X0
x
Xx
A
xol
2A
x 2
In the usual case of (generator of the Lie algebra), Xo vector we obtain the usual expression as flow coordinate
/N 1, 1,...,1) T)T to be the unit
マ log(2 + +x2 + .... xxx
(20)
B.3 X= S0(33) x R;
The dilation part is solved in the previous section, so we actually just focus on the action of SO(3) on R3, The orbits given by spheres centered at the origin, and we can decompose the action of SO(3)
16

---

by variying the azimuthal or the polar angle defined by vector x. Namely, we have the two actions
COs ip
sin +p cos up
0
xp)
sin
0
1
2
0
0
y
(2)
+ sin 0
0
0
sin 4
008
yp
sin 4
cos? yp
cos ypsin s
<
+(1 cos 0)
cos y sin yp
sin 1
(21)
0
we take the differentials
sin y0
cos up
0
dpea (xx, x)
cOs up
sin yp
0
1
0
0
A.x
0
f=
0
COS 4
dpza (0, x)=
cos 0
sin up
CoS y
sin
cos² 4
cos y sin y
00
sin
cos sin yp
sin x 0
A=0
0
0
CO8 yp
x
0
0
sin yp
y/
(cos pA,
sin pAx)xx
(22)
cos p
sin
y
where
10
0
0
0
0
1
0
-1
0
A₂ =
0
-1
Ay
=
0
A_ =
1
(23)
0
0
0
0
0
0
0
form a basis for (3). The corresponding differential operators are
Cpp - x0, yy
Lo
Ix zyy,, (x² +
(24)
1.2 +2
and it is an easy calculation to show that they commute [Cep Lel = 0. The attentive reader might have noticed that the commutation does not hold at the matrices level. While this is expected, since there is no 2-dimensional commuting subalgebra in so(3). it is nonetheless quite puzzling since everything works out the level of differential operators. This reflect the fact that the commutation properties are necessary at the level of the action of on X, and not necessarily at the Lie algebra level. In this case, however, we elegantly resolve the puzzle, we found matrix representation for the action ap (0)x which does commute with the action. To do this we note that we can rewrite
Co sin 0 x2 cos f
cos f
sin 0
sin ydy cos 0 =0=
(25)
which corresponds to simultancous dilations, with diffcrent coefficient, in the z axis and y-plane. The finite action takes the form
0
0
(0
70
0 0
Pg (0, x) = exp log sin
0
1
0
+ log cos
0
0
(26)
0
0
0
0
0
1
,
and computing the first order term we obtain
coss
sin 0
dpps (8, x)
0
sin 0
6
x
(27)
0
0
sin cos
This matrix is diagonal and it trivially commutes with Ag. The price we had to pay to realize system of commuting matrices that in P the flow parameter appear non-linearly, thus we traded-off commutativity at the level of the Lie algebra matrices for the linearity of the flow parameters at the group level. We remark that both give rise the same differential operator on X, which the relevant object for our purposes.
17

---

(a)
(b)
x
Figure 9: (a) The coordinates Ny are the coordinates in the coordinate system defined by X1, the orthogonal projection of X2 with respect to T1. X2 1I X1 = X2 X1 X2, and X₂ x X1. (b) Graphical depiction of the global symmetry transformations parametrized by the three angles (22, 02,00,1101 P11.
B.4 X = =R3N and global SO(3)
Let X = RN be parametrized by X;=I...,,NN We can describe a global SO(3) action as follows
X1 Re, Re, (01)
D22
X2 (e,((2)) Re, (01) Ro. (122)
0
Xp Jy....... N Ro, Roy (0)) Ro. (422) )
(28)
where Rg (w) represents a rotation of an angle w around the axis a. We can then derive the operator II JR3Nx3N as follows. Let R' (w) be the matrix where take the partial derivative with respect to w of all elements of R. Then
n1 = ([A,x1]T [A,x²]T ...
[AgxN]T)*
(cos 401 Ay - sin 4p1 Ag X1
(cos 9p Ay sin 1p Ar Xx2
Na
(cos 1p sin 42 A1AA)NN/
(sin 01 cos 1o Ar + sin 0g sin A1 + cos A,
(29)
(sin 0n cos A1 sin 0 sin PA cos A,NNN
Notice that these do represent global rotations since it is easy to see that (sin 0g cos Ap sin 0 sin Ay + cos , (xA,)x1 = 0. Formally, the true Lie algebra elements are x 3 matrices of the form
A 0
0
A2
0
A
Ag
0
(30)
A
18

---

and similarly for the other operators. Now, for the inverse relations we have
21 (0f + + 2
0 =
arccos
21 (2] + 1/)
P1 = sgn y1) arcoos
22 arctan
(31)
22
where X2 (01) Re, 112 Re (-0 Re, (-11)
B.5 = (0(4) I
Now we look at the case of a higher dimensional Lie group, namely G SO(4) x Rt. The parametrization given by
T1 e" COS 91
T2 e" sin yp co8 y22
T3 e" sin 1 sin p2 (P3
F4 e" sin y sin 922 sin yp3
(32)
The Lie algebra elements corresponding to the SO(4) flow coordinates are
COS SP
sin COS sin (2 sin PP
Ap1
cos (P2
0
sin cos 4P
0
0
0
sin P22 sin PP
0
0
0
O
Ap2
0
CO8 4P3
sin 4P3
0
cos yPa
0
0
sin
0
0
0
0
0
0
0
0
0
Aps
(33)
0
0
-1
0
0
Next, we compute the three non-trivial commutators (note that an operators with itself always commute). First, we list the differential operators
Cup
[11222 + x133ds + 21 - (x² + +2 +1))0
122
+
+
x
Cxp
[222z0s + 12
(4 +
23 +
x²2
Cps x3 ug03.
(34)
where we used the notation 0 dz.- These follow directly from and using the relations
33 together with Lp. = A,,, V.
D33
22
v +
sin
COS
COS 422
sin 22
x
V V x
V
10
+
x
x
v 12 + 2 x
(35)
Specifically, we have
1
TTzT4 + 2443
[Cyp2: Lypal
22 Og + x223304]
[ + +r (1g + x2)02
(a² + 2/2
[220303
T2 + 20z
V23
+ x
1r1300 x114d3]
2xzzz x1 + 2xzx1
Vah
x3 + x4
2 x
0.
(36)
19

---

( Lppll
110341
x1
-I334 + T433
[2i220] +110d + +.aa4ds - -22 + +3 +1)00
(v3 + u + x2))//2
22 + 13 +21
1
2xzz1 11zzdd
x1 + 20zz1011
22 V
+ 33 +
1
x10403]
o1ozd4 22zz4 - xII143 + 2rzz401
V12
43 21
v x2 + x3 +
0,
(37)
[pp1 Cppll
Σz4g x404
21
1 (x) + x2)3//2 [I22zds+ + x20,01 - (x3
-2 2,
Va2 23 22
V 23 xi
v 1 +
01021383 - 2 - - +
V n2 + +
(z) + 1/2
-22
2243
22)02
[イزl2d] + I13304 + pyzze - (rz + 23 + x1)0
[x] + x2)1//2
(x) + 23
+ 3//3
1
[11220401
21z170 + + - 222330 (0) + 21)(0112 - 2020r)]
(u3 + x2)
22 + +
-xn (rz + x
[22340 +222404 - (2) + x9)0z
v 12 + 13 + 24
(1) + 2//3/2
1
1
F1222388 - 2 2x, 201r 210221011
22 + + +
(2) + 2/2
1
2m
2rzzz0
(2 (2 2rr01]]
(x) + 2/2
13 22 +
11
[- 02
[2x
2x 210,]
V Va + 13
+ (13 + 2211
v 1 +
n + 2 (xg +
x
[ + 2 2112
(2] 2
0.
(38)
B.6 G = (NNN)
We present here the formalism for the G SO(N) for any N 2 4. The parametrization given by (Blumenson) 1960)
01 cº COS (P1
32 = e" sin pi CoS ,
T3 c" sin sin (22 cos pps
2j = c" sin 1o sin 92 sin pp .ii....iinngn p 100-0000011 ()
Fn-1 1 = e'sin 1 sin yo2 sin pp ......800 sin Pp C08 fn-
In = c" sin sin و2 sin (p3 sin -n sin pn-1
(39)
20

---

The corresponding Lie algebra elements are given by
0
0
A Pp-
0
0
-1
Pn-2
Pa-1
sin pp
COS pp-1
sin Pn-1
0
0
0
Apn-3
COS Pn-
sin ypn 2
008 pp-1
-ii pn 2
sin Pa-1
008 4n-
0
0
sin nn- COS nn
0
sin 22 sin n
0
0
0
0
1
11+1
Ap,
2i+1
0
Jjj
23 22+2
En
0
23
App 21
T3
(40)
0
0
0
0
0
C Proofs of condition for suitable Lie group
Here we provide the statements with proofs of the results in Section 2.2
Proposition C.1. The linear operator induced by II is complete if is the local frame of a vector bundle F. over X whose rank is n. 2 dim X almost everywhere. If rank F. = n everywhere, then E = TX, the tangent bundle of X.
Proof. We start by noting that, given the expression of the fundamental fields as derivations, we can write C(x) [xx V. Let
-> X be the projection map, then rank π -x) min(rank [I)),, n), since rank V = n. Now, consider L log p(x) = L Loog (xx,, which implies
U has measure g(x) = c, constant Vx U. Now, p(x) and q(x) are probability densities by assumption, thus 1, which proves the claim. O
C log pix) x(x) O. Let U X such that rank II 2 Vx E U, and by assumption
zero. Then the above holds if and only V log
p(x) xxx = which implics
p(x)
Proposition C.2. The operator II induced by g complete if and only if the subspace U S succh that dim G G~ for x E U, where = dim X, has measure zero in X.
Proof. First, we recall that the dimension of an orbit Ox of x E X equals the dimension of the image of the map dpx
TxX: A + [xx).. Suppose first that II is complete. Then, from Proposition C.1 the rank [xx is 2 n almost everywhere, and therefore dim G/Gx almost everywhere, which implies direction of the claim. The reverse is quite straightforward. Assume that the rank
21

