Published as a conference paper at ICLR 2026
Q UOTIENT-S PACE D IFFUSION M ODELS
arXiv:2604.21809v1 [cs.LG] 23 Apr 2026
Yixian Xu1∗, Yusong Wang2,5∗ , Shengjie Luo1 , Kaiyuan Gao3 , Tianyu He4 , Di He1†, Chang Liu5† 1 State Key Laboratory of General Artificial Intelligence, Peking University, Beijing, China 2 State Key Laboratory of Human-Machine Hybrid Augmented Intelligence, Institute of Artificial Intelligence and Robotics, Xi’an Jiaotong University 3 Huazhong University of Science and Technology, Wuhan, China 4 Microsoft Research Asia, Beijing, China 5 Zhongguancun Academy, Beijing, China
A BSTRACT Diffusion-based generative models have reformed generative AI, and have enabled new capabilities in the science domain, for example, generating 3D structures of molecules. Due to the intrinsic problem structure of certain tasks, there is often a symmetry in the system, which identifies objects that can be converted by a group action as equivalent, hence the target distribution is essentially defined on the quotient space with respect to the group. In this work, we establish a formal framework for diffusion modeling on a general quotient space, and apply it to molecular structure generation which follows the special Euclidean group SE(3) symmetry. The framework reduces the necessity of learning the component corresponding to the group action, hence simplifies learning difficulty over conventional groupequivariant diffusion models, and the sampler guarantees recovering the target distribution, while heuristic alignment strategies lack proper samplers. The arguments are empirically validated on structure generation for small molecules and proteins, indicating that the principled quotient-space diffusion model provides a new framework that outperforms previous symmetry treatments.
1
I NTRODUCTION
Diffusion models have emerged as the dominant approach for modeling distributions in highdimensional spaces. Building on their success in real-world domains such as images (Ho et al., 2020; Song et al., 2021), audios (Kong et al., 2021; Evans et al., 2024), and videos (Ho et al., 2022; Li et al., 2023), diffusion models are now increasingly adopted in scientific applications, ranging from fluid field solving (Bastek et al., 2025), electronic structure prediction (Kim et al., 2025), molecular structure generation (Xu et al., 2022; Abramson et al., 2024; Hassan et al., 2024; Geffner et al., 2025), and thermodynamic ensemble modeling (Zheng et al., 2024; Lewis et al., 2025). Compared with general tasks, scientific applications often exhibit inherent symmetry structures, wherein objects that can be related through specific transformations are regarded as equivalent. Consider molecular structure generation as a representative example. A molecular structure can be represented as a vector in R3N by concatenating the 3D coordinates of its N atoms. However, because the choice of coordinate system is arbitrary, vectors in R3N that differ only by a global 3D translation or rotation of all atoms correspond to the same underlying structure. Mathematically, such transformations typically form a Lie group — for example, the special Euclidean group SE(3) in the case of molecular structures, which formally characterizes the symmetry. The common treatment is putting the target distribution in the original space but assigning the same probability to equivalent objects, resulting in a distribution that is invariant under group action. This can be implemented by augmenting training data by applying randomly chosen group actions (Abramson et al., 2024), or using a group equivariant model (Xu et al., 2022; Hoogeboom et al., 2022b), which guarantees invariance if the starting prior distribution is invariant (Köhler et al., 2020). Nevertheless, we shall show that this treatment still has room to improve, as the neural network model, which is intended for updating the sample in each diffusion simulation step, still needs to ∗ †
Equal contribution. Correspondence to: Di He <[email protected]>, Chang Liu <[email protected]>.
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Table 1: Comparison among different training strategies in presence of a symmetry group. Learning difficulty is measured by whether the need to predict in the equivalent degrees of freedom (DoFs), induced by the group actions, is removed, and (if not) whether the variance on the equivalent DoFs is removed. Sampling compatibility means whether there is a sampler that exactly reproduces the target distribution. The denoising form of diffusion model Dθ is used to express the loss functions, where Ay (x) (Eq. (12)) represents aligning x towards y, and θ̄ denotes treating θ as constant (i.e., stop-gradient). The conclusions hold using either an equivariant architecture or a general architecture with data augmentation. See Sec. 3.4 for details. Reduction of learning difficulty Sampling Removal of Removal of variance compatibility equivalent DoFs on equivalent DoFs
Training strategy for Dθ
Optimal solution of Dθ
Conventional loss 2 E∥Dθ (xt , t) − x1 ∥
E[x1 |xt ]
✗
✗
✓
GeoDiff alignment loss 2 E∥Dθ (xt , t) − Axt (x1 )∥
E[Axt (x1 )|xt ]
✗
✓
✗
AF3 alignment loss 2 E Dθ (xt , t)−ADθ̄ (xt ,t)(x1 )
g · E[Axt (x1 )|xt ] for arbitrary g ∈ G
✓
✓
✗
✓
✓
✓
E[Pxt (x1 )|xt ] + vV quotient-space diffusion loss 2 E∥Pxt (Dθ (xt , t) − x1 )∥ for arbitrary vV ∈ Ker(Pxt )
learn a specific movement within the equivalence class (e.g., rotating a molecular structure), which is unnecessary as any such a movement does not update the intrinsic system state (e.g., the shape of a molecular structure) hence is acceptable. In hope to remove this redundancy, there are a few heuristic treatments using alignment, i.e., adjusting the prediction target within its equivalence class according to a reference to remove these equivalent degrees of freedom (Xu et al., 2022; Abramson et al., 2024). But we find that the corresponding sampling process becomes incompatible with such training strategies, even with heuristic fix attempts (Wohlwend et al., 2025). In this work, we develop a principled approach to building a diffusion model considering the intrinsic symmetry of the system. In particular, we leverage the concept of quotient space, in which a set of equivalent objects (equivalence class) are treated as one element. It is the formal mathematical construction that reflects the intrinsic variability of the system. We first derive the diffusion process on a general quotient space based on the correspondence between the Wiener processes on the two spaces. Considering that the quotient space is generally not Euclidean, hence it is hard to directly carry out a simulation on it, we further leverage the mathematical construction of horizontal lift to induce a diffusion process back in the original space that can easily implement the quotient-space diffusion process. The resulting process effectively amounts to projecting the update vector in the original diffusion process onto the subspace that does not induce a movement within the equivalence class (e.g., rotation). We show that this process guarantees producing the correct target distribution, meanwhile reduces learning difficulty by removing the necessity to learn a specific movement within an equivalence class. A visualization example in the 2-dimensional plane with SO(2) symmetry is shown in Fig. 1. In this example, the lifted process only has radial movements (Fig. 1(Left)) as the quotient space R2 /SO(2) is isomorphic to the half real line and recovers the correct target distribution as conventional equivariant diffusion models (Fig. 1(Middle, Right)). A conceptual comparison with existing methods is shown in Table 1. The quotient-space diffusion admits either an equivariant model or a general model with data augmentation. As a representative application, we deduce the specific training and sampling algorithms in the R3N /SE(3) scenario for molecular structure generation, which relaxes the model from learning a translation and rotation movement, while the sampling process keeps the structure with constant position and orientation. We study the empirical performance of quotient-space diffusion models on small molecule structure generation and protein backbone design tasks. The results show that our methods can consistently improve the generation performance in these applications over conventional equivariant diffusion models and using alignment strategies. Our method achieves 9%23% relative improvements of ET-Flow (Hassan et al., 2024) on GEOM-QM9 and GEOM-DRUGS datasets, surpassing previous heuristic alignment methods. For the protein structure generation task, our method surpasses the state-of-the-art Proteı́na model (Geffner et al., 2025) with the same parameter scale (60M) in a large margin and also outperforms the much larger model (200M) on most key distributional metrics. 2
Published as a conference paper at ICLR 2026
Figure 1: A motivative illustration highlighting the behavior of the quotient-space diffusion model against the conventional equivariant diffusion model for modeling a distribution on R2 (as M) with SO(2) (as G) symmetry, whose density is represented by the gray scale. (Left) SDE sampling trajectories by the two diffusion models. The same color indicates the same starting point (the round dot). The quotient-space diffusion model moves each sample only along the ray from the origin, which can be understood as only traversing the quotient space R2 /SO(2), i.e., traversing over origin-centered concentric circles, without moving within an equivalence class, i.e., an origincentered circle. The conventional equivariant diffusion model moves each sample over the whole R2 space, requiring subtler simulation treatment. (Middle) Samples generated by the conventional equivariant diffusion model. (Right) Samples generated by the quotient-space diffusion model, which also recovers the data distribution as guaranteed. Moreover, the quotient-space diffusion simplifies learning difficulty: the neural network does not need to learn anything in the output subspace that is responsible for intra-equivalence-class movement (Eq. (11)).
2
BACKGROUND
2.1
D IFFUSION - BASED G ENERATIVE M ODELS ON E UCLIDEAN S PACE
The main idea of diffusion models is to construct a step-by-step transformation from a simple prior distribution to a complex target distribution. In this paper, we follow the Stochastic Interpolant framework (Albergo et al., 2023), which unifies diffusion models and flow matching models (Lipman et al., 2023; Liu et al., 2023). Let ptarget (x) be the target distribution. The following linear interpolation is constructed: xt = αt x0 + βt x1 + γt ϵ,
(x0 , x1 ) ∼ pjoint ,
ϵ ∼ N (0, I),
t ∈ [0, 1]
(1)
where pjoint is a pre-defined joint distribution of (x0 , x1 ) with marginals x0 ∼ pprior and x1 ∼ ptarget . The coefficients αt , βt , γt satisfy the boundary conditions α0 = 1, β0 = 0, γ0 = 0, and α1 = 0, β1 = 1, γ1 = 0. Under these conditions, the following ordinary differential equation (ODE) can transform pprior to ptarget (Albergo et al., 2023, Cor. 2.18): dxt = v(xt , t) dt,
where
v(xt , t) := E[α̇t x0 + β̇t x1 + γ̇t ϵ | xt ],
(2)
and the dot decoration denotes time derivative. The velocity vector field v(xt , t) is typically trained with the objective: L(θ) := Ep(t) w(t)Epjoint (x0 ,x1 )p(ϵ) ∥vθ (xt , t)−(α̇t x0 +β̇t x1 +γ̇t ϵ)∥2 , where the prime denotes the time derivative, and p(t) and w(t) control the sampling distribution and weighting over time. There is also a stochastic process for sample generation, given by : p dxt = (v(xt , t) + ηt s(xt , t)) dt + 2ηt dwt , where s(xt , t) := ∇xt log pt (xt ) (3) is called the score function, and ηt ≥ 0 is a non-negative smooth function (Albergo et al., 2023, Cor. 2.10). In the special case where pprior = N (0, I) (the one-sided stochastic interpolant (Albergo et al., 2023, Def. 3.4)), contributions of x0 and ϵ can be combined as xt = α̂t ϵ + βt x1 , where α̂t = p t v(xt ,t) αt2 + γt2 , and the score function can be expressed by the velocity field: s(xt , t) = β̇α̂t x(α̂t˙−β . β −α̂ β̇ ) t
t
t
t
t
A convenient variant to formulate the learning task is to define the vθ (xt , t) model with a neural network Dθ (xt , t) which reformulates the objective: α̂˙ t xt − (α̂˙ t βt − α̂t β̇t )Dθ (xt , t) , (4) vθ (xt , t) := α̂t (α̂˙ t βt − α̂t β̇t )2 L(θ) := Ep(t) w(t) Ep(x1 ,xt ) ∥Dθ (xt , t) − x1 ∥2 , (5) α̂t2 3
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where p(x1 , xt ) is derived from Eq. (1) by integrating out x0 and ϵ. This objective conveys the intuition of recovering the clean-data sample x1 from a noisy sample xt , hence Dθ (xt , t) is called a denoising model and suits prevalent architectures. We adopt this form of a diffusion model below. 2.2
F ROM E UCLIDEAN S PACE TO Q UOTIENT M ANIFOLD
Tasks in scientific domains often involve inherent symmetry, where objects related by certain transformations are considered equivalent. A formal and inclusive description of symmetry in a system requires both the geometry of the configuration space and the algebraic structure of the transformations, which leads to the concepts of manifolds and Lie groups. Manifold and Lie groups. A (smooth) manifold is a geometric object that generalizes the Euclidean space to allow spatial heterogeneity. Typically, a manifold is endowed with a Riemannian metric, i.e., an inner product in each tangent space, which leads to common concepts like curve length, distance, measure, gradient, Laplacian, and Wiener process on the manifold (Appx. B.1). Symmetries are formally represented by transformations that connect equivalent (i.e., symmetric) objects, which constitute a group. A continuously-parameterized group that is also a manifold is called a Lie group. We consider the general case where the configuration space of the system is an M -dimensional Riemannian manifold M. The symmetry of the system is represented by a G-dimensional Lie group G acting on M. A distribution p on M is said G-invariant if p(g · x) = p(x), ∀g ∈ G, x ∈ M. This invariance implies that all equivalent points {g · x | g ∈ G}, collectively called an equivalence class, are assigned with the same probability. Quotient space. The symmetry group defines an equivalent relation in M, i.e., x1 and x2 are equivalent, if there exists a group action g ∈ G such that g · x1 = x2 , which is indeed an equivalent relation due to properties of a group. The quotient space Q := M/G treats equivalent objects under the action of G as one element, hence reflects the intrinsic variability of the system. There is a natural mapping called the projection connecting the two spaces: π(x) := {g · x | g ∈ G}. Under appropriate conditions, the quotient space is a smooth manifold with dimension M − G (Appx. C). However, defining a diffusion process on this space is non-trivial, necessitating the extension of “velocity” and Wiener process from Euclidean space to the manifold. Tangent vector. On a manifold M, the velocity of a process at a certain point x is represented as a tangent vector at x, intuitively representing an infinitesimal movement. All tangent vectors at x constitute a linear space Tx M called the tangent space at x. Since a manifold is typically curved, tangent spaces at different points are regarded as different linear spaces, but with a transformation on the manifold, e.g., a group action Lg : x 7→ g · x, an associated mapping (Lg )∗x : Tx M → Tg·x M between the tangent spaces can be defined by linking infinitesimal movements around x and around g · x by Lg (Appx. B). With this construction, we can define a G-equivariant vector field on M if it is unchanged under the group action: (Lg )∗x (vx ) = vg·x . The projection mapping π naturally induces a projection of tangent vectors onto the quotient space by π∗x : Tx M → Tπ(x) Q. Wiener process on a manifold. In Euclidean space, the Wiener process is generated by the Laplacian operator 21 ∆. The Laplace-Beltrami operator, defined from a Riemannian metric, serves as a counterpart on a manifold, and defines the Wiener process to the manifold. Under a symmetry group G, we require a meaningful stochastic process on the manifold M as G-invariant, meaning that its marginal distribution is G-invariant at any time step. See Appx. B for details.