---

of [x(x) is 2 almost everywhere. As II represent the action of the infinitesiamal transformations of G, it means that locally G cannot fix points in thus proving the claim.
D Proof of main theorem
Here we provide the full proof of Theorem 3.1
Theorem D.I. Let G be a Lie group acting on X satisfying the conditions of Section 2.2 and let g be its Lie algebra. The pair of SDEs
dx (x)) + 2()²
2 px (1) dt +x(t)
(41)
22)
dx [()II())(()
2 PX (S2) - (X) II(x)
( dt + (x)
(42)
where B, : IR. - R are time denedeenn functions, II : - RRux the fundamental vector fields, f IR - TR." the drift, Q E, A? is the quadratic Casimir element of g. and C II(x) V
such that
1. The forward-time SDE 41 is exactly solvable, with solution
x(0) II [o(T)))) ) (Ii
n
x(1)=
7. x(0),
(43)
where O₂ (t)(A): is the finite group action and (()) is the solution the SDE dr(x) ((t)f(xx)dt (t)dW
(44)
2. The SDE (6 is the reverse-time process of 5
3. The Langevin dynamic of the above SDEs decomposes direct sum of infinitesimal actions T.. each defining an infinitesimal transformation along the flows E
Proof. We start by proving 3. We start by rewriting 41 in terms of the fundamental flow coordinates _ (x) X + IR For this we employ Itô's Lemma for the multivariate case: given the and transformation Txx), it is given by
SDE 41
dr(x) (Vx)) (())II()((()) 22(1)
px((()) dt 22(1) Tr [I()]T (1xxT) IIx)) dt 2
+v(1((xx)) xT) (I(O(())TI(((xxl IX())dW
= 3())((x) + 72(1)
2 ( Tr [I() (HxT)
di + +t]dW
(45)
since X II (x) matrices. Now, the second term can be rewritten in components as {(Vx7) A + Tr [I[()) (HxT)
[xx]]
Σ Ox, dx; θT
Ox, Or, 8º
ΣΣ
OT
Ox, θn
Or:
Ora On 0r,0x;
0x, ①
Ox; dr
OTk Ing
On Oe.
Σ
Ox; ØT
0Tk
Om Oxy
Ô dT
1Tk Ōm
-H,T,
(46)
22

---

which vanishes. Thus we proved that
dx(x) (()) r(t)dW
(47)
and provided that chosen so that f((x(T)) = f((T)), this corresponds to a system of independent SDEs, claimed.
Now, prove let (t) be solution
447 and x(t as 77. Then Taylor expansion yields
x(t)=
I - Σ (1( 2
Σ (((()Ai) + O(()))
x(0)
(48)
since [A, A,1 = and where ( represents terms of third order T²'s. Then taking the differential and dropping higher order terms
27
dx(()) =
Σ (t)
Σ dri(t)
x(0)
2
Σ [8(1)E(x)dt +0 )7
Σ + A,
xx()
Σn E2(dawA)) ) x(0)
1 ((() (() +
2
((O)())))))))))))))))))))))))))) + 21)2
Σ Addt x(0)
2
(())I())((() 2()2 px (12) di (()
(49)
which in the forward SDE proving claim, where we used the relations dW? dt and dW, dWW = for j + i.
Finally, we prove 2. To do this suffices to apply Anderson's 's result (Andrson 1982
dx = (x)(I (² px (2) (() ((x))
(x) Vx lx) dt +
(()
(50)
and note that II(x) Vx = L, the generalized score, and
Vx (IIx)II((X)))
)) Oz
(I1.IIks)
θ
i TT.
(T.;)) IT + Thi Dakk
Ozk
Oek 0 (3x:
+ -1 [x)]];
dar ÔT.
Ô Oz:
+ I1is I[x)];
(Tr (x)),, H, VT I(x)),
(51)
where we recall that the divergence of a matrix is vector whose components are the divergence of its rows. Recalling the relationship between the trace of the Hessian and the Laplacian we can write in operator form
Vx (x)) ) = V I[(x) + Px))),
(52)
Plugging this back in into the previous expression we obtain our claim
22(t)
dx [())(())(() - 2 px ( - (x) II(x)
(xx dt (x)
(53)
23

---

Algorithm Training with seriance-preserving A Algorithm 2 Sampling with variance-preservinng scheduler
scheduler
I: repeat
Tr ((
Xo ~ q(x))
2: Mg (T:)
b Cartesian coordinates
3: >~ Uniform((1... (f1...,,...
3: for t T T...., do
4: 1)
n (0,, ift else n
5: To Mx(x))
Flow coordinates. Mc is
group-dependent
5: Va. E ( The (x,, 1):) A.)x.
T: 7:7o + o.
b Sample from p(T: (xxo))
7:
(Tt) (T)
Cartesian coordinates
Dynamies induced by drift and generalized scores
EB
8: Take gradient descent step
6: Px Px( = Ve,r (,, A,)x, b Dynamics induced by quadratic Casimir elements
Vo (xe: t)
9: until converged
7: Va, Σ, V. [A.x.]
Dynamics
ER
induced by divergences
8: Vi = Va,1 + 1 Va. + Vat
9: x,-1 = Xx + 1ev
Update state based on
velocity
10: X;-1 x v MAAA
Stochastic
dynamics
11: Ti-1 MM((x-11
12: end for
13: return
E
Experiments
Practical implementation. In this section we list practical implementations for training and inference of proposed Algorithm [ and Algorithm 2 assuming a variance-preservingg SDE for the flow-coordinates, see Eq. (55), because we know that this standard SDE is exactly solvable and related the forward SDE Cartesian space as stated in the main Theorem 3.1
The implementation showcase the examples for Gg = (SO(2) x TR+) (see the second paragraph in 3.1 for data living in IR2 and G1 (SO(3) R.) for E R from Appendix B.3
The flow-maps for Go and G can be computed by leveraging the bijection from Cartesian polar (r, 0) for Go and spherical r (r, 0, o) for G,, respectively. As stated in the main text and Appendix, we obtain
Mc. (x) = (V x2 y²
cos(0) MT. = ( (rsin(f)
(arctan( (2)
As mentioned in 13). the Lie algebra basis are A, and Ag 0
- yieldinggg quadratic
Casimir operator A3 + A3 = 0, such that the dynamics induced by the Casimir elements in line
Alg. 2 vanishes, i.e. v. 0. The dynamics induced by the divergences (line 7 in Alg. 2
V A,x = V x E Uz. 24 Σ = and V Agx = V (-#2,
returns
a
(-I2)
0 2 0. Therefore, the divergence dynamics returns the velocity component
Sa,
J0g 0A = 2x.
Va
2A,x
For G2 the bijection flow- and Cartesian coordinates is well-known
V2² + Mc, (x)= aarctan( Vz*to²
rr (() cos(o) (iin(()) sin( cos(())
MT)
arctan( ())
With A, I, Ag (cos ΦAy sin ΦA²) and As A, defined in 22 the quadratic Casimir elements A3 are left multiplied with the vector representation x, can distinguish each
24

---

group component as follows
A,x x'x Ix x Tz)
16x = (cos² (6) + sin (o) cos sin ΦA,A, cos AososinoAAAA,) -
(2,,,
A3x -
(21, ( 0)T,
defining the Casimir dynamics in line in Algorithm
The dynamics induced by the divergences are computed in the same manner as shown in the SO(2)
examples. Specifically, we obtain the (scalar) divergences
V A = x = 3
0
a
a
X V
A V
(-02, 21, 0)
(-22) +
+
0 = 0
dx
Ox2
A&x
i +
where the last divergence is point dependent.
In practice, it suffices to compute the quadratic Casimir elements directly using GPU-accelerated frameworks when these are point dependent as in Ag, pre-compute them should they be constant matrices. The divergences can be computed using automatic differentiation libraries from modern deep learning frameworks.
Experimental details. In this final section we present some further details regarding our experiment in Section 5 We provide the code to replicate our experiments in the Supplementary Information (SI). Following publication we will open-source source our code.
E.1
2D and 3D toy datasets
Table 3: Comparison of GSM and Fisher Score matching on 2D and 3D synthetic datasets. Best
results are in bold. When numbers are two close we consider them par.
Dataset
Group
W2
MoG (2D)
SO(2) R
0.34
MoG (2D)
((2)
0.15
Concentric Circles (2D)
SO(2) x R+ T(2)
0.19
Concentric Circles (2D)
0.17
Line (2D)
SO(2) R T(2)
0.33
Line (2D)
0.56
MoG (3D)
SO(2) R +
0.40
MoG (3D)
T(3)
0.44
Torus (3D)
SO(3) R+ T(3) 0.35
0.14
Torus (3D)
Möbius Strip (3D) Möbius Strip (3D)
SO(3) R+
0.06
T(3)
0.16
We perform quantitative evaluation using the Wasserstein-2 (W2) distance synthetic 2D and 3D datasets, comparing standard (Fisher) score matching (G = T(2), G T(3)) with our proposed approach based on Lie groups (G SO(2) x RT) and (G- SO(3) x R+). A strong bias in such experiments arises from the similarity between the prior and target distributions. The considered toy datasets often symmetric with respect to the origin in 2,3 in the standard Gaussian prior in Fisher score matching. The similarity of the prior distribution to the target one affects decisively the performance of the generating process.
To account for this, we report normalized W2 metric, dividing the W2 distance between samples and target by the W2 distance between target and the corresponding priors. We observe that generalized
25