3
M ETHODS
As the quotient space represents the “essential states” of a system with symmetry, a principled diffusion model for the system is expected to be built on it. In this section, we unroll the development of the quotient-space diffusion model by deriving the projected diffusion process onto the quotient space, then lift it back into the total space (i.e., the original space) for convenient implementation. We then derive the specialization in the R3N /SE(3) case for molecular structure generation, followed by training and sampling algorithms. Finally we highlight the merit of the quotient-space diffusion in reducing training difficulty and sampler validity with a comparative analysis with existing symmetry treatments.
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3.1
D IFFUSION P ROCESS ON A G ENERAL Q UOTIENT S PACE
When viewed in the total space, due to the intrinsic symmetry, the distribution should assign the same probability to elements in each equivalence class, formally regarded as a G-invariant distribution. To guarantee this, a common approach is to start from a simple G-invariant distribution pprior and let it undergo a G-equivariant generation process, which translates to the G-equivariance of the vector field bt (the drift term) for a diffusion process (the Wiener process defined under the Riemannian metric is G-invariant since the group G acts on the Riemannian manifold M isometrically). This guarantees that each intermediate distribution along the generation process is also G-invariant (Köhler et al., 2020; Hoogeboom et al., 2022a). As G-invariant distributions vary only across equivalence classes, they correspond to distributions on the quotient space Q := M/G, and the generation process can also be exactly represented on the quotient space, as explicitly given by the following theorem. Theorem 1. Assume {xt }t∈[0,T ] is a diffusion process on M, specified by the following SDE: dxt = bt (xt ) dt + σt dwt ,
x0 ∼ pprior ,
(6)
where bt is a G-equivariant time-dependent vector field on M, wt is the Wiener process on M, and pprior is a G-invariant distribution. Then the projected process {yt := π(xt )}t∈[0,T ] onto the quotient space Q := M/G is the solution to the following SDE: σt2 dyt = (π∗ bt )(yt ) − h(yt ) dt + σt dωt , y0 ∼ π# pprior , (7) 2 where π∗ bt is the projected vector field of bt onto Q induced by π, h(y) is the mean curvature vector field of Q reflecting the geometry of Q, ωt is the Wiener process on Q, and π# pprior is the pushed-forward distribution of pprior (i.e., y0 = π(x0 ) where x0 ∼ pprior ). See Appx. D.1 for formal definitions of the concepts and the proof. Thm. 1 shows that the projected process on the quotient space is indeed a diffusion process, and running the original process followed by projection gives the same process by running Eq. (7) on the quotient space. The quotient-space process Eq. (7) is expressed using the projected vector field, the Wiener process on Q, and perhaps unexpectedly, an additional vector field h reflecting the curvature of Q. As the quotient space squeezes an equivalence class as one point, a process viewed on the quotient space should accommodate for the change of the volume of the equivalence class along the movement. This additional vector reflects the change rate of the volume of the equivalence class with respect to spatial movement.
Figure 2: Illustration of the relation beAlthough the diffusion process on the quotient space is tween the total space M and the quowell defined, it is not convenient to directly simulate it tient space Q and the correspondence of in the quotient space due to the difficulty in represent- tangent vectors among them. ing this quite abstract space and its non-trivial geometric structure. We hence work towards deriving a diffusion process in the total space M, which would be much easier to simulate, while inheriting key features of a quotient-space diffusion process. A key observation from Thm. 1 is that if b(1) = v + b(2) where vx ∈ Ker π∗x := {v ∈ Tx M | π∗x (v) = 0} at each x ∈ M, then the corresponding SDE in Eq. (6) induces the same quotient-space diffusion Eq. (7) after projection. Hence, vector components in this subspace are not really necessary for updating a sample point, as it only induces an infinitesimal movement within the equivalence class π(x). Under the view of the illustration in Fig. 2, this subspace is regarded as the vertical space Vx := Ker π∗x , which is G-dimensional. Correspondingly, its completion, referred to as the horizontal space Hx , identifies vectors that leads to essentially necessary movements. On a Riemannian manifold M, the inner product in Tx M provides a convenient way to specify a horizontal space by orthogonal completion: Hx := (Ker π∗x )⊥ . Any tangent vector v ∈ Tx M hence admits a unique decomposition: v = vV + vH , where the horizontal component vH identifies the essentially necessary component. Since both removes intra-equivalence-class movements and represents an essential update direction, there is a correspondence from a quotient-space tangent vector u ∈ Ty Q to a total-space horizontal 5
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vector ũx ∈ Hx for any x ∈ π −1 (y), called the horizontal lift of u at x (Def. 32). Since it is horizontal, it preserves the gist of the quotient-space vector u of an essential update direction, we can leverage this notion to move the desired quotient-space diffusion Eq. (7) back to the total space M, which admits a straightforward representation and simulation. Theorem 2. The horizontal lift of Eq. (7) has the following explicit expression: σt2 dx̃t = Px̃t (bt (x̃t )) − h̃(x̃t ) dt + σt dw̃t , x̃0 ∼ pprior , (8) 2 where Px (v) := vH is the horizontal projection in the tangent space of M, and h̃ is the horizontal lift of the mean curvature vector h, w̃t is the horizontal lift of the Wiener process of Q. See Appx. D.2 for proof. Comparing Eq. (6) and Eq. (8), the lifted process is not simply given by projecting the vector field and lifting the Wiener process in Eq. (6), but an additional vector field h̃ arises as the reflection of the spatial change rate of equivalence-class volume, which can be thought of as the compensation for the disability to change the spread of mass within an equivalence class. As mentioned, Eq. (8) only has horizontal movements. As a result, the length of sampling paths is reduced. This is one of the key advantages over existing equivariant diffusion models, which still traverse within each equivalence class following a certain rule. A representative illustration is given in Fig. 1. This advantage is formally stated in the following corollary (proof in Appx. D.2). Corollary 3. The resulting state x̃1 by the lifted diffusion process Eq. (8) and the resulting x1 by the original diffusion process Eq. (6) follow the same distribution in the total space. When σt ≡ 0 and starting from the same x0 ∈ M, Eq. (8) leads to a shorter trajectory than Eq. (6) does. 3.2
S PECIAL C ASE : THE S HAPE S PACE
The abstract results in the previous section give the direction for practical implementation. In this subsection, we focus on the space of 3-dimensional coordinates of N points under the symmetry defined by the special Euclidean group SE(3), composed of the 3-dimensional translation group and the 3-dimensional rotation SO(3). By definition, an element of this space is structured as x := (⃗x(1) , · · · , ⃗x(N ) ), where each ⃗x(n) ∈ R3 , and the SE(3) group acts on x by translating and rotating each ⃗x(n) . Since the translation group is not compact, there does not exist a translational invariant distribution. We (as well as many others (Yim et al., 2023; Lin et al., 2024)) hence represent the quotient space with respect to this group by considering the center-of-mass(CoM)-free subspace PN M := {x ∈ R3N | N1 n=1 ⃗x(n) = ⃗0}, and consider the SO(3) action on it.1 The resulting quotient space Q := M/SO(3), as the concrete construction for R3N /SE(3), is a smooth manifold under certain conditions (Appx. C.1). Each element in Q represents N -point configurations that are equivalent under altogether translation and rotation; therefore, Q is regarded as the “shape space” reflecting the intrinsically different states of the N points. Now we can develop the correspondence between the diffusion process in M (Eq. (6)) and the its horizontal lift from the quotient space projection (Eq. (8)). The results are summarized in the following theorem. Theorem 4. Assume xt is a diffusion process in the CoM subspace M ⊂ R3N , specified by the SDE: dxt = bt (xt ) dt + σt dwt , where bt (xt ) is SO(3)-equivariant, x0 ∼ pprior follows an SO(3)-invariant distribution, and wt is the Wiener process on M. Then the lifted process of its quotient-space process on Q := M/SO(3) (representing R3N /SE(3)) onto the total space M is given by σt2 h̃(x̃t ) dt + σt Px̃t dwt , dx̃t = Px̃t (bt (x̃t )) − (9) 2 where P and h̃ are explicitly given by: Px (v) =
⃗v
(n)
−K
−1
N X
! ⃗x
(n′ )
× ⃗v
(n′ )
! × ⃗x
(n)
n′ =1
,
∀v = (⃗v (n) )n ∈ Tx M, and
h̃(x) = − tr(K−1 )I − K−1 ⃗x(n) , n
where K :=
N X n=1
∥⃗x(n) ∥2 I −
N X
⃗x(n) ⃗x(n)
⊤
∈ R3×3 ,
n=1
and (·)n means concatenating the 3-dimensional vectors by running through n. 1
(10)
n
Technically, to guarantee proper structures, M needs to exclude a negligible subset; see Appx. C.1.
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See Appx. D.3 for proof. On M, a vertical vector is induced from a total angular momentum of the point cloud, which induces a rigid-body rotation without deformation; correspondingly, a horizontal vector leads to a pure deformation with zero total angular momentum (Appx. C.1), keeping the “total angular orientation” fixed. As a projection onto the horizontal space, Eq. (10) essentially removes the total rigid-body angular momentum of v, leaving only deformation which exactly corresponds to the movement on the shape space Q. This is similar to the conventional treatment for translation symmetry by zeroing out the total (linear) momentum of v. Together, the projected update only focuses on updating the shape of the point cloud (deformation), without incurring any rigid-body movement. Cor. 3 guarantees that this treatment still produces the same target distribution. 3.3
P RACTICAL I MPLEMENTATIONS
Previous results describe how we can construct a diffusion process in the quotient space using the coordinates in the total space. If we have a diffusion process on the total space, we can construct the horizontal lift of its projection process, which has no vertical velocity along its trajectory and the two processes are the same on quotient space. This fact implies that the vertical components of the original diffusion process are not dispensable and enables us to design a more efficient training and sampling algorithm of the diffusion model based on the quotient structure. In practice, we often set the total space as the Euclidean space. Next, we show the training and sampling methods for the special case pprior = N (0, I), and the general case is shown in Appx. E. Training objective. The diffusion model on the total space M is trained by the objective Eq. (5). Since the vertical components of the velocity are not strictly needed, we propose to supervise the model only on the horizontal components and allow arbitrary vertical output of the model. We leverage the horizontal projection operator Px (Thm. 4) and construct the horizontal training objective: L(θ) := Ep(t) w(t)Ep(x1 ,xt ) ∥Pxt (Dθ (xt , t) − x1 ) ∥2 .