---

score matching (GSM, ours) performs on par or better in most datasets, particularly where symmetry provides a clear inductive bias as indicated in Table the MoG datasets, standard (Fisher) score matching (G T(N)) outperforms the Lie group model (G = (O(N) x TRM), which is expected since no rotational symmetry present, while translation symmetry effectively helps locate the Gaussian modes. The performance gap becomes even more pronounced in 3D, where GSM shows stronger advantages. We hypothesize that in higher dimensions, memorizing the target distribution becomes more difficult, and models that incorporate symmetry more explicitly benefit increasingly from this inductive bias.
E.2 MNIST
We parametrize the noising process through the SDE
dr V 3(0dW.
(54)
where set the drift term to zero. Notice that this choice is consistent with 2d-rotation of function over the grid Xi.j given by (xx,,) = fú,j. denoting the value of the pixel of image f at the location j. We train a convolutional neural network (CNN) with three convolutional layers followed by fully connected layers that outputs a single value, being the score for the flow coordinate T, For the specific details of the implementation we refer to the code-base in the SI. In sampling, we apply a smoothing function to compensate interpolation artifacts due to rotations on discretized grid. We choose T 100 time-steps in training but only need T 10 time-steps during sampling.
E.2.1 BBDM
We implement the Brownian Bridge Diffusion Model (BBDM) (Li et al. 2023) and train it on the rotated MNIST dataset. The BBDM operates on the full pixel space lk 88 of the 28 MNIST digits and indicates continuous time stochastic process conditioned on the starting x x(0) and end point x() which are pinned together as paired data. In this case, we assume x(T) ~ p(xT) to be randomly augmented MNIST digit obtained from an original MNIST digit (0). During training, we sample an intermediate point x() ~ N (=))), (T)), Σx) where the mean function e4 ()) x()) is a linear interpolation between the endpoints (x),), x(T) and use the score-network to predict the original data point x() = sy (xy, t, (T) as opposed to the noise or difference paramterization proposed in the original BBDM paper. We noticed that predicting the original data point led to better sampling quality including the inductive bias that MNIST digits are represented as binary tokens. Furthermore, we observe that the sampling quality is also better when the prior image Fy is input as context into the score network, enforcing a stronger signal throughout the trajectory. As opposed to our model, we trained the BBDM on T 1000 diffusion timesteps using the sin-scheduler from BBDM.
To evaluate the quality of generated MNIST samples, we train convolutional neural network (CNN) classifier to predict digit labels. This classifier provides a reliable metric for assessing the reconstruction accuracy by comparing the predicted labels of original (unrotated) and generated (rotated) images. The architecture consists of two convolutional blocks, followed by fully connected layers to predict the 10 MNIST digit labels. All convolutional layers use 3x3 kernels with padding I, and max pooling uses 2x2 kernels. The model is trained using Adam optimizer (Ir=0.001), cross- entropy loss, batch size 64, for 10 epochs the standard MNIST training set (60,000 samples). The trained classifier achieves greater than 99% accuracy on the MNIST test set, providing a reliable metric for evaluating reconstruction quality.
To calculate the FID scores, we extract the embedding after the second convolutional block.
E.3
QM9 & CrossDockedd220
QM9. The conformer generation tasks is about learning a conditional probabilistic map x ~ Po MM), where x E RRV for a molecule with N atoms. We implement variant of EQGAT (Le) et al., 2022b) as neural network architecture where input features for the nodes consist of atom types and atomic coordinates, while edge features are encoded to indicate the existence of a single-, double, triple or aromatic bond based on the adjacency matrix. We use L = 5 message passing layers with Sdim = 128, Vaim 64 scalar and vector features, respectively. To predict the scores for each atom, concatenate the hidden scalar and vector embeddings s E R 128
. 3x3x61 into output
26

---

embedding o = R 1284 which is further processed by a 2-layer MLP with three output units. Notice that the predicted scores per atom are neither invariant nor equivariant since the scalar and vector features are transformed with an ML..
We choose the drift/ f with its scaling 33 and the diffusion coefficients y in such way that the forward SDE for the flow coordinates T in (8) has the expression
dr 1 -- -()()t + v B(tddW
(55)
where for clarity we have omitted the dependency between the flow coordinates and the original data in Cartesian coordinates, i.e. T(x), since the coordinate transformations with Lie algebra representation are described in B.3 The forward SDE in (55) is commonly known as variance-
preserving SDE (Song et al. 2020b). We use the cosine scheduler proposed by Dhariwal & Nichol (2021) and T = 100 diffusion timesteps.
CrossDocked. For this experiment we adopt again an SDE of the form (55) for the three SO(3) flow coordinates 01, (1,, PP and the three T(3) center of mass Cartesian flow coordinates. The SO(3) flow coordinates are always computed and applied in the ligand center of mass. In this way there is no ambiguity regarding the non-commutativity of SE(3), as rotation around the origin commute with translations of the system. We train variant of EQGAT as in the QM9 case, but now including also node and edge features of the protein pocket. Specifically, the adjacency matrix for the GNN is computed dynamically at each time step, according to the relative distance between ligand and protein. For this, we choose a cut-off of Å. We also use in this experiment a cosine scheduler and T = 100 diffusion timesteps. Since this learning problem is 6-dimensional, we aggregate the last layer's node embeddings from the ligand atoms into global representation through summation. This embedding is fed as input into 2-layer MLP to predict the six scores.
E.3.1 RSGM and BBDM CrossDocked
RSGM We utilized the framework of Riemannian Score- Based Generative Models (RSGM)
by (De Bortoli et al., 2022) to model rigid-body motions on G (SO(3) x T(3)), in sim- ilar fashion (Corso et al., 2023 Yim et al.) [223) by choosing a variance exploding SDE for the rotation dynamics and variance preserving SDE for the global translations. The ter- minal distribution for the rotation designed to converge to an isotropic Gaussian distribu- tion on SO(3) (Leach et al., 2022), while the terminal distribution for the translation compo- nent converges to an isotropic Gaussian in T To obtain the tractable scores for rotation and translation, we use the code by the authors from DiffDock and Sif3)-Diffusion for Protein Backbone Modeling so3 .py and httpts:s//gtthhtee data/ stl//ee___d_ fid__diffueeeereeeeeee py and make sure that the score outputs for rotation and translation are SO(3) equivariant using the same EQGAT model architecture. The prreeerierr)r))ngg) scheduler for the translation dynamics is chosen in similar fashion to our experiment using the cosine scheduler, while the (variance-cxploding increasing sequence in log10 space with amin = 0.001 and maa = 2.0 and T = 100 discretized diffusion steps (t) 10 for & (log (F 1ogio(0maz) mm
di fflfffuuuo www.fonssnooooooo httob//maaeerreeeeeee
scheduler for the rotation dynamics is implemented as linear
BBDM In similar fashion to the MNIST experiment, we train and evaluate a standard Euclidean diffusion model on CrossDocked2022 for rigid docking, We sample rotated ligand endpoint ar using the Riemannian Score- Based Generative Models (RSGM) scheduler, with the original ligand To. and sample intermediates as 3₁ = -2 + (1 m4 m.) + ave, where ~ N 1) and m, = 4, og = 2(m, - m3) using T = 100 diffusion steps. As the Euclidean bascline, we train an equivariant Fisher score network with 3.V degrees of freedom to predict the ground- truth pose 2o- In this setting intermediate perturb ligand coordinates I; do not resemble ligands due to the lincar interpolation and addition of Gaussian noise, while the learning task is to predict the ground-truth ligand coordinate, given the static protein pocket. The output prediction head in BBDM is 3N, compared to to GSM and RSGM which model 6 dimensions accounting for global rotation and translation.
To compare all modeling approaches with respect to the dynamics using the same network architecture, we perform 5 dockings per protein-ligand complex in the CrossDocked test dataset comprising 100 complexes and compute the mean RMSD between ground-truth coordinates and predicted coordinates.
27

---

F Lie group-induced flow matching modeling
We briefty summarize the formalism of flow matching. Given target distribution (x) and a vector field ue generating the distribution Pe(x), i.e. if satisfies px(x) [x)]+Po(X) where [ue]= is the push- forward map, the flow matching objective is defined as
Xa
(0) = E, xx (x,) ug (xx))².
(56)
(x,
Marginalizing over samples Xo ~ (x) we obtain the conditional flow matching objective
(x(x))
(C)
[U;,((x)) un (xx (57)
(x))
Now, under the assumptions for learning the generalized score through the objective 10 we have that (x)))
((T)u((T()) (t)), where T(O) (0) Then the solution of the SDE from Theorem 3.1
Figure 10: so(2) (green and (58)
T II
x(1)
(.)A x(0),
blue) vs. t(2) (orange) in-
duced flows.
is flow inducing the distribution Pt (x)). Thus, the vector field that generates the conditional probability path is obtained by differentiating the path above with respect to f, yielding
4. ,x(x(t/x(()) = dt
x(t) x((1)=
d
Σ @x(t) Ör.
ØT, 0t
Σ Axx(t) (((
=II(x(t) (2((T(())) +((
d(()((0)) oz(T((0)) (T(t) - ((((()))]
(59)
where we used the fact that
- + (
(60)
where 77 (0, 1). Thus, proportional to the fundamental vector field II(x) of the Lie algebra g of G. In figure 10 we illustrate the flow generated by formalism in the matching of (2). The orange path depicts the linear (in Euclidean metric) displacement given by the traditional flow matching, assuming G = T(2). In green and blue depicted the orbits trajectories resulting from generalized flow matching with = 50(2) x I Although the start and end points the same, the path is decomposed into transformations along the orbits of the two group factors. This is particularly useful when these correspond to meaningful degrees of freedom in the system. For example, when flowing between conformers the same molecule, the intermediate states produced by traditional flow matching are often unphysical, as they involve linear interpolation between the Cartesian coordinates of the atoms. However, generalized score matching, following the degrees of freedom given by bond and torsion angles as described in Section 3.1, would not only yield efficient learning but also produce chemically meaningful intermediate states, as the path is broken down into updates of chemically relevant degrees of freedom.
that the unique vector field that defines the flow is again
of SO(2) in comparison with the traditional flow
G Further Related Work
In the context of interpreting the latent space of diffusion models, Park al. 2023) explores the local structure of the latent space (trajectory) of diffusion models using Riemannian geometry. Similarly, Haas et al. (2024) propose a method uncover semantically meaningful directions in the semantic latent space (h-space) (Wang et al. 2023) of denoising diffusion models (DDMs) by PCA. Wang et al. (2023) propose method to learn disentangled and interpretable latent representations
diftusion models in unsupervised way. We note that the aforementioned works aim to extract meaningful latent factors in traditional DDMs, often restricting to ienereitteerretabble semantic features and focusing image generation.
28