(11)
We can see that Dθ + vV has the same loss value with Dθ , where vV is an arbitrary vertical vector. Thus, our objective does not require the diffusion model to match the vertical component of the target, and reduce the difficulty of learning. ODE sampler. After the training stage, Pxt (Dθ (xt , t)) is an approximation of the ground truth denoiser in the horizontal subspace. For the ODE sampler, we simulate the horizontal lift of the t projected ODE, which is given by dx dt = Pxt vθ (xt , t)dt, where vθ (xt , t) is given by Eq. (4). In practice, the ODE process is approximated by numerical solvers. SDE sampler. For the stochastic sampler, the we need to simulate the horizontal lift of the projected original SDE in Eq. (3). According to Thm. 1 and Thm. 4, the lifted process is given by p dxt = Pxt (vθ (xt , t) + gt sθ (xt , t)) dt + γηt h(xt )dt + 2γηt Pxt dwt , where sθ (xt , t) = − xt −βtα̂D2θ (xt ,t) and we introduce the hyperparameter γ for protein generation t following Geffner et al. (2025). The details are summarized in Alg. 1 and 3 in Appx. E. 3.4
A NALYSIS ON E XISTING T REATMENTS FOR S YMMETRY
In this section, we make a detailed analysis on existing methods that handle symmetry, and verify the conclusions in Table 1. In contrast to our quotient-space diffusion, we find that they either have not fully leveraged the symmetry to reduce model-learning difficulty, or do not have a proper sampler. Conventional equivariant diffusion models and data augmentation. A common treatment is by assigning equal probability to equivalent objects, resulting in an invariant target distribution p(x1 ). This can be implemented by augmenting data samples by applying randomly chosen group actions, mimicking sampling from the invariant distribution, or using an invariant prior distribution and an equivariant architecture securing Dθ (g · x, t) = g · Dθ (x, t). The training strategy is the same as modeling a general distribution in the original space following Eq. (5), and the standard samplers by Eqs. (2, 3) remain valid. For each value of xt , this objective asks the model to minimize the 2 average of ∥Dθ (xt , t) − x1 ∥ terms where x1 come from p(x1 |xt ), so the optimal solution is the conditional expectation E[x1 |xt ]. Fig. 3 shows an example and reveals characteristics of the training strategy. The example considers generating the structure of a diatomic molecule, where the target distribution p(x1 ) concentrates on
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Figure 3: Illustration of denoising-model learning target using conventional training and using GeoDiff alignment. (Left) The example considers the structure distribution p(x1 ) of a diatomic molecule, which concentrates on a single structure x⋆ up to a uniform random orientation. (Middle) Given an xt sample, the corresponding x1 samples distribute with a variance, and their expectation E[x1 |xt ] is the conventional learning target, which is not equivalent to x⋆ (the bond is shorter). (Right) Given an xt sample, all the x1 samples after alignment coincide with x⋆ posed in the orientation of xt , which is also the learning target of GeoDiff E[Axt (x1 )|xt ]. a single structure x⋆ up to a uniform random orientation (Left). For a given xt , samples of p(x1 |xt ) are x⋆ structures posed in orientations distributed around the orientation of xt (Middle). Indeed, an x1 sample more closely oriented with xt would have a higher probability to produce the given xt in the diffusion process, so there is a specific orientation correspondence between the learning target E[x1 |xt ] and xt . So the model is still asked to learn a correspondence in the equivalent degrees of freedom (DoFs) (i.e., rotation of the output), in contrast to the quotient-space case in Eq. (11) where the model is unconstrained in the vertical space (i.e., tangent space of the rotation group). Moreover, the x1 samples are not all posed in the orientation of xt because x⋆ in other orientations can also generate this xt through the diffusion process. So the model learns the correspondence in the equivalent DoFs from samples with a variance, leading to another aspect of learning difficulty. GeoDiff alignment. To reduce the learning difficulty, some heuristic treatments are proposed based on alignment. The first representative alignment used in GeoDiff (Xu et al., 2022) uses the following 2 training loss: Ep(x1 ,xt ) ∥Dθ (xt , t) − Axt (x1 )∥ , where the alignment operation is defined as: Ay (x) := argminx′ ∈{g·x|g∈G} d(x′ , y),
(12)
where d(·, ·) is the distance metric on M. With an illustration in Fig. 3(Right), the learning task can be understood as that for a given value of xt , the model output needs to fit Axt (x1 ) samples, which are all posed in the orientation of xt , and they all coincide with the x⋆ structure in the orientation of xt . This supervises the model to the target E[Axt (x1 )|xt ] from samples with no variance in the equivalent DoFs (i.e., rotation of the output), hence reduces certain learning difficulty. Nevertheless, this target still requires the model to learn a specific mapping in the equivalent DoFs, hence does not enjoy the learning advantage in the quotient-space case that relaxes the learning in the DoFs. A caveat of this alignment approach is that a proper sampler needs to be developed, as the conventional samplers still require a model targeting E[x1 |xt ], which is different from E[Axt (x1 )|xt ]. Fig. 3 illustrates this difference: E[x1 |xt ] averages diversely oriented x⋆ structures, resulting in a different shape than x⋆ (the bond is shorter), while E[Axt (x1 )|xt ] is just x⋆ in the orientation of xt . AF3 alignment. Another alignment approach, which is used in Alphafold 3 (AF3) (Abramson et al., 2 2024), aligns the x1 samples towards the model output: Ep(x1 ,xt ) Dθ (xt , t) − ADθ̄ (xt ,t) (x1 ) , where θ̄ is treated constant in optimization. This loss function allows the model output to differ by an arbitrary group action (e.g., rotation), hence removes the need to learn a specific target in the equivalent DoFs. Indeed, for an arbitrary group action gxt ,t , a new denoising model gxt ,t · Dθ (xt , t) achieves the same loss since ∥gxt ,t · Dθ (xt , t) − Agxt ,t ·Dθ̄ (xt ,t) (x1 )∥2 = ∥gxt ,t · Dθ (xt , t) − gxt ,t · ADθ̄ (xt ,t) (x1 )∥2 = ∥Dθ (xt , t) − ADθ̄ (xt ,t) (x1 )∥2 , where the last equality holds since the group preserves metric (Appx. C). Up to this DoF, the learning target is the same as GeoDiff’s E[Axt (x1 )|xt ], since all the x1 samples are averaged after aligned with the same reference. In the sampling process, the arbitrariness in the equivalent DoFs (e.g., orientation) of the learned model Dθ (xt , t) leads to an arbitrariness2 in the vector field vθ (xt , t) through Eq. (4). Hence there is no guarantee of recovering the target distribution using conventional samplers. This problem is also noted by Boltz-1 (Wohlwend et al., 2025), which proposes to align the prediction Dθ (xt , t) towards xt in the sampling process. As the AF3 target is the same as GeoDiff’s up to an arbitrary rotation, 2 This is not even an arbitrary group action (e.g., rotation) since xt does not vary together with the arbitrariness of Dθ (xt , t).
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Table 2: The effect of the quotient-space diffusion scheme for molecular structure generation on the GEOM-QM9 and the GEOM-DRUGS datasets using the ET-Flow(SO(3)) and ET-Flow(O(3)) architectures. We use the same sampling steps of 50 NFEs for fair comparison. Best results are marked in bold. Best results for the same architecture are underlined. Recall Datasets
GEOM-QM9 (Positive samples are within 0.5 Å RMSD.)
GEOM-DRUGS (Positive samples are within 0.75 Å RMSD.)
Precision
Methods
Coverage ↑ mean median
AMR ↓ mean median
Coverage ↑ mean median
AMR ↓ mean median
CGCF GeoDiff GeoMol Torsional Diff. MCF
69.47 76.50 91.50 92.80 95.0
96.15 100.00 100.00 100.00 100.00
0.425 0.297 0.225 0.178 0.103
0.374 0.229 0.193 0.147 0.044
38.20 50.00 87.60 92.70 93.7
33.33 33.50 100.00 100.00 100.00
0.711 1.524 0.270 0.221 0.119
0.695 0.510 0.241 0.195 0.055
ET-Flow(SO(3)) + Geodiff alignment + AF3 alignment + Quotient-space diffusion
95.98 95.71 92.67 96.40
100.00 100.00 100.00 100.00
0.076 0.085 0.131 0.069
0.030 0.040 0.070 0.024
92.10 95.20 84.38 93.30
100.00 100.00 100.00 100.00
0.110 0.098 0.205 0.096
0.047 0.050 0.146 0.036
GeoDiff GeoMol Torsional Diff. MCF - S (13M) MCF - B (62M) MCF - L (242M)
42.10 44.60 72.70 79.4 84.0 84.7
37.80 41.40 80.00 87.5 91.5 92.2
0.835 0.875 0.582 0.512 0.427 0.390
0.809 0.834 0.565 0.492 0.402 0.247
24.90 43.00 55.20 57.4 64.0 66.8
14.50 36.40 56.90 57.6 66.2 71.3
1.136 0.928 0.778 0.761 0.667 0.618
1.090 0.841 0.729 0.715 0.605 0.530
ET-Flow (8.3M) + reproduction + Quotient-space diffusion
79.53 78.94 79.86
84.57 84.24 85.71
0.452 0.489 0.459
0.419 0.472 0.433
74.38 66.24 72.70
81.04 70.42 79.63
0.541 0.651 0.565
0.470 0.595 0.501
ET-Flow(SO(3)) (9.1M) + reproduction + Geodiff alignment + AF3 alignment + Quotient-space diffusion
78.18 74.91 75.11 71.66 78.50
83.33 80.90 80.74 76.09 84.20
0.480 0.541 0.545 0.572 0.477
0.459 0.515 0.526 0.570 0.455
67.27 60.33 59.58 52.21 67.35
71.15 62.71 60.48 50.00 71.42
0.637 0.724 0.734 0.828 0.635
0.567 0.665 0.678 0.793 0.563
this amounts to using the GeoDiff model for sampling, which still cannot guarantee producing the target distribution as concluded above. These discussions are summarized in Table 1.
4
E XPERIMENTS
In this section, we study the empirical performance of our quotient-space diffusion model. We carefully conduct several experiments covering different types of data, scales and scenarios. To evaluate our quotient space diffusion model framework for real-world applications, we focus on the molecule structure generation protein backbone design tasks, in which we consider the diffusion models on R3N /SE(3) (Sec. 3.2). The details of all experiments are shown in Appx. G. 4.1
S TRUCTURE G ENERATION FOR S MALL M OLECULES
Datasets. First, we evaluate our framework on the molecule structure generation task. In this scenario, our goal is to generate the 3D coordinates of a molecule given the graph structure of the molecule. We conduct the experiments on the GEOM datasets (Axelrod & Gomez-Bombarelli, 2022), which provides structure ensembles generated by metadynamics in CREST (Pracht et al., 2024) and we focus on the GEOM-QM9 and GEOM-DRUGS datasets. Following the data processing and splits from Hassan et al. (2024), we use the random splits with train/validation/test of 243473/30433/1000 for GEOM-DRUGS and 106586/13323/1000 for GEOM-QM9. In addition, data with disconnect molecule graph are removed for GEOM-DRUGS. Setting. We primarily follow the setting in Hassan et al. (2024). We use an equivariant graph transformer architecture from ET-Flow (Hassan et al., 2024) and set the Gaussian distribution as prior distribution on GEOM-QM9 and use the harmonic prior for GEOM-DRUGS (Volk et al., 2023). We fix the architecture as ET-Flow(SO(3)) for experiments on GEOM-QM9, and use the ET-Flow(O(3)), ET-Flow(SO(3)) architecture on the GEOM-DRUGS dataset. Following Jing et al. (2022); Xu et al. (2022), we report the RMSD-based metrics, e.g. Coverage and Average Minimum RMSD (AMR) between the generated and ground truth structure ensembles. Results. The results are presented in Table 2 for the GEOM-QM9 and GEOM-DRUGS datasets. As shown, our proposed quotient-space diffusion framework consistently outperforms prior methods and alignment techniques in terms of generation quality on both datasets. Our framework reduces learning difficulty by removing redundant components, enabling us to further improve the perfor9
Published as a conference paper at ICLR 2026
Table 3: The effect of the quotient-space diffusion scheme for protein structure generation using the Proteı́na model. Best results are marked in bold. Settings
Methods
Designability (%)↑
FPSD vs.
fS
fJSD vs.
PDB↓
AFDB↓
(C/A/T)↑
PDB↓
AFDB↓
FrameDiff FoldFlow (base) FoldFlow (stoc.) FoldFlow (OT) FrameFlow ESM3 Chroma RFDiffusion Proteus Genie2
65.4 96.6 97.0 97.2 88.6 22.0 74.8 94.4 94.2 95.2
194.2 601.5 543.6 431.4 129.9 933.9 189.0 253.7 225.7 350.0
258.1 566.2 520.4 414.1 159.9 855.4 184.1 252.4 226.2 313.8
2.46/5.78/23.35 1.06/1.79/9.72 1.21/2.09/11.59 1.35/3.10/13.62 2.52/5.88/27.00 3.19/6.71/17.73 2.34/4.95/18.15 2.25/5.06/19.83 2.26/5.46/16.22 1.55/3.66/11.65
1.04 3.18 3.69 2.90 0.68 1.53 1.00 1.21 1.41 2.21
1.42 3.10 2.71 2.32 0.91 0.98 1.08 1.13 1.37 1.70
SDE Sampling
Proteı́na Msmall FS , γ = 0.35 + Quotient-space diffusion small Proteı́na MFS , γ = 0.45 + Quotient-space diffusion Proteı́na Msmall FS , γ = 0.50 + Quotient-space diffusion
96.0 97.6 92.2 92.6 89.2 90.2
386.5 274.7 332.9 244.5 306.2 228.0
378.2 277.1 320.4 246.3 290.8 228.7
1.77/4.97/17.78 2.24/6.69/20.99 1.83/5.01/20.22 2.24/6.68/23.47 1.86/4.92/21.15 2.25/6.59/25.24
2.17 1.68 1.93 1.43 1.81 1.32
1.73 1.55 1.49 1.28 1.36 1.17
ODE Sampling
Proteı́na MFS Proteı́na Msmall FS + AF3 alignment + Quotient-space diffusion
19.6 13.8 3.8 15.6
85.4 83.2 229.0 69.9
21.4 21.9 82.4 17.6
2.51/5.65/27.35 2.45/5.63/31.76 2.18/4.30/14.28 2.57/6.40/32.14
0.59 0.58 1.35 0.41
0.09 0.12 0.36 0.11
Representative References
mance of the ET-Flow framework3 on both datasets. On the GEOM-QM9 dataset, our quotient-space diffusion model framework surpasses strong baselines such as MCF (Wang et al., 2023) and the ETFlow framework with other heuristic alignment methods among most of the RMSD-based metrics. On the GEOM-DRUGS dataset, our framework not only significantly surpasses the ET-Flow baseline with heuristic alignment methods, since these methods are incompatible with training, but also achieves competitive performance against the larger MCF-L (242M) model (Wang et al., 2023) on the Precision metrics. 4.2
P ROTEIN BACKBONE D ESIGN
Setting. To demonstrate the advantage of our quotient-space diffusion model for larger and more relevant molecules, we perform a comparative analysis on the task of protein structure generation against the state-of-the-art Proteı́na model (Geffner et al., 2025). We select their most efficient variant Msmall FS , a 60M parameter transformer trained on the Foldseek AFDB clusters (DFS ) that forgoes triangle layers and pair representation updates, as a strong and relevant baseline. We train the quotient-space diffusion model from scratch using the identical architecture on the identical dataset. For evaluation, both our model and the officially released Proteı́na checkpoint are sampled using 400 steps with self-conditioning. We explore the designability-diversity trade-off by testing a range of noise scales, γ ∈ {0.35, 0.45, 0.5}4 . To faithfully evaluate the distributional metrics proposed by Geffner et al. (2025), we utilize ODE sampling. Results. The results in Table 3 highlight the superiority of our quotient space framework, which, unlike alignment-based strategies (adapted from AF3 and Boltz-1), provides a theoretical guarantee for sampling the correct target distribution. The alignment-based methods fail to recover this distribution, with performance metrics falling short of even data-augmented, semi-equivariant baselines. We attribute this failure to a fundamental incompatibility between their samplers and the learned models. Furthermore, our formulation effectively reduces learning difficulty by removing the need to learn a specific target in redundant spatial transformations, enabling the model to capture key structural features more efficiently than standard semi-equivariant baselines. This advantage of efficiency leads to significant results: our 60M parameter model not only surpasses its direct baseline across both SDE at all noise scales and ODE sampling setting, but also outperforms the much larger 200M MFS model on most key distributional metrics. This provides compelling evidence that a quotient space framework ensuring both sampling fidelity and learning efficiency is key to advancing generative protein models. 3
We reproduce the results using the released configurations: https://github.com/ shenoynikhil/ETFlow. Due to changes in the data processing pipeline, our reproduced results do not exactly match those reported in the original paper. 4 Due to a known bug in a previous version of Foldseek (Daras et al., 2025, Appendix B), our comparative analysis in the main text is focused solely on the designability. More comprehensive metrics evaluating our self-sampled structures are provided in Table 6.
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5
C ONCLUSION
In this work, we formally construct a framework for building diffusion models on the quotient space over a group, in hope for a principled approach to handle symmetry in a generative task. We explicitly give the expression of the diffusion process on the quotient space, then also construct a corresponding diffusion process in the original space for easier implementation. The resulting training algorithm reduces learning difficulty by removing the need to predict the tangent vector in the direction along group action, and the resulting sampling process guarantees producing the target distribution while removes the unnecessary movement in the group-action direction. We instantiate the method in the case of R3N /SE(3) for molecular structure generation. Empirical results on structure sampling for small molecules from the GEOM-QM9 and GEOM-DRUGS datasets and protein backbone generation demonstrate the better generation quality and design success rate over existing conventional equivariant diffusion models and alignment-based approaches given equal or fewer training epochs, demonstrating the practical advantages from this principled framework to handling symmetry in diffusion models. ACKNOWLEDGMENTS This work is supported by Zhongguancun Academy (Grant No. C20250506). DH is supported by National Science Foundation of China (NSFC62376007), National Science Foundation of China (under Key Project No. 92570203), Beijing Natural Science Foundation (Z250001) and Beijing Major Science and Technology Project under Contract no. Z251100008425004.
6
E THICS S TATEMENT
This work adheres to the ICLR Code of Ethics. Our study does not involve human subjects, personal data, or sensitive demographic information. All experiments are conducted on publicly available benchmark datasets, which are widely used in the machine learning community. No new data collection or human/animal experimentation was performed.
7
R EPRODUCIBILITY S TATEMENT
To facilitate the reproducibility of our research, we provide comprehensive details throughout the paper and its supplementary materials. We begin by establishing the necessary foundational knowledge in Sec. 2.1 and Appx. B. For all theoretical claims and proofs presented in the main text, we offer detailed step-by-step derivations in Appx. D. Our experiments are thoroughly documented; the datasets, training procedures, and evaluation protocols are carefully described in Sec. 4 and Appx. G. Upon acceptance of this paper, we commit to making our full codebase and all model checkpoints publicly available to ensure that the community can fully reproduce our results.
8
T HE USE OF L ARGE L ANGUAGE MODELS (LLM S )
In the preparation of this manuscript, LLMs were employed as a writing assistant to refine the language and improve the grammar. Furthermore, we utilized LLMs to assist in verifying our mathematical formulas for notational consistency. Following this process, all textual and mathematical content was meticulously reviewed, revised, and validated by the authors, who assume full responsibility for the final work presented.
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A PPENDIX The organization of the appendix are as follows. In Appx. A, we briefly discuss the related work relevant to our research. In Appx. B, we review some background knowledge of Riemannian geometry and stochastic calculus on the manifold. In Appx. C, we give the details of the Riemannian structures of the quotient space. In Appx. D, we give all the proofs of the theorems in the main text. In Appx. E, we show our methods for the general case. In Appx. F, we give some additional results and discussions. Finally, the details of the experiments are given in Appx. G.
A
R ELATED W ORK
Diffusion models on Riemannian manifolds. As the quotient has the Riemannian manifold structure, several previous works construct the diffusion model on the Riemannian manifolds. De Bortoli et al. (2022) constructs diffusion models using different overlapping local coordinate systems of the manifold and requires geodesic random walk to simulate the forward process. Huang et al. (2022); Chen & Lipman (2023) construct diffusion models in an embedding space which allows a global representation but requires explicit geodesic formula of the manifold. Zhu et al. (2024) constructs the reverse of kinetic Langevin dynamics on a Lie group to perform generative modeling. Such an approach is not designed for and not readily applicable to the quotient space, which has a different geometric structure from the Lie group. In our quotient space case, the specialty with a quotient structure enables us to construct diffusion models using the coordinate systems of the total space without relying on an embedding of the quotient in the total space (unnecessarily an embedding space), which is more practical to implement yet still general. Geometric diffusion models. To ensure physical symmetry in the generation process, a mainstream strategy integrates fundamental physical constraints, such as SE(3) equivariance, directly into the diffusion-model architecture. This approach, pioneered by models like EDM (Hoogeboom et al., 2022a), typically employs an EGNN to operate directly on atomic coordinates, using techniques like zero center of mass adjustments to guarantee translational invariance. This foundational concept was subsequently extended in several directions. For instance, the approach was adapted for Diffusion Bridges in models like EDM-Bridge (Wu et al., 2022) and for diffusion in a latent space in models like GeoLDM (Xu et al., 2023). These equivariant diffusion techniques have been successfully applied across a range of molecular tasks. For structure generation, models like GeoDiff (Xu et al., 2022) predict 3D structures from molecular graphs. In molecular optimization, methods such as DiffHopp (Torge et al., 2023) refine existing molecules to enhance desired properties. For de novo design, a key advancement has been to combine discrete diffusion models (D3PM) (Austin et al., 2021) for 2D topology with continuous equivariant diffusion for 3D geometry, enabling joint generation as seen in models like DiffSBDD (Schneuing et al., 2024) and MUDiff (Hua et al., 2024). A similar problem has also been considered in crystalline structure generation, where the intrinsic periodic translation invariance is an intrinsic symmetry. Lin et al. (2024) highlighted the intrinsic periodic translation symmetry that has been omitted for a long time in the field of periodic crystalline structure generation. The work designed a modified diffusion process that induces a transition kernel that is invariant under periodic translation, leading to a learning target for the score model that is invariant under periodic translation. Cornet et al. (2025) proposes a novel method that generalizes the Trivialized Diffusion Model framework for fractional coordinates to model the intrinsic periodic translation symmetry using flat coordinates. The proposed method considers the process with the velocity restricted to the CoM-free linear subspace. They have achieved the removal of variance on equivalent DoFs, but still asks the neural network model to learn to predict a specific target in the equivalent DoFs. Learning with alignment To reduce learning difficulty, some heuristic treatments (learning with alignment) have been proposed in hope to reduce the DoFs corresponding to the symmetry group action. The alignment strategy used in GeoDiff (Xu et al., 2022) aligns the target structure to the noisy input by finding an optimal rigid transformation that minimizes the distance between them. Another approach, proposed in AlphaFold 3 (AF3) (Abramson et al., 2024), aligns the target samples to the model output structure. As discussed in the main text, these two alignment-based training frameworks lack a definite guarantee for recovering the correct target distribution, and is incompatible with the sampling process. Boltz-1 (Wohlwend et al., 2025), an open-source replication of AF3, noticed this issue and proposed a modification in sampling to align the denoised structure to the 15
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structure in the current generation step before updating. Nevertheless, as discussed in Sec. 3.4, this, together with the training protocol of AF3, amounts to the operation of GeoDiff, still questioning the sampling process.