---

H Further Outlook
In the context of generative chemistry, particularly for modeling interactions within protein pockets. our methods could be employed to decouple the intrinsic generation of ligands from the global transformations required fit the ligand into the pocket. This approach can also be extended beyond 3D coordinates, for instance, by working with higher-order representations, such as modeling electron density (Rackers et al. 2023).
Moreover, for more complex problems, is feasible that an optimal generation process can be achieved by combining different choices of G along the trajectory. In the context of ligand generation, we propose a time-dependent group action G, (T(3 (1-t) x R+) at the beginning of the diffusion process, when the point cloud is still far from forming recognizable conformer, we can leverage the properties of a true Gaussian prior. As the point cloud is gradually optimized to "resemble molecule". we progressively transition allows us to fine-tune chemically relevant properties, such as bonds and torsion angles, ensuring that the intermediate and final conformers are chemically valid and accurate. This will be the focus of our forthcoming work.
generalized score-gae ecore-guided process. This shift
potential limitation of our work is that it currently does not extend representations of finite groups. While finite groups also admit a rich representation theory, it remains an open question how adapt our framework to those settings. Another limitation lies in our assumption that X a vector space, whereas Lie groups can act on more general manifolds. Although the conditions discussed in Section 2.2 hold for arbitrary manifolds, our main theoretical results are restricted to actions on vector spaces. Extending the full analysis to curved spaces would be a compelling direction for future work, potentially enabling a general theory of diffusion on Riemannian manifolds via Lie group representations.
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Reviews:
[{"review_id": "qWSOWB61yw", "reviewer": "Reviewer_FJCA", "summary": "The paper introduces a new method for applying diffusion models to Lie group representations. It employs a score-based diffusion process with generalized score matching, enabling modeling on any Lie groups, including non-Abelian ones, which have been challenging to handle until now. The diffusion process stays in the original Euclidean space but is guided by Langevin dynamics through Lie algebras. Several examples of Lie algebras are provided, including those with real-world applications, such as generating molecular conformations. The experimental section demonstrates the performance of this model across tasks from different domains, including generating point clouds, predicting molecular conformations or docking poses, and rotating images.", "questions": "1. The rotated MNIST images in Figure 6 are blurred for the proposed model. Do you think it is because of the expressivity of the neural network used to generate these images or other problems with the diffusion process?\n2. For molecular conformation generation, why do you use Lie groups other than those introduced in Section 3.1, which were well-motivated in that section? For the docking example, it would be interesting to compare your approach with DiffDock for semi-flexible docking, which employs a similar strategy of moving diffusion to a different manifold of torsional rotations. For the QM9 example, could you justify using this particular group? If the space is spanned at each atom position, what is the role of the dilation transformation? Does it not cause problems with scaling bond lengths?\n3. You mentioned that you use the Casimir element to compensate for the deviation of the tangent vector from the orbit. In practice, do you sometimes observe errors that accumulate throughout the diffusion process?", "limitations": "The limitations are explained in Appendix G. The authors do not mention any broader impacts, nor do they justify why their research is considered to have no societal impacts in the checklist.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 4, "strengths_and_weaknesses": "Strengths:\n- The paper has solid theoretical foundations, with theorems proven in the Appendix (although I did not review the proofs thoroughly).\n- The core concept of the paper is presented in figures, which are helpful in understanding the proposed diffusion process.\n- The diffusion model on Lie group representations may have interesting applications in different domains, including molecular geometry prediction, as described in Section 3.1.\n- The presented approach is original and unique because generalized score matching has not yet been applied to Lie algebras. This development required working through the mathematical foundations of the model and providing Langevin dynamics for this new SDE.\n\nWeaknesses:\n- The experiments lack baselines to motivate the use of the proposed new diffusion model. For example, the generated distributions in Figure 5 should be both quantitatively and qualitatively compared to diffusion models using standard score matching. \n- Similarly, can docking poses be predicted as a simple regression task with a similar neural network architecture? It would be recommended to compare these results with other neural docking models, such as DiffDock.\n- The paper might be hard to follow for readers unfamiliar with Lie algebras. Many new terms are briefly defined in Section 2 without providing the intuition behind them. \n- The labeling of rows in Figure 6 may be confusing, especially for panel (b), where two labels correspond to two rows, and the middle label does not correspond to any row.\n- Although the method is very interesting, its significance in real-world applications is unclear. The authors mention modeling molecular geometry using dihedral angles, but then in the experimental section, they employ different, simpler Lie groups, such as rigid body movements.", "was_revised": true, "final_justification": "The Authors adequately addressed all my concerns. The paper is strong theoretically, and the experiments have now been significantly improved. I hope that all feedback from the discussion period will be integrated into the final version. Consequently, I increased my score to 5.", "rebuttal": "We thank the reviewer FJCA for their time reading and reviewing our paper and for their insightful questions and comments.\n\n---\n\n## Weaknesses\n\n- We performed several extra experiments to further increase the benchmarking evaluation of our strategy for three of our experiment setups. In summary:\n\n    - **Quantification of performance on the synthetic datasets** (for the **table of results please refer to the answer to reviewer d94q**): We conducted a quantitative evaluation using the Wasserstein-2 (W2) distance on synthetic 2D and 3D datasets, comparing standard (Fisher) score matching ($G = T(2)$) with our proposed approach based on Lie groups ($G = SO(2)\\times\\mathbb{R}^+$ and $SO(3)\\times\\mathbb{R}^+$). A strong bias in such experiments arises from the similarity between the prior and target distributions. To account for this, we report a normalized W2 metric, dividing the W2 distance between samples and target by the W2 distance between target and the corresponding priors. We observe that **GSM performs on par or better in most datasets**, particularly where symmetry provides a clear inductive bias. In the MoG datasets, standard score matching ($G = T(N)$) outperforms the Lie group model ($G = SO(N)\\times \\mathbb{R}^+$), which is expected since no rotational symmetry is present, while translation symmetry effectively helps locate the Gaussian modes. The performance gap becomes even more pronounced in 3D, where GSM shows stronger advantages. We hypothesize that in higher dimensions, memorizing the target distribution becomes more difficult, and models that incorporate symmetry more explicitly benefit increasingly from this inductive bias.\n\n   - We **performed a new experiment on CrossDocked2020** (see **Response to Reviewer DmbC below for details of the implementation**): We *evaluated a (equivariant) Brownian Bridge Diffusion Model (BBDM) on CrossDocked2020 and compared it to our Lie algebra-based method*. Both achieved similar RMSD (Lie: $2.91\\pm1.05$, BBDM: $2.92\\pm1.57$), but BBDM suffers from unphysical poses and fails to preserve global SO(3) rotation (despite being an equivariant network), yielding a rotation $\\text{MAE}(D(x_0, x_0), D(\\hat{x}_0, \\hat{x}_0)) = 0.43\\pm0.21$ (computed between the pairwise internal distance matrices of the two point clouds) versus $0.0$ for our method, indicating that our Lie algebra-induced diffusion offers a clear advantage over standard Diffusion models in this bridging problem. \n\n   - **Quantitative MNIST evaluation**:  We evaluated the FID as well as the classification accuracy on the generated images from both our model and the BBDM one. While our model is only slightly better on the FID, it is far superior in the classification accuracy (the classifier is a simple network trained on the original (unrotated) MNIST dataset, whose embeddings we also extract for the FID computation). This happens because BBDM sometimes generate bridges between different classes, as it cannot enforce a strict \"rigid\" rotation. We showed one instance of this occurrence in Figure 6 of the manuscript.  We report the evaluation in the following table (best results in bold):\n\n      | Model | Average Accuracy (\\$\\uparrow\\$) | Average FID (\\$\\downarrow\\$) |\n      | ----- | ------------------------------- | ---------------------------- |\n      | GSM   | **$\\boldsymbol{0.93 \\pm 0.04}$**           | **$\\boldsymbol{130.8 \\pm 22.0}$**       |\n      | BBDM  | \\$0.80 \\pm 0.10\\$               | \\$133.4 \\pm 19.0\\$           |\n\n-  We acknowledge that Lie group and representation theory may be unfamiliar to many in the machine learning community. In response to the reviewer’s suggestion, we added a table in the appendix summarizing key notation, definitions, and intuitive explanations. The **full version is available in the response to Reviewer DmbC**.\n\n- We understand the reviewer's point. We will just have one label per column to improve consistency of notation. \n\n- We understand the reviewer’s concern regarding the real-world significance of our method, and to this regard we wish to share with the reviewer our point of view on the matter and our current/future plans. When preparing this paper, we faced a strategic choice. One option was to focus on a single high-impact application, such as molecular design or protein docking, and demonstrate competitive (out)performance in that specific context. The other was to present a more methodological contribution: a general framework grounded in rigorous theory, validated across multiple data types to illustrate its versatility and correctness. We deliberately chose the second path. Our motivation is that the relevance of symmetry and Lie group structure in data extends far beyond individual applications such as molecular conformer generation or protein docking. By establishing the theoretical foundations and practical applicability of our method in a variety of controlled settings, we aim to provide the machine learning community with a robust and general-purpose tool. While specific applications often require extensive domain-specific tuning, these can obscure the contribution of the underlying method itself. Nonetheless, we agree that a focused, real-world application would further underscore the utility of our approach. We are currently working on a follow-up study dedicated to Lie group-aware molecular generation in protein binding pockets.\n\n---\n\n## Questions\n\n- The blurriness of the images is due to a interpolation layer we added to the sampling process due to the fact that the pixels do lie on a integer grid, but the diffusion process is in continuous angle space. Unless the angle is a multiple of 90 degrees, the transformed pixels will lie outside the grid, and we necessitate an interpolation step to assign values on the original grid. Performing this procedure for each step of the diffusion process (T=10) results in this smoothness effect. To address the blurriness in visualization, we initialize the rotation angle change $\\delta_\\theta = 0$ and iteratively update it during each diffusion sampling step by adding the score network output (properly scaled by the diffusion scheduler) and Brownian noise. At the final timestep, we apply the accumulated rotation change $\\delta_\\theta$ to the original (perturbed) MNIST image. The trajectory figures will be updated in the final manuscript.\n    \n- That's a very good question. Given the technical and methodological nature of the paper, the main purpose of the experiment is to show the validity and the effectiveness of the method. A full implementation of the method for semi-flexible docking would go beyond the scope of this work, but it is indeed the topic of a follow-up focused on a very similar use-case as discussed here. With the QM9 experiment, we aim to validate our formalism (as outlined in Sections 2 and 3) by demonstrating that it holds even in the absence of an exploitable symmetry structure. We compare two group actions, $G_1 = T(3N)$ (standard diffusion) and $G_2 = (\\mathbb{R} \\times \\mathrm{SO}(3))^N$ (generalized score matching), and show that both models solve the task equally well, with no significant differences in performance or energy. This supports our claim that, as long as the setup $(G, X, \\rho)$ satisfies the conditions in Section 2.2, it can be used to learn any unconstrained, unconditional distribution, regardless of whether it aligns with specific degrees of freedom. So we choose on purpose two groups with no data-related symmetry to demonstrate the universality and correctness of the approach, rather than claiming some particular advantage in this case. \n    \n- For the Casimir, we begin by making a small clarification. We did not add such term explicitely by hand. The term arises naturally in both backward and forward SDEs when imposing the condition that they are paired, namely, that they generate the same time-marginal probability distributions $p_t(x)$ at all times $t$. In the manuscript, we described the intuitive meaning of such term, namely, that it compensates for the orbit deviation due to the curvature, since it is not obvious from just looking at the formulas. To better demonstrate its effect, we have sampled from the trained model with and without the Casimir term in the backward process for the $SO(2)$ components in the 2d synthetic datasets. We then report the average radial distribution $r=\\sqrt{x_1^2 + x_2^2}$ of the sampled points and of the original dataset\n\n\n  | Dataset                 | With Casimir | Without Casimir | Original data |\n  | ----------------------- | ------------ | --------------- | ------------- |\n  | GMM (2D)                | 8.3          | 9.6             | 8.3           |\n  | Concentric Circles (2D) | 6.4          | 7.3             | 6.4           |\n  | Lines (2D)              | 2.0          | 2.5             | 2.0           |\n\n   As we can see from the table, with Casimir the model captures perfectly (at least the mean) of the true radial distribution. The sampling with no Casimir always leads to a systematic overshooting of the radial component (we recall that the \\$\\mathrm{SO}(2)\\$ Casimir is purely radial, as depicted in Figure 3 of the manuscript).\n\n---\n\nWe hope that our responses have adequately addressed the reviewer’s questions and concerns. We would be grateful if the reviewer could acknowledge the efforts we have made, particularly in generating new benchmarks, by considering to increase the overall score when updating their review. Should any issues remain unresolved, we would be more than happy to provide further clarification."}, {"review_id": "lt7tW33L1X", "reviewer": "Reviewer_DmbC", "summary": "The authors present a framework for constructing diffusion models on data spaces equipped with Lie group actions. By leveraging the infinitesimal action of Lie algebras, the authors define forward diffusion processes that respect the symmetries of the data space. These dynamics yield fundamental vector fields and associated differential operators. A generalized score matching framework is proposed minimizing the generalized Fisher divergence, which includes the standard score matching as a special case. Experiments were conducted on rotated MNIST, molecular conformer generation (QM9), and molecular docking (CrossDocked2020), demonstrating the framework's applicability to structured data with symmetries.", "questions": "Please refer to weaknesses.", "limitations": "Yes.", "rating": 5, "confidence": 1, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n- The proposed framework is mathematically grounded, extending denoising score matching to data with Lie group symmetry using infinitesimal generators.\n- A set of sufficient conditions for suitable Lie groups for score matching and Langevin dynamics are given.\n- The paper introduces a class of solvable SDEs that govern the Lie group diffusion via Euclidean coordinates.\n-Choosing a suitable group G, the learning process is significantly simplified, effectively reducing the dimensionality of the problem.\n- While related to previous works on diffusion models on Riemannian manifolds and latent spaces, this work articulates how the Lie algebra induces vector fields that guide the forward SDE. The link to fundamental vector fields and their role as a Jacobian-like structure is well presented.\n- Experiments were run on three different datasets.\n\n\nWeaknesses:\n\n- The paper is densely packed and hard to follow, specially for some one not familiar with the topic. While there are figures provided to help understand concepts used  in the proposed approach, they are not well explained in the main text and their captions are not very explanatory either. Having a table in the appendix with a summary of the notation would be helpful.\n\n- It would be helpful to clarify when Lie algebra-induced diffusion offers a clear advantage over standard diffusion models. A direct comparison with conventional Euclidean diffusion -- both with and without built-in equivariance-- would strengthen the motivation and help identify specific scenarios where the proposed approach is most beneficial.", "was_revised": true, "final_justification": "We would like to thank the authors for the detailed responses clarifying many points raised by all the reviewers. We believe that the table will help readers to follow the paper. We also encourage the authors to try to improve the figures presentation and their captions. Thanks also for the extra experiments, which significantly strengthen the paper.", "rebuttal": "We thank the reviewer DmbC for their time reading our manuscript and for their insightful comments, which we will address below.\n\n---\n\n- We understand that Lie group and representation theory is very technical and not common knowledge amount the machine learning community. We appreciate the reviewer's suggestion and we added a table in the appendix with the notation of the main concepts used in this work, together with their definition and a more intuitive explanation of the ideas behind them. The full table is presented here\n\n| **Symbol**         | **Name**                            | **Definition**                                                   | **Intuition**                                                                                                                                     |\n|--------------------|-------------------------------------|-------------------------------------------------------------------|--------------------------------------------------------------------------------------------------------------------------------------------------|\n| $G$                | Lie group                           |                                                                   | A continuous symmetry group, e.g., rotations ($\\mathrm{SO}(N)$), translations ($T(N)$), scalings($\\mathbb{R}^+$). Encodes the structure of transformations acting on the data. |\n| $e$                | Identity element of $G$             | $e \\cdot g  = g$ $\\forall g\\in G$                                   | The identity transformation leaving everything unchanged.                                                                                         |\n| $\\mathfrak{g}$     | Lie algebra of $G$                  | $T_eG$                                                            | Tangent space at the identity; represents infinitesimal group transformations.                                                                  |\n| $X$                | Data manifold                       |                                                                   | The space where the data lives, often $\\mathbb{R}^n$, but can be more general or even discrete (e.g., graph for molecules, grid for images, etc.). |\n| $\\rho_X$           | Group action                        | $\\rho_X: G \\times X \\to X$                                       | Specifies how each abstract group element $g \\in G$ transforms data points in $X$ via matrix multiplication.                                                  |\n| $G \\cdot x$      | Orbit of $x$ under $G$            | $ \\{ \\rho_X(g)(x),~ g\\in G \\}$                                 | The set of all points reachable from $x$ via group actions. Captures the “symmetry class” of $x$.                                                |\n| $G_x$            | Stabilizer subgroup at $x$        | $ \\{ g\\in G \\mid \\rho_X(g)(x) = x \\}$                        | Subgroup of $G$ that leaves $x$ unchanged. Describes residual symmetries at that point.                                                        |\n| $d\\rho_X$          | Infinitesimal action                | $d\\rho_X : \\mathfrak{g} \\to \\text{Vect}(X)$                       | Maps infinitesimal transformations to vector fields on $X$; captures how a tiny \"step\" in $G$ moves a point in $X$.                              |\n| $\\exp$             | Exponential map                     | $\\exp(A) = \\gamma_A(1)$, where $\\gamma_A \\colon \\mathbb R \\to G$ | Point on the geodesic path on $G$ determined by the direction and length given by the vector $A\\in \\mathfrak{g}$.                                |\n| $\\xi_A$            | Flow on $X$ induced by $\\rho_X$     | $\\rho_X\\left(\\exp(\\tau A)\\right)(x)$                           | Path on $X$ corresponding to a geodesic path on $G$ determined by $A$.                                                                           |\n| $\\Pi_A(x)$       | Fundamental vector field from $A$   | $[\\frac{d}{d\\tau} \\rho_X (\\exp(\\tau A)) (x)]_{\\tau=0}$       | A vector field on $X$ generated by a direction $A$ in the Lie algebra; describes how $x$ moves under an infinitesimal group transformation.       |\n\n\n- Additional experiments/benchmarks:\n\n    - We perform **additional experiments on CrossDocked2020** using a Brownian Bridge diffusion model (BBDM) similar to the MNIST experiment in Section 5. We sample a rotated ligand endpoint $x_T$ using the Riemannian Score-Based Generative Models (RSGM) scheduler, with the original ligand as $x_0$, and sample intermediates as $x_t= \\alpha_t x_0 + (1 - \\alpha_t) x_T + \\sigma_t \\epsilon$, where $\\epsilon \\sim N(0, I)$ and $m_t = t/T, ~~ \\sigma_t = 2(m_t - m_t^2)$ using $T=100$ diffusion steps. As the Euclidean baseline, we train an **equivariant Fisher score network** with $3N$ degrees of freedom to predict the ground-truth pose $\\hat{x}_0$. Similar to MNIST, intermediate BBDM samples display unphysical 3D poses due to linear interpolation and noising. We use the same network architecture as in the GSM and RSGM experiments to learn the correct $SO(3)$ rotation.\n    Unlike existing experiments (our method and RSGM), the Euclidean BBDM in this setting attempts to learn only global $SO(3)$ rotation, neglecting translation. Since the problem is implicitly 3-dimensional but the equivariant score network predicts all $3N$ ligand atom coordinates, final samples with implausible coordinate trajectories tend to have higher energies due to unphysical poses including bond stretching, non-planar aromatic rings, and deformed rings.\n    In terms of mean/std RMSD on the CrossDocked2020 testset, our method (Lie algebra: $2.91\\pm 1.05$) is comparable with BBDM ($2.92\\pm1.57$). However, since BBDM models all $3N$ atomic coordinates, the overall dynamics does not follow global SO(3) rotation, achieving $\\text{MAE}(D(x_0, x_0), D(\\hat{x}_0, \\hat{x}_0)) = 0.43\\pm0.21$, while RSGM and our method achieve $0.0$ by design. This indicates that Lie algebra induced diffusion offers a clear advantage over standard Diffusion models in that particular bridging problem. We will include the new experiment in the final version of the manuscript.\n\n    - **Synthetic datasets** (please **refer to the response to reviewer d94q for the full table of results**): We ran a quantitative evaluation using the Wasserstein-2 (W2) distance on synthetic 2D and 3D datasets, comparing standard (Fisher) score matching ($G = T(2)$) with our proposed method based on Lie groups ($G = \\mathrm{SO}(2)\\times\\mathbb{R}^+$ and $\\mathrm{SO}(3)\\times\\mathbb{R}^+$). One challenge in these experiments is that the similarity between the prior and target distributions can affect the results. To reduce this bias, we report a normalized W2 metric, which divides the W2 distance between the samples and the target by the W2 distance between the prior and the target. We find that GSM performs as well or better in most datasets, especially in cases where symmetry gives a useful inductive bias. In the MoG datasets, standard score matching ($G = T(N)$) performs better than the Lie group model ($G = \\mathrm{SO}(N)\\times \\mathbb{R}^+$), which makes sense since these data don’t have rotational symmetry, and translation symmetry helps the model find the centers of the Gaussian modes. The difference becomes even clearer in 3D, where GSM shows stronger performance. We believe that in higher dimensions, it is harder for a model to memorize the distribution, and the benefits of using symmetry-aware biases become more important.\n\n     - **Quantitative MNIST evaluation** (**table with numerical results can be found in the response to Reviewer d94q**):  We evaluated the FID as well as the classification accuracy on the generated images from both our model and the BBDM one. While our model is only slightly better on the FID, it is **far superior in the classification accuracy (93% vs 80%)** (the classifier is a simple network trained on the original (unrotated) MNIST dataset, whose embeddings we also extract for the FID computation). This happens because BBDM sometimes generate bridges between different classes, as it cannot enforce a strict \"rigid\" rotation. We showed one instance of this occurrance in Figure 6 of the manuscript. \n\n---\n\nWe hope our responses have effectively addressed the reviewer’s questions and concerns. We would sincerely appreciate it if the reviewer could take into account the additional efforts made, particularly in generating new benchmarks, by updating the overall score in their review. If any issues remain unclear or need further discussion, we would be happy to provide additional clarification."