B
BACKGROUND IN R IEMANNIAN G EOMETRY AND S TOCHASTIC C ALCULUS
B.1
R IEMANNIAN G EOMETRY
In this section, we review some background on differential geometry and Riemannian geometry. For a systematic treatment of the subject, please refer to standard textbooks Lee (2003; 2018). First, we give the formal definition of the smooth manifold. A manifold is a general topological space that locally has a Euclidean structure. Definition 5. An M -dimensional topological manifold is a topological space M such that: • M is locally Euclidean, i.e. locally homeomorphic to RM . Formally, ∀x ∈ M,5 there exists an open neighborhood x ∈ U ⊂ M that is homeomorphic to some open set V ⊂ M. We call the homeomorphism ϕ : U → V ⊂ RM a coordinate system or a chart. • M is a Hausdorff topological space. • M has a countable basis for its topology. A smooth manifold is a topological manifold with an additional smooth structure, which is defined as follows. Definition 6. A smooth structure on a M -dimensional topological space M is a collection of coordinate systems C = {(U (α) , ϕ(α) ) : α ∈ A} which satisfies the following properties: • The collection C covers M:
S
α∈A U
(α)
= M;
• For any α, β ∈ A, the transition function ϕ(α) ◦ ϕ(β)
−1
is a smooth map;
• C is a maximal collection, i.e. if (U, ϕ) is a coordinate system such that for all α ∈ A that −1 the maps ϕ ◦ ϕ(α) and ϕ(α) ◦ ϕ−1 are smooth, then (U, ϕ) ∈ C. The pair (M, C ) is called a smooth manifold of dimension M . If there is a coordinate system (U, ϕ) around a point x ∈ M, then in this neighborhood of x, the manifold admits a coordinate chart xi (x) := ϕi (x) and a manifold point in the neighborhood can be expressed as a vector x(x) = (x1 (x), · · · , xM (x))⊤ . With the smooth structure, we can define a smooth function on the manifold and a smooth mapping between smooth manifolds. Definition 7. Let M, N be smooth manifolds with dimensions M, N respectively. • A function f : M → R is called a smooth function if its vectorized form f ◦ ϕ−1 : ϕ−1 (U) → R is smooth on ϕ−1 (U) ⊂ RM for all smooth coordinate systems (U , ϕ) of M. Denote all the smooth functions on M as C ∞ (M). • A map F : M → N is called a smooth map if its vectorized form ψ ◦F ◦ϕ−1 : ϕ−1 (U) → ψ(V) is smooth for all smooth coordinate systems (U, ϕ) of M and (V, ψ). A smooth map F : M → N which is invertible and whose inverse is smooth is called a diffeomorphism. In this case we say that M and N are diffeomorphic manifolds. To define movement on a smooth manifold M, we need to define tangent vectors on the manifold. 5 On an abstract manifold, a point is an abstract object and may not be a vector by itself, so we do not use a boldface notation. A vector representation as the coordinates is available after choosing a (local) coordinate system.
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Definition 8. Let M be a smooth manifold, and x ∈ M is a point, and U is a neighborhood of it. A linear map v : C ∞ (U) → R is called a derivative at x if it satisfies v(f g) = f (x)v(g) + g(x)v(f ),
∀f, g ∈ C ∞ (U).
The set of all the derivatives of C ∞ (U) in x, denoted by Tx M, is a vector space called the tangent space to M at x. An element of Tx M is called a tangent vector at x. Definition 9. Let M, N be smooth manifolds and F : M → N be a smooth map. Let x ∈ M and V ⊆ N be a neighborhood of F (x). Then F induces a push-forward map over the tangent spaces, F∗x : Tx M → TF (x) N , is defined as: F∗x (v)(f ) := v(f ◦ F ),
∀f ∈ C ∞ (V), v ∈ Tx M.
When a coordinate system (U , ϕ) around x and (V, ψ) around F (x) are chosen, the coordinate expression for F∗x is just the Jacobian matrix of its vectorized form ψ ◦ F ◦ ϕ−1 , i.e., ∇(ψ ◦ F ◦ ϕ−1 )(x). So F∗ is also called the differential of F and also admits the notation dF . The tangent F bundle T M of a smooth manifold M is the union of the tangent spaces of each points, i.e. T M := x∈M Tx M. Similar to the total derivative of the smooth map in Euclidean space, the differential of a smooth map between smooth manifolds is a linear map between tangent spaces. A vector field v on a smooth manifold M is a correspondence that associates to each point x ∈ M a vector vx ∈ Tx M. The vector field is smooth if the mapping v : M → T M is smooth. Denote all the smooth vector fields on M by X (M). With the definition of a vector field, we can define the solution of ordinary differential equation (ODE) on the manifold. The idea is similar to the definition in Euclidean space, the solution of the ODE is a curve whose velocity at each point is the same as the vector field. Definition 10. Let v be a smooth vector field on the smooth manifold M. An integral curve of v is a differentiable curve γ : [0, T ] → M whose velocity at each point is equal to the value of v at that point: γ̇t = vγt ,
∀t ∈ [0, T ].
at x. The cotangent Let Tx∗ M be the dual space of Tx M, which is called the cotangent space of MF bundle T ∗ M is the union of the cotangent space of each points, i.e. T ∗ M := p∈M Tx∗ M. Definition 11. A 1-form Θ on smooth manifold M is a correspondence that associates to each point x ∈ M a covector Θx ∈ Tx∗ M. The 1-form is smooth if the mapping Θ : M → T ∗ M is smooth. With the definition of a smooth manifold, we can define a continuous group with good properties. Definition 12. A Lie group is a smooth manifold G that is also a group with the property that the multiplication map G × G → G, (g, h) 7→ g · h and the inversion map G → G, g 7→ g −1 are both smooth. Define the left multiplication mapping Lg (h) = gh, which is introduced to differentiate g as a Liegroup element and as an action on a group element. A vector field v on G is said to be left-invariant if it is invariant under all left multiplications, i.e. (Lg )∗g′ (vg′ ) = vgg′ . Definition 13. A Lie algebra is a real vector space g endowed with a map called the bracket [·, ·] : g × g → g that satisfies the following properties for all X, Y, Z ∈ g: • Bilinearity: ∀a, b ∈ R, [aX + bY, Z] = a[X, Z] + b[Y, Z], [Z, aX + bY ] = a[Z, X] + b[Z, Y ]; • Antisymmetry: [X, Y ] = −[Y, X]; • Jacobi Identity: [X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y ]] = 0. The Lie algebra of all smooth left-invariant vector fields on a Lie group G is called the Lie algebra of G, which has the same dimension with G. Example 14. The Lie algebra of the group SO(3), denoted by so(3), is given by all the 3dimensional antisymmetric matrices so(3) = {A ∈ R3×3 | A + A⊤ = 0}. 17
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Smooth manifold is a topological structure. If we want to define the ”length of the velocity” and distance between two points on the manifold, a metric on the tangent space is required. Such a metric endows the metric with an additional geometry structure. The formal definitions are as follows. Definition 15. A Riemannian metric on a smooth manifold is a correspondence which associates to each point p of M an inner product ⟨·, ·⟩x that varies smoothly on M. In other words, for any two smooth vector fields u, v, ⟨u, v⟩ is a smooth function on M. A smooth manifold with a given Riemannian metric is called a Riemannian manifold. To define the ”difference” between tangent space at different points, we need to introduce a concept called affine connection. Definition 16. An affine connection ∇ on a Riemannian manifold is a mapping ∇ : X (M) × X (M) → X (M) which is denoted by (u, v) → ∇u v which satisfies the following properties: • ∇u v is linear over C ∞ (M) in u: ∀f (1) , f (2) ∈ C ∞ (M) and u(1) , u(2) ∈ X (M), ∇f (1) u(1) +f (2) u(2) v = f (1) ∇u(1) v + f (2) ∇u(2) v; • ∇u v is linear over R in v: ∀a(1) , a(2) ∈ R and v(1) , v(2) ∈ X (M), ∇u(1) (a(1) v(1) + a(2) v(2) ) = a(1) ∇u v(1) + a(2) ∇u v(2) ; • ∇ satisfies the following product rule: ∀f ∈ C ∞ (M), ∇u (f v) = f ∇u v + (uf )v. A connection is called the Levi-Civita connection if satisfies the following additional properties: • ∇ is compatible with metric: ∇u v(1) , v(2) = ∇u v(1) , v(2) + v(1) , ∇u v(2) ; • ∇ is torsion-free: ∇u v − ∇v u = u(v(·)) − v(u(·)). The Levi-Civita connection is the connection with nice properties. Its existence and uniqueness is a fundamental result of Riemannian geometry. Theorem 17. (Fundamental Theorem of Riemannian Geometry (Lee, 2018, Thm. 5.10)) Assume (M, ⟨·, ·⟩) is a Riemannian manifold. Then there exists a unique Levi-Civita connection. As the end of this subsection, we introduce the Laplace-Beltrami operator on the manifold, which is used to define the Wiener process on the manifold. Definition 18. Let ∇ be the Levi-Civita connection on M. The Hessian of f ∈ C ∞ (M) is defined by Hess(f )(u, v) := v(u(f )) − (∇v u)(f ),
∀u, v ∈ X (M).
The Laplace-Beltrami operator ∆ is defined as the trace of Hessian. In other words, ∆f := PM i=1 Hess(ei , ei ) where {e1 , ..., eM } is an orthonormal basis for Tx M. B.2
S TOCHASTIC C ALCULUS ON A M ANIFOLD
With the Riemannian structure defined in the previous section, we can consider the definition of stochastic differential equations (SDE) and diffusion processes on the manifold. For a systematic treatment of the subject, please refer to standard textbooks Hsu (2002); Thalmaier (2023). First, we recall the definition of SDE and diffusion process in Euclidean space. Definition 19. The infinitesimal generator of a stochastic process (xt )t for a function ϕ(x) is E[ϕ(xt+s )|xt = x] − ϕ(x) , s where ϕ is a suitably regular function. For an Itô process defined as the solution to the SDE dxt = f (xt , t) dt + Σ(xt , t) dwt , the generator is Lt ϕ(x) = lim
s→0+
Lt =
D X i=1
D
f i (x, t)∂i +
ij 1 X Σ(x, t)Σ(x, t)⊤ ∂i ∂j . 2 i,j=1 18
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On the other hand, the diffusion process can also be defined by its generator. Definition 20. A D-dimensional stochastic process xt with continuous sample path defined on a probability space (Ω, F , P) is called a diffusion process generated by a smooth second-order elliptic operator Lt if the following hold: ∀f ∈ C ∞ (RD ), the process Z t f Mt = f (xt ) − f (x0 ) − Ls f (xs ) ds 0
is a Ft -martingale. To generalize the definition of SDE to a Riemannian manifold M, we need to define the secondorder differential operator on the manifold. Let M be an M -dimensional Riemannian manifold. A second-order partial differential operator on M is of the form L = v(0) +
R X
2
where v(k) ∈ X (M),
v(k) ,
k=1 +
for some R ∈ N . The square of a vector field is understood as the decomposition of derivatives: 2
v(k) (f ) := v(k) (v(k) (f )),
∀f ∈ C ∞ (M).
The vector fields can be generalized to the time-dependent case. Now we can extend the definition of a diffusion process on a Riemannian manifold. Definition 21. (Thalmaier, 2023, Def. 1.1.3) Let (Ω, F , P; (F )t⩾0 ) be a probability space equipped with increasing sequence of sub-σ-algebra Ft ⊆ F . An adapted continuous process xt taking values in M, is called Lt -diffusion if for all test functions f ∈ Cc∞ (M), the process Z t Ntf := f (xt ) − f (x0 ) − (Ls f )(xs ) ds, t ⩾ 0, 0
is a martingale, i.e. E[Ntf − Nsf | Fs ] = 0,
∀s ⩽ t.
For a special case, we can define the Wiener process on the Riemannian manifold M. Definition 22. A Wiener process wt on M is a diffusion process with generator 12 ∆, where ∆ is the Laplace-Beltrami operator of M, i.e. wt is a continuous stochastic process on M such that for any f ∈ C ∞ (M), Z 1 t f (xt ) − ∆f (ws ) ds, 2 0 is a local martingale up to a valid time period. For stochastic differential geometry, the Stratonovitch integral is more convenient than the Itô Integral. The Stratonovitch differential effectively subsumes the deterministic second-order effect of the Wiener process from the quadratic variation into the drift term, so that it satisfies the ordinary chain rule of calculus. This property enables a clear correspondence between the diffusion process under a diffeomorphism between two Riemannian manifolds. Next, we give the definition of the Stratonovitch integral on the Euclidean space and its generalization to Riemannian manifolds. Definition 23. For continuous real-valued semimartingales x and y, let x ◦ dy := x dy + 21 d[x, y] be the Stratonovitch differential. Here x dy is the usual Itô differential, and d[x, y] := dxdy is the quadratic co-variation of x and y. The integral Z t Z t 1 x ◦ dy = xdy + d[x, y]t 2 0 0 is called the Stratonovitch integral of x with respect to y. Proposition 24. (Itô-Stratonovitch formula (Thalmaier, 2023, Prop. 1.2.10)). Let x be a continuous RD -valued semimartingale and f ∈ C ∞ (RD ). Then df (x) = ⟨∇f (x), ◦dx⟩. The Itô-Stratonovitch formula shows the advantage of the Stratonovich differential: it satisfies the usual chain rule of classical calculus. So at least formally, classical differential calculus can be applied in calculations involving Stratonovich differentials. 19
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Proposition 25. (Thalmaier, 2023, Prop. 1.2.11) Solutions to the Stratonovitch SDE dxt = b(xt , t) dt + Σ(xt , t) ◦ dwt
(13)
define L-diffusions for the operator D
L = v(0) +
1 X (k) 2 , v 2
where v(0) = b, v(k) =
D X
Σik ∂i .
i=1
k=1
From this result, we can see that Eq. (13) describes the same diffusion process as the following Itô SDE: D 1 X (k) (k) v∗ (v ) dt + Σ(xt , t) dwt , dxt = b(xt , t) + 2 k=1 PD j i (k) where v∗ (v(k) ) := i,j=1 v(k) (∂j v(k) )∂i . Now we can generalize the definition of SDE to the Riemannian manifold case. An SDE on manifold M can be defined by vector fields v(0) , v(1) , ..., v(M ) on M. Let w be the RM -valued Wiener process and x0 be an M-valued random variable serving as the initial value of the solution. The equation is symbolically written as dxt = v(0) (xt , t) dt +
D X
v(k) (xt , t) ◦ dwtk .
(14)
k=1
Definition 26. An M-valued semimartingale xt defined up to a proper stopping time τ is a solution to the SDE Eq. (14) up to τ if for all f ∈ C ∞ (M), ! Z t D X (k) k (0) v (f )(xs , s) ◦ dwt , 0 ⩽ t < τ. f (xt ) = f (x0 ) + v (f )(xs , s) ds + 0
k=1
PD 2 Proposition 27. (Thalmaier, 2023, Cor. 1.2.19) Let L = v(0) + 21 k=1 v(k) and xt be the solution ∞ to the SDE Eq. (14). Then for all f ∈ C (M), Z t Ntf := f (xt ) − f (x0 ) − (Ls f )(xs )ds, t ⩾ 0, 0
is a martingale. In other words, the solution of SDE Eq. (14) is a L diffusion to the operator PD 2 L = v(0) + 12 k=1 v(k) .
C
C ONSTRUCTION OF Q UOTIENT S PACE
In this section, we describe a rigorous construction of the quotient space and endow it with a manifold structure. Please refer to the standard textbooks Lee (2018) for the systematic treatments. Assume that the total space M is a Riemannian manifold and G is a compact Lie group. First we give the formal definition of the group action. Definition 28. Let G be a group and M is a Riemannian manifold. A left action of G on M is a map G × M → M, (g, x) 7→ g · x, satisfying g1 · (g2 · x) = (g1 g2 ) · x and e · x = x, ∀g1 , g2 ∈ G, x ∈ M. An action is smooth if its defining map G × M → M is smooth. We also reload the notation Lg (x) := g · x for distinguishing g from its action on the manifold. In the case where the Lie group acts on a Riemannian manifold, to draw meaningful conclusions, we would expect some compatibility between group action and the Riemannian metric, which is the concept of an isometric action. Moreover, to ensure the topological structure of the quotient space so as to define useful constructions on the quotient space, concepts of a free action and proper action are introduced. Definition 29. (1) A smooth action is said to be an isometric action if the map Lg : M → M, x 7→ g · x is an isometry for any g ∈ G, i.e., ⟨u, v⟩x = ⟨(Lg )∗x (u), (Lg )∗x (v)⟩g·x . 20
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(2) A smooth action is said to be free if for any x ∈ M, g · x = x indicates g = e. (3) A smooth action is said to be proper if the map G × M → M × M, (g, x) 7→ (g · x, x) is a proper map, meaning that the preimage of every compact set is compact. For the properness, there is a convenient characterization. Proposition 30. (Lee, 2018, Prop. C.15) Assume G is a Lie group acting smoothly on the smooth manifold M. The action is proper if and only if the following condition is satisfied: if {p(n) }n is a sequence in M and {g (n) }n is a sequence in G such that both {p(n) }n and {g (n) · p(n) }n converge, then a subsequence of {g (n) }n converges. Particularly, every smooth action by a compact Lie group on a smooth manifold is proper. The group action typically represents a symmetry in the sense that points that can be transformed to each other by a group action are regarded as symmetric, i.e., they are equivalent. Therefore, we can define an equivalence relation ∼ on M as x ∼ x′ if ∃g ∈ G, x′ = g · x. The equivalence class with representative x is defined as the set of all points that are equivalent to x. The quotient space Q := M/G (as a set) is defined under this equivalence relation, which consists of equivalence classes under the relation ∼. The original space M is referred to as the total space. There is a natural mapping called projection that connects the total space and the quotient space, which maps any x ∈ M to the equivalence class it represents. In this case where the equivalence is defined by a Lie group, the projection mapping can be written as: π : M → Q, π(x) := {g · x | g ∈ G}. Due to this expression, the equivalence class in such a case is the orbit of the Lie group G at x. Therefore, it can be understood that an equivalence class is a “representation” (literal meaning; not the mathematical concept) of the Lie group, hence can also adopt manifold structures of G under the mentioned “good” conditions. Also, the (M, Q, π) structure forms a fiber bundle, in which context the equivalence class is also called a fiber at π(x), and this special fiber bundle induced from a Lie group action is called a principal G-bundle. Moreover, under certain conditions, the quotient space inherits the Riemannian structure of the total space M through the projection mapping. Theorem 31. (Lee, 2018, Cor. 2.29) Let M be a Riemannian manifold, and G be a Lie group acting smoothly, freely, properly, and isometrically on M. Then the quotient space M/G has a unique smooth manifold structure and Riemannian metric such that π is a Riemannian submersion. We will assume the conditions, i.e., G be a Lie group acting smoothly, freely, properly, and isometrically on M, in the following development. Given that Q is a smooth manifold, the projection mapping induces a linear mapping π∗ between the tangent spaces of the two manifolds. It introduces more structures in the total space M. In each tangent space Tx M, we can define a subspace of it, called the vertical space, by the kernel of π∗ : Vx := Ker π∗x . By this definition, tangent vectors in the vertical space can be understood that it does not move x in a way that alters the projection onto Q by π, so the movement stays within the equivalence class. The vertical space can then be understood as the tangent space of the equivalence class. As mentioned above, in this case where the quotient space is induced from the Lie group G, the equivalence class is a “representation” of the Lie group, hence the vertical space is a “mirror” of the tangent space of the Lie group, which is in turn isomorphic to the Lie algebra g of the Lie group G. To complete the whole tangent space, a concept of horizontal space Hx is expected. In general, the horizontal space Hx is a linear subspace of Tx M that makes up Tx M by direct sum with Vx : Tx M = Vx ⊕ Hx . Under this direct-sum construction, any tangent vector v ∈ Tx M can then be uniquely decomposed into the vertical and horizontal components, v = vV + vH . Correspondingly, a vector field on M is called a vertical/horizontal vector field if it takes a vertical/horizontal tangent vector at every point. Every smooth vector field v on M can be expressed uniquely in the form v = vV + vH , where both the vertical and horizontal vector fields are smooth (Lee, 2018, Prop. 2.25). For future reference, we assign a convenient notation to the horizontal projection within Tx M itself: Px (v) := vH , 21
∀v ∈ Tx M.
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Nevertheless, the horizontal space Hx as a subspace that makes up the tangent space Tx M by the direct sum with Vx is not unique. Therefore, a smooth correspondence from x to such an Hx is an independent structure, referred to as the “connection” in the fiber-bundle context. In the current specific case where M endows a Riemannian structure, we can uniquely define the horizontal space as the orthogonal complement under the inner product in the tangent space: ⊥(Tx M,⟨·,·⟩x )
Hx := Vx
,
(16)
which gives a canonical “connection”. As would be expected, in contrast to vertical tangent vectors, a horizontal tangent vector represents a movement through different equivalence classes, corresponding to a movement on the quotient space Q. Therefore, we can construct the concept of horizontal lift which establishes a correspondence from a vector field on Q to a horizontal vector field on M. Definition 32. Given a vector field u on Q, a vector field ũ on M is called a horizontal lift of u, if ũ is a horizontal vector field, i.e., ũx ∈ Hx for all x ∈ M, and ũ is π-related to u by π∗x (ũx ) = uπ(x) . Proposition 33. (Lee, 2018, Prop. 2.25) Given a smooth connection x 7→ Hx and assuming π : M → Q is a smooth submersion, every smooth vector field on Q always has a unique smooth horizontal lift to M. If the connection is induced from the Riemannian structure of M by Eq. (16) and if the group action is isometric, then a nice compatibility can be derived. For a quotient-space tangent vector u ∈ Ty Q at some y ∈ Q, consider two ways to construct a tangent vector at some point x ∈ π −1 (y) in the equivalence class. The first way is directly by the horizontal lift, which gives ũx , which is the unique horizontal tangent vector such that π∗x (ũx ) = u. The other way is to first horizontal-lift u to another point x′ ∈ π −1 (y) in the equivalence class, then push it forward to the tangent space at x by a transformation that maps x′ to x. Since both points lie in the same equivalence class and the Lie group acts on the manifold freely, there exists a unique group action g ∈ G such that x = g · x′ = Lg (x′ ), so the resulting tangent vector is (Lg )∗x′ (ũx′ ). Noting that (Lg )∗x′ preserves the metric between Tx′ M and Tx M (see Eq. (15)), we know that it also preserves the horizontal spaces, i.e., (Lg )∗x′ (Hx′ ) = Hx , so (Lg )∗x′ (ũx′ ) ∈ Hx . Moreover, π∗x (Lg )∗x′ (ũx′ ) = (π◦Lg )∗x′ (ũx′ ) = π∗x′ (ũx′ ) = u also projects to the quotient-space tangent vector u (noting that π ◦ Lg = π for any g ∈ G, and recalling the definition of horizontal lift), by the uniqueness of the horizontal tangent vector that projects to u, we have ũx = (Lg )∗x′ (ũx′ ), or equivalently, ũg·x = (Lg )∗x (ũx ),
∀x ∈ π −1 (y), g ∈ G.