}, {"review_id": "ksJ9v17Nty", "reviewer": "Reviewer_d94q", "summary": "The authors present a new, more general way of presenting score matching, using representations of Lie groups (instead of diffusing on them directly), which enables them to retrieve usual score matching and generalise it to datasets with more structure. They demonstrate the effectiveness of their method on a variety of datasets.", "questions": "1. Are there any other datasets that could be using higher dimensional groups, and have you tried them out? It seems to me that in low dimensions, quite a lot of methods can work naturally, but, of course, it might be hard to find a good example. Perhaps some material sciences datasets, with space groups? I understand as well it can be out of the expertise of the authors. In general, I would also like to see more evidence that the method outperforms others that do not have such a geometric inductive bias. (Again, please point it out if I have missed it in the existing paper.)\n2. Are there any other low-dimensional groups you can try out, even on synthetic data? Other groups that are perhaps a bit more challenging than $\\mathrm{SO}(3)$ or $\\mathrm{SE}(3)$? (Feel free to reply to 1 and 2 together.)\n3. How \"difficult\" to satisfy are the conditions that you pose on $X$ and $\\Pi$? That is to say, are they high requirements in terms of regularity of $X$ and $\\Pi$?", "limitations": "I am mostly concerned about the scalability of the method in higher dimensions.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths:\n1. The paper is very well-presented. Neat, great-quality plots are available for all visualisations, and rather thorough introductions to Lie groups are given (as much as can fit in a page or so). \n2. The goal is very clear, and the paper is well written in general. Theoretically, the paper is sound, and the results are nicely presented.\n3. The experiments demonstrate the quite clearly.\n\nWeaknesses:\n1. Section 3 should be restructred slighlty, wherein I would include a subsection containing the theory, instead of jumping into it straight ahead (so to have a 3.1 and 3.2).\n2. The experiments are good, but sometimes lack easily understandable metrics, or they are sometimes not put forth enough. (And they should be, as the numbers are quite good.) But MNIST could use an FID evaluation, for instance. The synthetic experiments would benefit from numbers as well (perhaps, W2 distance?). For QM9 conformers, the standard metrics are reported, but none really exhibit the superiority of the method, I believe, at least in this scenario (which is fine). \n3. More experiments on higher dimension groups could be useful to verify that the method scales.\n\nThe paper is good as such, I believe, but I think a few improvements to its experimental section can make it stronger. Although, perhaps there are no useful examples of higher dimensional Lie groups?", "was_revised": true, "final_justification": "The authors clearly demonstrated that their method works on a variety of settings with a rigorous and thorough methodology. The thoroughness and rigour of the work is not to be doubted.", "rebuttal": "We thank the reviewer d94q for their time reading and reviewing our manuscript for their and insightful questions and comments. We are particularly pleased that the reviewer found the goal of the paper clear, theoretically sound and that the experiments demonstrate our claims. \n\n----\n\n## Weaknesses\n\n- The suggestion of the reviewer regarding the structure of Section 3 makes absolutely sense and we will implement it in the final version of the manuscript. \n\n- Additional experiments and quantitative benchmarks\n    - We include a **quantitative evaluation using the Wasserstein-2 (W2) distance on the synthetic 2D/3D datasets comparing standard (Fisher) score matching ($G=T(N)$) to our proposed approach** with Lie group $G=SO(N)\\times\\mathbb{R}^+$.\nOne strong bias of these distributions is that all the datasets considered are in fact symmetric with respect to the origin in $\\mathbb{R}^{2,3}$, as is the standard Gaussian which is the prior for the Fisher score matching models. The similarity of the prior distribution to the target one affects decisively the performance of the generating process. Indeed, both methods produce visually indistinguishable samples across all manifolds tested, as they only differ to the relative weighting given to the different modes (in the updated manuscript we will provide images of all the generated distributions). \nTo try to at least partially mitigate this effect, we report a normalized W2 metric, obtained by dividing the W2 distance for the sampled points with the W2 distance of the priors.\nWe noticed that **GSM performs on par or better in most datasets,** especially the ones where there is a clear inductive bias given by symmetry.  In the MoG datasets standard score matching (GSM with $G=T(N)$) does perform better than the group $G=SO(N)\\times \\mathbb{R}^+$, (which is to be expected since there is no rotation symmetry to be exploited and translation symmetry is a good inductive bias since it helps identify the center of the various Gaussian modes.\nWe also notice that outperformance of GSM is stronger in 3d than in 2d. We postulate that as the dimension grows, the model has a harder time to just \"memorize\" the distribution, and the symmetry-awareness advantage also increases. \n\n   **Table**: Comparison of GSM ($G=SO(N)\\times\\mathbb{R}^+$) and standard Fisher Score matching ($G=T(N)$) on 2D and 3D      synthetic datasets. Best results are in bold. When numbers are too close, we consider them on par.\n\n    | **Dataset**                | **Group**                          | **W2**                |\n    |----------------------------|------------------------------------|------------------------|\n   | MoG (2D)                   | $SO(2)\\times\\mathbb{R}^+$ (GSM)         | 0.34                   |\n   | MoG (2D)                   | $T(2)$ (standard)                             | $\\boldsymbol{0.15}$    |\n   | Concentric Circles (2D)    | $SO(2)\\times\\mathbb{R}^+$         | $\\boldsymbol{0.19}$    |\n  | Concentric Circles (2D)    | $T(2)$                              | $\\boldsymbol{0.17}$    |\n   | Line (2D)                  | $SO(2)\\times\\mathbb{R}^+$         | $\\boldsymbol{0.33}$    |\n   | Line (2D)                  | $T(2)$                              | 0.56                   |\n   | MoG (3D)                   | $SO(3)\\times\\mathbb{R}^+$         | $0.40$    |\n   | MoG (3D)                   | $T(3)$                              | $\\boldsymbol{0.24}$    |\n   | Torus (3D)                 | $SO(3)\\times\\mathbb{R}^+$         | $\\mathbf{0.14}$        |\n  | Torus (3D)                 | $T(3)$                              | 0.35                   |\n   | Möbius Strip (3D)          | $SO(3)\\times\\mathbb{R}^+$         | $\\boldsymbol{0.06}$    |\n  | Möbius Strip (3D)          | $T(3)$                              | 0.16                   |\n\n- **We performed additional experiments on CrossDocked2020 comparing our Lie-algebra induced generalized score matching against standard equivariant Euclidean diffusion (BBDM)** using identical network architectures (please **see Reviewer 2 DmbC response for implementation details**). While RMSD metrics are comparable, GSM generates 3D poses that faithfully preserve $SE(3)$ transformations, whereas **BBDM produces implausible final ligand poses** (with deformed geometries, e.g. non-planar aromatic rings, or wrongly streched bonds) that furthermore deviate from proper global rotations. In particular, to measure this effect we computed the mean absolute error on the distance matrices, i,e., $\\text{MAE}(D(x_0, x_0), D(\\hat{x}_0, \\hat{x}_0)) = 0.43 \\pm 0.21$, while RSGM and our method achieve $0.0$ by design. Thus, our approach offers a clear advantage over other strategies, as it keeps the RMSD of unconstrained diffusion while keeping the perfect structure due to the symmetry inductive bias, and outperforming Lie group diffusion. We will include the new experiment in the final version of the manuscript. Also note that the group is $G=SE(3)$ with $\\dim G = 6$ (so it is of high dimensionality in a way), and the representation on which it acts has dimensionality $3N$ where $N$ is the number of atoms in the ligand, also of very high dimension, showing scalability of the method.\n\n- We also perform **additional benchmarking in the MNIST experiment,** as the reviewer suggested. In particular we evaluted the FID as well as the classification accuracy on the generated images from both our model and the BBDM one. While **our model is only slightly better on the FID, it is far superior in the classification accuracy** (the classifier is a simple network trained on the original (unrotated) MNIST dataset). This happens because BBDM sometimes generate bridges between different classes, as it cannot enforce a strict \"rigid\" rotation. We showed one instance of this occurrance in Figure 6 of the manuscript. \n\n    | Model | Average Accuracy (\\$\\uparrow\\$) | Average FID (\\$\\downarrow\\$) |\n    | ----- | ------------------------------- | ---------------------------- |\n    | GSM   | $\\boldsymbol{0.93 \\pm 0.04}$           | $\\boldsymbol{130.8 \\pm 22.0}$       |\n    | BBDM  | \\$0.80 \\pm 0.10\\$               | \\$133.4 \\pm 19.0\\$           |\n\n---\n\n## Questions\n\n- In the manuscript we already presented a **synthetic dataset in 4D with group $G=SO(4)\\times \\mathbb{R}^+$, showing that the method straightforwardly applies to higher dimensional groups**. The results can be found in Figure 5h of the manuscript. In appendix A.6 we provided the necessary mathematical derivation for applying our method to $SO(N)$. We did search in the short time we had at our disposal for the rebuttal if there is some easy to implement dataset from material science, but unfortunately we did not find a standard one on which to run our method (material science does not lie unfortunately within our area of competence, and it seems it would require some further knowledge about the data and experience to find \"the right questions to ask\"). \n\n- Unfortunately, most of the continuous groups actions relevant for common data type are (product of) rotations ($SO(N)$), translations $T(N)$, dilations $\\mathbb{R}^+$. These are also the groups that are tackled in virtually all the papers on Lie group diffusions. Other groups like the unitary groups $U(N)$ are relevant for quantum mechanics applications, but they are complex groups and the diffusion process needs to take place on a complex manifold, which has not been tackled yet systematically in the literature. This is however a very interesting direction for future work, but would be out of scope for the current manuscript. \n\n- We thank the reviewer for raising this important point. While the formal conditions on the space $X$ and the fundamental fields $\\Pi$ may initially appear technical, they are in fact natural and routinely satisfied in common settings involving Lie group actions. We explicitly verified these conditions in every experiment presented in the paper, including those involving $SO(2), SO(3), SO(4), T(N)$, and $SE(3)$.\n\n   The results of the conditions can be formulated in a practical series of steps that the practitioner must follow/check when setting up the theoretical framework for learning on a new use case:\n\n     - **Identify the space $X$ where diffusion is performed.** This is not necessarily the space  where the data resides, but rather the space where the network outputs. For instance: $X$ has dimension 1 for the MNIST experiment, while dim data-space $=28 \\times 28$. (Pre-condition)\n     - **Dimension matching of the group action:** Ensure that the dimension of the image of the group action satisfies $\\dim \\rho_X(G) \\geq \\dim X$. This step is often straightforward, as it involves computing the kernel of the group action (i.e., the group elements whose action is the identity element on $X$) and ensuring the dimension of its complement matches or exceeds $\\dim X$. (Condition 1)\n     - **Group coverage of the space:** Verify that the group \"covers\" the entire space . For example, $T(N)$ generates all of $\\mathbb{R}^N$, and $SO(2)$ generates all rotations of an image. This property is usually so obvious that it doesn’t require explicit mention. (Conditions 1 and 2)\n     - **Lie algebra differential operator commutators:** Compute the action of the Lie algebra differential operators and check that the commutators vanish. While this calculation can sometimes be lengthy, it follows systematically from the representation of the group action. (Condition 3)\n\n     We hope that this clarifies the intuition behind the conditions and makes it clear that they are natural and mostly very easy to satisfy and check.