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The unique existence of the correspondence from Tπ(x) Q back to Tx M by horizontal lift (Prop. 33) allows us to introduce a Riemannian structure on Q from that on M. For any y ∈ Q and u(1) , u(2) ∈ Ty Q, define: (1) (2) M ⟨u(1) , u(2) ⟩Q y := ⟨ũx , ũx ⟩x ,
(18)
for any x ∈ π −1 (y). This is well-defined since the right hand side is independent the choice of x due to the horizontal-lift–push-forward compatibility (Eq. (17)) and isometry (Eq. (15)): (1) (2) (1) (2) M (1) (2) M ⟨ũg·x , ũg·x ⟩M g·x = ⟨(Lg )∗x (ũx ), (Lg )∗x (ũx )⟩g·x = ⟨ũx , ũx ⟩x . Subsequent constructions on the quotient space Q can be induced from this Riemannian structure. Due to its compatibility with the original Riemannian manifold M, these constructions have direct connections to their counterparts on M. Particularly, the Levi-Civita connections on Q and M follow the relation below. Proposition 34. (Lee, 2018, Exercise. 5.6) Let ∇M and ∇Q denote the Levi-Civita connections on M, Q, respectively, where ∇Q is constructed from the Riemannian metric induced from that of M by Eq. (18). Then for any vector fields u(1) , u(2) on Q, denoting their horizontal lifts to M as ũ(1) , ũ(2) , we have: V 1 ^ (2) (2) Q (2) + ∇M ũ = ∇ u L ũ , (1) ũ(1) u(1) 2 ũ 22
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^ where ∇Q u(2) denotes the horizontal lift of the vector field ∇Q u(2) on Q, and Lũ(1) ũ(2) is the u(1) u(1) d Lie derivative (or commutator) of vector field ũ(2) under ũ(1) , defined as dt (Φt )∗ ũ(2) where t=0 (1) Φt is the flow of vector field ũ . The Lie derivative adopts an explicit expression Lũ(1) ũ(2) (f ) = ũ(1) (ũ(2) (f )) − ũ(2) (ũ(1) (f )). Particularly, ^ (2) H ∇Q u(2) = (∇M ) . ũ(1) ũ u(1) C.1
T HE S HAPE S PACE R3N /SE(3)
For a concrete and practically highly concerned example, we consider the shape space R3N /SE(3). In this example, each R3N element is structured as: x = (⃗x(1) , ⃗x(2) , · · · , ⃗x(N ) ) ∈ R3N ,
with each ⃗x(n) ∈ R3 ,
which represents the 3-dimensional coordinates of N points in R3 (point cloud). The SE(3) group is composed of the 3-dimensional translation group and the 3-dimensional rotation group SO(3). Since the translation group is not compact, there does not exist a probability distribution that is translation invariant. We (as well as many others (Yim et al., 2023; Lin et al., 2024)) hence represent the quotient space w.r.t this group by suppressing these equivalent DoFs by choosing a canonical translational position by anchoring the center of mass (CoM) of the point cloud at the origin, and PN consider the resulting CoM-free subspace M◦ := {x ∈ R3N | N1 n=1 ⃗x(n) = ⃗0} 6 and consider the SO(3) action on it. Since the constraint is linear, this space M◦ is a linear subspace of R3N , and it is naturally a Riemannian manifold with the standard inner product of R3N . An element of the SO(3) group is given by a 3 × 3 rotation matrix for which we reload the notation g. The natural action of g on x is defined as g · x = g⃗x(1) , g⃗x(2) , · · · , g⃗x(N ) (g here represents the rotation matrix), i.e. the rotation is applied on each point of the system. Unfortunately, SO(3) does not act freely (see Def. 29) on M◦ in some degenerate cases, e.g. all the points lie on a straight line. So we define the subset D ⊂ M◦ that SO(3) does not have free action on it; i.e., for any x ∈ D, there exists a nontrivial action g ̸= e ∈ SO(3) such that g · x = x, indicating that D contains points that have a higher symmetry beyond SO(3). For a converging sequence {x(n) }n in D which converges to x ∈ M◦ as n → ∞, there exists a sequence {g (n) }n in G such that g (n) ·x(n) = x(n) . Since the group SO(3) is compact and the group action is continuous, {g (n) }n has a convergent subsequence that converges to g ∈ G, which satisfies g · x = x. Hence x ∈ D, and therefore, D is closed. Subsequently, M := M◦ \ D is still a smooth manifold. As D is measure-zero in M◦ (since any g ∈ SO(3) is non-singular, the equation g · x = x reduces degrees of freedom of x), it is unlikely for a real simulation in M◦ to hit the set D, making negligible difference algorithmically. By removing the degenerate set D, SO(3) can now act freely and smoothly on M. Moreover, since the SO(3) action is isometric in the Euclidean space and M inherits the same metric, SO(3) also acts isometrically on M. Since SO(3) is a compact group, by Prop. 30, the action is also proper. Now that the action is smooth, free, proper, and isometric, by Thm. 31, the quotient space Q := M/SO(3) is a Riemannian manifold and the projection π : M → Q is a Riemannian submersion. Since the each element in this quotient space Q is an equivalence class containing equivalent point-cloud configurations, we refer to this space Q as the “shape space”. By the projection mapping π : M → Q, the vertical space Vx := Ker π∗x can already be defined, which reflects the infinitesimal movements in M by group actions, which amounts to movements within the equivalence class π(x) (Appx. B). Since M is a Riemannian manifold (tangent space inner product is inherited from the standard Euclidean inner product), we can define the horizontal space Hx := (Vx )⊥Tx M as the orthogonal complement of Vx in Tx M. Since Vx and Hx recover Tx M by direct sum, any tangent vector v ∈ Tx M can thus be uniquely decomposed as the addition of a vertical component and horizontal component. On this concrete example, the vertical and horizontal spaces can be expressed explicitly. Since the vertical space is induced from group action, which acts freely, so this space is isomorphic to the 6 Here we choose a simple form of CoM by treating atoms equally weighted to avoid unnecessary notation complexity. In fact, any choice to determine one point in R3 from the N points suffices the reduction of the translation DoFs (as long as proper permutational invariance is guaranteed).
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tangent space of the Lie group, i.e., the Lie algebra. For G = SO(3), the Lie algebra so(3) is the set of antisymmetric 3 × 3 matrices. So the vertical space is given by: Vx = {(A⃗x(1) , A⃗x(2) , · · · , A⃗x(N ) ) | A ∈ so(3)}. Using the 3-dimensional representation ⃗a = (⃗a1 , ⃗a2 , ⃗a3 )⊤ ∈ R3 for so(3), any antisymmetric 3 × 3 matrix can be represented as: 3 0 −⃗a3 ⃗a2 X 0 −⃗a1 = A = ⃗a3 ⃗ai Ji , (19) 2 1 −⃗a ⃗a 0 i=1 ! ! ! 0 −1 0 0 0 1 0 0 0 (20) where J1 := 0 0 −1 , J2 := 0 0 0 , J3 := 1 0 0 . 0 0 0 −1 0 0 0 1 0 The a vector is the Euler vector (or rotation vector) representation of so(3). Its direction represents the axis of rotation, and its length represents the rate of rotation; therefore, the a vector can be thought of as an angular velocity/momentum vector. Eq. (19) suggests that {J1 , J2 , J3 } forms a basis for so(3). They are also regarded as the basis for the angular momentum operator. Following this representation, we have A⃗x(n) = ⃗a × ⃗x(n) , where “×” denotes the usual cross product on R3 , so the vertical space can also be written as: Vx = {(⃗a × ⃗x(1) , ⃗a × ⃗x(2) , · · · , ⃗a × ⃗x(N ) ) | ⃗a ∈ R3 }.
(21)
This can be thought of as the (linear) velocity on each atom induced from the common angular velocity a as if all the atoms are on a rigid body which rotates following the angular velocity a. As a rigid-body movement, this (linear) velocity does not deform the shape of the body, hence the “shape” of the point cloud is reserved, and the resulting configuration would still be within the equivalence class. This is the intuition behind the concept of the vertical space. For the horizontal space, which is the orthogonal complement of the vertical space, is given by N N o n X X ⃗v (n) = ⃗0, ⃗x(n) × ⃗v (n) = ⃗0 , (22) Hx = v = (⃗v (1) , · · · , ⃗v (N ) ) ∈ R3N n=1
n=1
since the ⃗v (n) vectors should keep the CoM fixed, and as the orthogonal complement, they should PN PN also satisfy n=1 ⃗v (n) ·(⃗a ×⃗x(n) ) = ⃗a ·( n=1 ⃗x(n) ×⃗v (n) ) = 0 for any ⃗a ∈ R3 . Intuitively, the first PN constraint n=1 ⃗v (n) = ⃗0 means the total (linear) momentum of the system is zero, meaning that the movement keeps the CoM of the point cloud, i.e., the total/rigid translational degree of freedom PN is fixed. The second constraint n=1 ⃗x(n) × ⃗v (n) = ⃗0 means the total angular momentum of the system is zero, meaning that there is no net rotation as a whole, i.e., the total/rigid rotational degree of freedom is fixed. Therefore, vectors in the horizontal space correspond to movements that does not move within an equivalence class, but purely across equivalence classes. Given the construction of the horizontal space (i.e., a “connection”), the horizontal lift for a quotientspace tangent vector (and vector field) can be derived. As SO(3) acts smoothly, freely, properly, and isometrically on M, a Riemannian structure can be induced for Q from that of M, which is inherited from the standard Euclidean metric. Since the horizontal-lifted vector field is horizontal, by the above intuition, it induces a movement purely across equivalence classes, making it a perfect correspondence to a vector field on the quotient space, and avoiding “unnecessary” movements that are within an equivalence class.
D
P ROOFS
D.1
P ROOF OF T HEOREM 1
Theorem 1’. Assume {xt }t∈[0,T ] is a diffusion process on M, specified by the following SDE: dxt = bt (xt ) dt + σt dwt ,
x0 ∼ pprior ,
(6’)
where bt is a G-equivariant time-dependent vector field on M, wt is the Wiener process on M that is G-invariant, and pprior is a G-invariant distribution. Then the projected process {yt := π(xt )}t∈[0,T ]
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onto the quotient space Q := M/G is the solution to the following SDE: σ2 dyt = (π∗ bt )(yt ) − t h(yt ) dt + σt dωt , y0 ∼ π# pprior , 2
(7’)
where: (1) π∗ bt is the pushed-forward vector field of bt induced by π, i.e., (π∗ bt )(yt ) := π∗xt bt (xt ), which is the same for any xt ∈ π −1 (yt ) due to the G-equivariance of bt ; (2) h(yt ) := PM π∗xt ( i=M −G+1 ∇ei ei ) for any xt ∈ π −1 (yt ) is the mean curvature vector at yt , where {ei }M i=1 is an orthonormal basis of Txt M such that Vxt = span{ei }M ; (3) ω is the Wiener process t i=M −G+1 on Q; and (4) π# pprior is the pushed-forward distribution of pprior , i.e., its samples can be produced by y0 = π(x0 ) where x0 ∼ pprior . Proof. As xt is a diffusion process on M given by the the SDE dxt = bt (xt ) dt + σt dwt , by Prop. 27 and Def. 22, xt is an Lt -diffusion and the generator is σt2 M ∆ . 2 M −G For any x ∈ M, let {ei }M i=1 be an orthonormal basis of Tx M such that Hx = span{ei }i=1 , M and Vx = span{ei }i=M −G+1 . Then by the Riemannian submersion construction of π : M → Q −G (see Appx. C), {ēi := π∗x ei }M is an orthonormal basis of Tπ(x) Q. Such a basis as a smooth i=1 function of x always exists in a neighborhood, so locally each ei can be viewed as a vector field. Let ∇M and ∇Q be the Levi-Civita connections on M, Q, respectively, where ∇Q is induced from the Riemannian metric inherited from M. Using the local expression of the Laplace-Beltrami operator (Def. 18), the generator is given by ! M M M σt2 X σt2 X M σ2 X 2 σt2 M M ∆ = bt + (ei (ei (·)) − ∇ei ei ) = bt − ∇ei ei + t e . Lt = bt + 2 2 i=1 2 i=1 2 i=1 i Lt = bt +
Then the process is the solution to the following Stratonovitch SDE dxt = v(0) (xt , t)dt +
M X
v(i) (xt , t) ◦ dwti ,
i=1
where
v(0) := bt −
M σt2 X
By Def. 26, for all f ∈ C ∞ (M), Z t f (xt ) = f (x0 ) +
2 i=1
∇M ei ei ,
and v(i) := σt ei for i = 1, · · · , M.
v(0) (f )(xs , s)ds +
0
M X
! v(i) (f )(xs , s) ◦ dwsi
.
i=1
Let f¯ ∈ C ∞ (Q), then f := f¯ ◦ π ∈ C ∞ (M), then ! Z t M X f¯(π(xt )) = f¯(π(x0 )) + v(0) (f¯ ◦ π)(xs , s)ds + v(i) (f¯ ◦ π)(xs , s) ◦ dwsi , 0
= f¯(π(x0 )) +
i=1
Z t (π∗ v
(0)
M X )(f¯)(π(xs ), s)ds + (π∗ v(i) )(f¯)(π(xs ), s) ◦ dwsi
0
! ,
i=1
by Def. 9. Since f¯ is arbitrary, by Def. 26, yt := π(xt ) is the solution to: dyt = π∗ v(0) (yt , t)dt +
M X
π∗ v(i) (yt , t) ◦ dwti .
i=1
We first need to check that the projected vector field is well defined. In fact, we only need to check that π∗ b is well defined. Since b is G-equivariant, then for any g ∈ G, (Lg )∗ bt (x) = bt (g · x). Then π∗ (bt (g · x)) = π∗ ((Lg )∗ bt (x)) = (π ◦ Lg )∗ (bt (x)) = π∗ (bt (x)), where we have used the chain rule in the second-last step, and the last step holds since π ◦ Lg and π projects to the same equivalence class (x and g · x are in the same equivalence class). By a notational equivalence that π∗ (bt (x)) = π∗ bt (π(x)), we know that π∗ bt (y) is the same on the equivalence class regardless of the choice of x in π −1 (y), which implies that the projected vector field π∗ bt is well defined. 25
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Next, we calculate the expression of the projected vector field. Since Hx = span{e1 , · · · , eM −G }, Vx = span{eM −G+1 , · · · , eM }, we have ēi , if i ⩽ M − G, π∗x ei = 0, if i ⩾ M − G + 1, so π∗x (v(i) ) = σt ēi for i = 1, · · · , M − G and π∗x (v(i) ) = 0 for i ⩾ M − G + 1. Moreover, since each ei for i ∈ {1, · · · , M − G} lies in the horizontal space and π∗ ei = ē by definition, by ^ ˜i = ei . By Prop. 34, which means π∗ ∇Q the uniqueness of horizontal lift, we know ē u(2) = u(1) Q ∇Q u(2) = π∗ (∇M ũ(2) )H = π∗ (∇M ũ(2) ), we know that π∗ (∇M ei ei ) = ∇ēi ēi . Therefore, ũ(1) ũ(1) u(1) for the drift term, we have: M
π∗ v(0) (y, t) = π∗ bt (y) −
σt2 X π∗ (∇M ei ei ) 2 i=1 M −G
= π∗ bt (y) −
= π∗ bt (y) −
= π∗ bt (y) −
σt2 X σt2 π∗ (∇M e ) − i ei 2 i=1 2 M −G σt2 X
2
i=1
M −G σt2 X
2
∇Q ēi ēi − ∇Q ēi ēi −
i=1
σt2 2
π∗ (∇M ei ei )
i=M −G+1
M X
σt2 2
M X
π∗ (∇M ei ei )
i=M −G+1
h(y).