\n\n---\n\nWe thank the reviewer for their thoughtful and constructive feedback, which has helped us improve the experimental section through the inclusion of new metrics. We hope that our additional work demonstrates our commitment to addressing the reviewer's concerns, and we would be grateful if this could be reflected in an updated review score."}, {"review_id": "QxoH6akBxy", "reviewer": "Reviewer_PLvS", "summary": "This paper proposes diffusion generative models on Lie groups using group representations based on the Generalized Score Matching. The idea is to separate the group actions (Lie groups) and the data space, and generate distributions on Lie groups that yield samples in the data space. The model works under some technical assumptions. Some numerical examples are provided for this method.", "questions": "1. The $\\rho_X$ defined in line 35 seems different from that in line 86.\n\n2. If $\\rho_X$ is defined as in line 35, then is the image of $d\\rho_X$ indeed $TX$ in line 56?\n\n3. In Figure 1, why is any point in $X$ generated from the origin $\\mathbf{0}$? Also, why is $g_1$ associated with $\\rho_X(g_0)(\\mathbf{0})$ instead of $\\rho_X(g_1)(\\mathbf{0})$?\n\n4. To make lines 86 and 87 consistent, it may be better to write $\\rho_X(e,x) = x,\\forall x\\in X$ in line 87.\n\n5. Line 90, $G_x$ should be $G_{\\mathbf{x}}$.\n\n6. Is there any empirical comparison to the work of Zhu et al. (2024) and Kong & Tao (2024) cited in line 287? (e.g., for Abelian groups.)\n\n7. Line 98, what is $\\xi_A$? I cannot find the definition of that before this line. Is $\\tau$ well-defined for any $(A,\\mathbf{x}_0, \\mathbf{x})$?\n\n8. Line 136, \"result\" $\\to$ \"results\".\n\n9. Line 160, \"Beyond this, the formalism remains fully applicable in the non-homogeneous case.\" I don't quite understand how the formalism remains fully applicable in the non-homogeneous case if $X$ is not homogeneous for $G$.\n\n10. What's the benefit or advantage of the proposed method compared to equivariant models? I understand that the model can be unconstrained on Lie groups, but it would also limit the diversity of generated samples, since you only create samples using group actions.\n\n11. What if $X$ is not Euclidean but a curved manifold, in addition to the curved Lie group?", "limitations": "I would suggest that the authors include a paragraph or a section of discussion about the limitations.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths: \n\n1. The method is explained in a good mathematical manner. Theories are provided for the method to work.\n2. The discussion of related work is thorough.\n3. Numerical experiments on different data types, including molecular conformer generation, are provided.\n\nWeaknesses: \n\n1. While the core idea appears to be interesting and straightforward, after reading the introduction, I found it difficult to clearly identify the paper’s contributions and novelty in comparison to prior work. I recommend revising the writing to improve clarity. \n2. Numerical experiments are not convincing or extensive enough to demonstrate the strength of the proposed methods. It can be improved by comparing it with a few recent methods cited in the related work (including equivariant diffusion models and some previous diffusion models on Lie groups), in particular, for the QM9 and CrossDocked2020 examples. Moreover, complexity of training and sampling should also be compared.", "was_revised": true, "final_justification": "I appreciate the efforts that the authors make to address my questions. The additional experiments are praiseworthy.", "rebuttal": "We thank the reviewer PLvS for their time for reading and reviewing our manuscript and for their insightful questions and comments.\n\n---\n\n## Weaknesses\n\n - *Paper's contribution and novelty.* **TL;DR.**: We propose a novel diffusion process governed by SDEs, which operates not directly on the Lie group itself, but on its space of representations. Specifically, the diffusion takes place in the (flat) space of linear operators (i.e., matrices) induced by the group action rather than on the (curved) group manifold. \n\n   Formally, given $x\\in X$ it can be written as $x=\\rho_X(g)(x_0)$ for some fixed $x_0\\in X$ and $g\\in G$ (in the manuscript we denoted $x_0=0$), where the above map is a matrix map, i.e., $x=\\rho_X(g_x)(x_0) = \\Omega(g) x_0$ where $\\Omega(g)$ is a $n\\times n$ matrix. For instance, for $G=SO(2)$, $g$ is the angle and $\\Omega(g)=[[\\cos g, \\sin g], [-\\sin g, \\cos g]]$ the known rotation matrix in 2d. Given this setup we can explain the three approaches:\n   - Lie group diffusion extracts $g$ from $x$ and performs diffusion on (the generally curved) group $G$;\n   - We propose to perform diffusion in the space of $\\Omega$'s for any group $G$. This matrix-valued diffusion process induces a stochastic flow on $X$ by applying the evolving matrices to points in $X$, $x_t \\rightarrow \\Omega_t(g) x_{t-1}$. In this way, the entire process remains within Euclidean space, as the representation elements are matrices acting on a vector space. \n    - Standard diffusion consists in taking $\\Omega$ to be the identity matrix (+ constant vector). This corresponds to the representation of the translation group $T(n)$, which makes it explicitly a subcase of our general formalism.\n\n   Compared to diffusion on the Lie group itself, our method is computationally simpler and more general. It avoids the geometric and optimization challenges of curved manifolds, does not require projections, and avoids the difficulty of sampling forward SDE trajectories for non-Abelian groups, where no closed-form solutions exists (but they do in our formalism!). We plan to clarify these points in the revised manuscript following the rebuttal.  Please let us know if the above explanation is helpful for the reviewer to understand our work and its novelty in the context of standard Lie group diffusion.\n\n- *Further experiments and comparison*. We performed extra experiments to further increase the benchmarking evaluation of our strategy:\n    - **Synthetic datasets** (for **table of results please refer to the answer to reviewer d94q**): We performed a quantitative evaluation using the Wasserstein-2 (W2) distance on the synthetic 2D/3D datasets comparing standard (Fisher) score matching ($G=T(N)$) to our proposed approach with Lie group $G=SO(N)\\times\\mathbb{R}^+$.\n        One strong bias of such experiments is due to how close the prior distribution is to the target one and this affect the performance of the generative process. To mitigate this effect we report a normalized W2 metric, where we divide the W2 distance for the sampled points with the W2 distance of the priors. We noticed that **GSM performs on par or better in most datasets, especially the ones where there is a clear inductive bias given by symmetry.** In the MoG datasets the standard score matching (with $G=T(N)$) does perform better, which is to be expected since there is no rotation symmetry to be exploited and translation symmetry is a good inductive bias since it helps identify the center of the various Gaussian modes. We also notice that outperformance of GSM is stronger in 3d than in 2d. We postulate that as the dimension grows, the model has a harder time to just \"memorize\" the distribution, and the symmetry-awareness advantage also increases. \n\n  - **Further CrossDocked benchmarking** (**for implementation details please refer to the answer to reviewer d94q**):  We conducted additional experiments on CrossDocked2020 using a BBDM model, finding that while BBDM and our Lie algebra-based method achieve comparable RMSD (Lie: $2.91\\pm1.05$, BBDM: $2.92\\pm1.57$), BBDM produces unphysical poses due to linear interpolation in Euclidean space. **Unlike our method and RSGM, BBDM fails to preserve global $SO(3)$ rotational structure, resulting in a higher rotation $\\text{MAE}(D(x_0, x_0), D(\\hat{x}_0, \\hat{x}_0)) = 0.43 \\pm 0.21$ between the two distance matrices, compared to $0.0$ for both our method and RSGM.** These results demonstrate that our method combines the best of both worlds: it maintains the symmetry structure of the data without sacrificing performance, matching unconstrained models in accuracy while outperforming Lie group diffusion approaches.\n   - **Quantitative MNIST evaluation**(please **refer to the answer to reviewer d94q for table of results**):  We evaluated the FID as well as the classification accuracy on the generated images from both our model and the BBDM one. While **our model is only slightly better on the FID, it is far superior in the classification accuracy (93% vs 80%)** (the classifier is a simple network trained on the original (unrotated) MNIST dataset, whose embeddings we also extract for the FID computation). This happens because BBDM sometimes generate bridges between different classes, as it cannot enforce a strict \"rigid\" rotation. We showed one instance of this occurence in Figure 6 of the manuscript.\n\n---\n\n## Questions\n\n1. The reviewer is correct that our use of notation is a bit sloppy here. In line 35 it is the usual definition of a linear representation on a vector space. In line 86 we generalized this to be a general group action (thus not needing the structure of a vector space for $X$), but more importantly the map in 86 is the map in 35 applied at a specific point $x\\in X$, using the fact that $GL(X) : X\\rightarrow X$. To reconcile the two, these definitions relate via $\\rho_X(g, x) := \\rho_X(g)(x)$. We will revise the notation and clarify the text accordingly.\n2. This simply follows from the fact that $GL(X) : X\\rightarrow X$, and the differential of this has indeed $TX$ as the image. \n3. In Figure 1 we wish to emphasize how elements in $X$ are obtained through the representation of the group action. Here $0$ is not necessarily the classical origin $0=(0,0,0,...)$ of $\\mathbb{R}^n$, but it is any point in the image of the identity element $e\\in G$, $0=\\rho_X(e)$. $g_1$ and $g_0$ are obviously swapped in the bottom side of the figure, we thank the reviewer for catching that! \n4. We appreciate the suggestion of the reviewer to make the notation clearer!\n5. Thank you for catching that!\n6. We did not compare our results with the works of Zhu et al. and Kong \\& Tao. While it is fair to cite these papers as they also approach group-aware diffusion using the structure of the Lie algebra, these methods are quite new and not been yet applied to real-world scenarios like molecule generations.\n7. $\\xi_A$ is defined in line 93, and as Figure 2 depicts, it corresponds to the flow on $X$ induced by the map $\\rho_X$, given a flow (geodesic) on $G$ corresponding to a Lie algebra element $A$. When the conditions of section 2.2 are satisfied, $\\tau$ is well-defined. It can still be that there might be an ambiguity, for instance for $SO(2)$, $\\tau=\\theta$ is the angle parameter, which is $2\\pi$ periodic. In such cases, we establish an interval for allowed values. \n8. Thank you!\n9. If $X$ is not homogeneous for $G$, then $G$ cannot connect all points in $X$, and the orbits partition $X$. In such cases, our diffusion process will be restricted to the orbit of the prior initial condition, which is not the standard case where each (prior) point can be moved by the dynamics to any target point. However, the framework (score estimation, Langevin dynamics, etc.) still applies on each orbit separately.\n10. Equivariant models restrict the score function to a fixed representation of the $G$, limiting its expressivity. In contrast, our method does not enforce equivariance on the score function, but it leverages the group action to guide the generative dynamics while allowing the score to be freely learned in $\\mathbb{R}^n$. For instance, in $X = \\mathbb{R}^2$, a strictly $SO(2)$-equivariant model cannot capture angular variation, as equivariance dictates how the score must transform under rotations. Our approach overcomes this by modeling dynamics under a product group (e.g., $SO(2) \\times \\mathbb{R}^+$), allowing us to capture both angular and radial dependencies. Crucially, the score function remains unrestricted: the group governs the direction of the Langevin dynamics but does not constrain the form of the learned function.\n11. This is a very good question! We address it already in lines 283-284. In short, our results for Generalized Score Matching hold also for a generic curved manifold. Our Theorem 3.1 requires the flatness of $X$ since it is the ultimate goal of our work to derive a diffusion process following Lie group trajectories but all in Euclidean space. Nonetheless, it would be a very nice theoretical extension to extend our Theorem 3.1 to more generic spaces. \n\n---\n\n### Limitations\n\nIn Appendix G, we discussed several limitations of our approach. Another potential challenge can lie in the flexibility of our framework. While its flexibility is actually a strength, as it enables accurate inductive bias modeling of the data structure, it can also require more extensive parameter tuning, particularly when working with novel datasets or symmetry groups as compared to standard score matching methods. We will make a separate section about the limitations. \n\n---\n\nWe hope that our responses have adequately addressed the reviewer’s questions and concerns. We would be grateful if the reviewer could acknowledge the efforts we have made, particularly in generating new benchmarks, by considering to increase the overall score when updating their review. Should any issues remain unresolved, we would be more than happy to provide further clarification."}]

Author rebuttal:

Meta-review: The paper introduces a new framework for diffusion-based generative modeling on Lie group representation spaces rather than on the curved Lie group itself. The contribution is theoretically sound, with careful derivations using generalized score matching and paired SDEs, and it provides a general-purpose formalism that includes standard score matching as a special case. Reviewers agreed that the work is original, mathematically rigorous, and of potential broad impact across domains where symmetry plays a central role.

During the rebuttal, the authors strengthened the paper with additional experiments, including quantitative benchmarks on synthetic datasets, docking poses, and rotated MNIST with FID and classification metrics. These additions addressed concerns about missing baselines and clarified the practical advantages of the method, particularly in preserving symmetry structures and avoiding unphysical samples. Reviewers noted that the explanations of notation and conditions, along with a new summary table, substantially improved clarity.

Remaining concerns centered mostly on accessibility and exposition, which the authors have committed to improving in the final version through clearer figures and expanded explanations. Overall, the consensus among reviewers is that the paper is technically solid, original, and timely, and it provides a framework likely to influence future work on generative modeling with symmetries.

Given its strong theoretical contributions, thorough responses, and improved experimental validation, I recommend acceptance. The work stands out as both foundational and broadly relevant, and I suggest it be nominated for a spotlight presentation.
Decision: Accept (spotlight)