So the generator of the process yt is M −G M −G σt2 X Q σt2 σt2 X 2 Ls = π∗ bt − ∇ ēi − h+ ē 2 i=1 ēi 2 2 i=1 i ! M −G M −G X σt2 σt2 X 2 Q = π∗ bt − h + ēi − ∇ēi ēi 2 2 i=1 i=1 σ2 σ2 = π∗ bt − t h + t ∆Q . 2 2
(23)
Then we can conclude that the projected process yt := π(xt ) is the solution to the following SDE σ2 dyt = (π∗ bt )(yt ) − t h(yt ) dt + σt dωt , 2 where π∗ bt is the push-forward vector field, h(yt ) is the mean curvature vector at yt and ωt is the standard Wiener process on the quotient space Q.
D.2
P ROOF OF T HEOREM 2
In Def. 32, we define the horizontal lift of a vector field that generates a deterministic flow. In fact, for a stochastic process on Q, we can define the horizontal lift for it similarly. First, we need to define the stochastic line integral, which is the integration of a one-form along the trajectory of a stochastic process. Definition 35. (Hsu, 2002, Prop. 2.4.2) Let Θ be a 1-form (Def. 11) on M and xt the solution to the equation dxt = v(0) (xt , t)dt +
D X
v(i) (xt , t) ◦ dwti .
i=1
Then Z t
Z t Θxs ds =
0
Θ(v(0) )(xs ) ds +
0
Z tX D 0 i=1
26
Θ(v(i) )(xs ) ◦ dwsi .
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Definition 36. (Baudoin et al., 2024, Def. 3.1.9) A semimartingale (xt )t on M is called horizontal Rt if for every 1-form Θ on M whose kernel contains the horizontal space H, one has 0 Θxs ds = 0, for all t ⩾ 0. Let (yt )t be a semimartingale on Q such that y0 is a point of Q. Then for a given starting point x0 ∈ π −1 (y0 ), there exists a unique horizontal semimartingale xt on M such that xt starts from x0 and π(xt ) = yt for all t ⩾ 0. This process (xt )t is called the horizontal lift of (yt )t from x0 . Theorem 2’. The horizontal lift of Eq. (7) has the following explicit expression: σt2 dx̃t = Px̃t (bt (x̃t )) − h̃(x̃t ) dt + σt dw̃t , x̃0 ∼ pprior , (8’) 2 where Px (v) := vH is the horizontal projection in the tangent space of M, h̃ is the horizontal lift of the mean curvature vector, and w̃t is the horizontal lift of the Wiener process of Q. Proof. We only need to check the definition of the horizontal lift (Def. 36). Again, for any M −G x ∈ M, let {ei }M i=1 be an orthonormal basis of Tx M such that Hx = span{ei }i=1 , and M Vx = span{ei }i=M −G+1 . Then by the Riemannian submersion construction of π : M → Q −G (see Appx. C), {ēi := π∗x ei }M is an orthonormal basis of Tπ(x) Q. Such a basis as a smooth i=1 function of x always exists in a neighborhood, so each ei can be viewed as a vector field in each neighborhood. Let ∇M and ∇Q be the Levi-Civita connections on M, Q, respectively, where ∇Q is induced from the Riemannian metric inherited from M. Now we calculate the generator of the SDE in Eq. (8’). From Eq. (23), the Wiener process ωt PM −G on the quotient space Q has generator 21 ∆Q = 12 i=1 ē2i − ∇Q ē . Since each ei for i ∈ i ēi {1, · · · , M − G} lies in the horizontal space and π∗ ei = ē by definition, by the uniqueness of ˜i = ei . By Prop. 34, we know that the generator of the horizontal-lifted horizontal lift, we know ē PM −G H Wiener process w̃t has generator 21 i=1 e2i − (∇M . So the generator of the horizontal ei ei ) lifted process in Eq. (8’) is: M −G σt2 σt2 X 2 M H L̃t = bH − h̃ + e − (∇ e ) . (24) i t i ei 2 2 i=1 Its projection under π∗ is given by: M −G σt2 X 2 σt2 h + ēi − ∇Q π∗ L̃t = π∗ bt − ēi ēi , 2 2 i=1 which coincides with the generator of Eq. (7) (see Eq. (23)). So the process π(x̃t ) is the same diffusion process as yt , where yt is defined in Eq. (7). Let Θ be a 1-form on M whose kernel contains the horizontal space H everywhere. From Eq. (24), x̃t is the following SDE dxt = v(0) (xt , t)dt +
M X
v(i) (xt , t) ◦ dwti ,
i=1
where v(0) =
M −G σ2 X H H h̃ − t (∇M bt − ei ei ) , 2 2 i=1 σt2
v(i) = σt ei .
Then the line integral Z t Θxs ds = 0
Z tX M
Θ(v(i) )(x̃s ) ◦ dwsi = 0,
0 i=0
since v(i) ∈ H and Θ(v(i) ) = 0. So we can conclude that x̃t is the horizontal lift of yt . Corollary 3’. The resulting state x̃1 by the lifted diffusion process Eq. (8) and the resulting x1 by the original diffusion process Eq. (6) follow the same distribution in the total space. When σt ≡ 0 and starting from the same x0 ∈ M, Eq. (8) leads to a shorter trajectory than Eq. (6) does: Z 1 Z 1 ⟨Px̃t (bt (x̃t )), Px̃t (bt (x̃t ))⟩ dt ⩽ ⟨bt (xt ), bt (xt )⟩ dt. 0
0
27
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Proof. Since x0 and x̃0 follow the same G-invariant distribution pprior , and by Thm. 1 and that the lifted process projects to the same quotient-space process before the lift, we know that π(xt ) follows the same diffusion process as π(xt ) on Q. Since bt is G-equivariant and the Wiener process is Ginvariant, we know that the transition kernel p(x1 | x0 ) is G-equivariant. Together with that pprior is G-invariant, we know that p(x1 ) is G-invariant (Xu et al., 2022, Prop. 1). On the side of the lifted quotient-space diffusion process, since the lifted process does not have a vertical movement, i.e., the process effectively does not apply any group action, we know that p(x̃1 | x̃0 ) is also G-equivariant, and together with the G-invariance of pprior , we know that p(x̃1 ) is G-invariant as well. Since p(x1 ) and p(x̃1 ) project to the same distribution on Q and they are G-invariant, we know that they are the same distribution on the total space M. dx̃t t When σt ≡ 0, the process becomes deterministic and solves the ODEs dx dt = bt (xt ) and dt = Px̃t (bt (x̃t )). Since they start from the same point x0 = x̃0 , and the vector fields bt and Px̃t (bt (x̃t )) differ only in a vertical movement, i.e., movement by a group action, we know that there exists gt ∈ G such that xt = gt · x̃t . By leveraging the G-equivariance of bt and the isometry of group action, the lengths of the two processes can then be related by: Z 1 Z 1 ⟨bt (xt ), bt (xt ))⟩xt dt = ⟨bt (gt · x̃t ), bt (gt · x̃t ))⟩xt dt 0
0
Z 1 ⟨(Lgt )∗x̃t bt (x̃t ), (Lgt )∗x̃t bt (x̃t ))⟩gt ·x̃t dt
= 0
Z 1 ⟨bt (x̃t ), bt (x̃t ))⟩x̃t dt =
= 0
Z 1
H
0
0
bt (x̃t )H , bt (x̃t )H ) x̃t + bt (x̃t )V , bt (x̃t )V ) x̃t dt
Z 1
bt (x̃t ) , bt (x̃t ) ) x̃t dt =
⩾
D.3
H
Z 1
0
⟨Px̃t (bt (x̃t )), Px̃t (bt (x̃t ))⟩x̃t dt.
P ROOF OF T HEOREM 4
Theorem 4’. Assume xt is a diffusion process in the CoM subspace M ⊂ R3N , given by the following SDE: dxt = bt (xt ) dt + σt dwt , where bt (xt ) is an SO(3)-equivariant vector field ∀t ∈ [0, T ], wt is the standard Wiener process on CoM. The horizontal lift of the process π(xt ) is given by the following SDE: σ2 (9’) dx̃t = Px̃t (bt (x̃t )) − t h̃(x̃t ) dt + σt Px̃t dwt , 2 where the Px is the horizontal projection operator at x and h̃(x) is the horizontal lift of mean curvature vector. The explicit expressions of P and h̃ are shown as follows: ! ! N X ′ ′ Px (v) = ⃗v (n) − K−1 ⃗x(n ) × ⃗v (n ) × ⃗x(n) , ∀v = (⃗v (1) , · · · , ⃗v (N ) ) ∈ Tx M, and n′ =1
n
h̃(x) = − tr(K−1 )I − K−1 ⃗x(n) , n
where K :=
N X n=1
∥⃗x(n) ∥2 I −
N X
⃗x(n) ⃗x(n)
⊤
∈ R3×3 ,
n=1
and (·)n means concatenating the 3-dimensional vectors by running through n. Proof. We unroll the proof by first deriving the general expression Eq. (9’) in the case of Euclidean total space, then deriving the expression for the projection operator Px (v) for G = SO(3), and the horizontal-lifted mean curvature vector field h̃(x) for G = SO(3). Expressions for the SO(3) case are expressed in the total space M defined in Appx. C.1, which can be embedded in the Euclidean space R3N . The general expression Eq. (9’). Again, for any x ∈ M, let {ei }M i=1 be an orthonormal basis −G M of Tx M such that Hx = span{ei }M , and V = span{e } x i i=M −G+1 . Then by the Riemannian i=1 28
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−G submersion construction of π : M → Q (see Appx. C), {ēi := π∗x ei }M is an orthonormal basis i=1 of Tπ(x) Q. Such a basis as a smooth function of x always exists in a neighborhood, so each ei can be viewed as a vector field in each neighborhood. Let ∇M and ∇Q be the Levi-Civita connection on M, Q, respectively, where ∇Q is induced from the Riemannian metric inherited from M.
As shown in Appx. D.2, Eq. (24), the horizontal lift of Eq. (8) has the generator M −G σt2 σ2 X 2 H H Lt = bt − . h̃ + t e − (∇M ei ei ) 2 2 i=1 i V = 0 for i = Since Lei ei = ei (ei (·)) − ei (ei (·)) = 0, by Prop. 34, we know that (∇M ei ei ) {1, · · · , M − G}, so M −G σt2 σt2 X 2 Lt = bH − h̃ + ei − (∇M t ei ei ) . 2 2 i=1 PM j j Since M is embedded in a Euclidean space, we have ∇M ei ei = j=1 ei (ei )∂j , where ei is the j-th component of ei and ∂j := ∂/∂xj is the derivative w.r.t a standard coordinate system of the Euclidean space (i.e., these {∂j }M j=1 are a standard orthonormal basis frame for tangent spaces of the Euclidean space, which are isometrically isomorphic to the Euclidean space itself). Since bH t (x) = Px bt (x), then the generator becomes M −G σt2 X 2 σt2 e − (∇M h̃(x) + Lt = bH (x) − ei ei ) t 2 2 i=1 i M −G M σt2 σ2 X X j ei (∂j eki )∂k + eji eki ∂j ∂k − eji (∂j eki )∂k = Px bt (x) − h̃(x) + t 2 2 i=1 j,k=1
M −G M σt2 σ2 X X j k = Px bt (x) − h̃(x) + t ei ei ∂j ∂k 2 2 i=1 j,k=1
M σ2 X σ2 (Px )jk ∂j ∂k = Px bt (x) − t h̃(x) + t 2 2 j,k=1
M σ2 X σ2 (Px Px ⊤ )jk ∂j ∂k , = Px bt (x) − t h̃(x) + t 2 2 j,k=1 PM −G where we have defined Px := i=1 ei e⊤ i as the projection operator. Then Lt is the generator of σ2 dx̃t = Px̃t (bt (x̃t )) − t h̃(x̃t ) dt + σt Px̃t dwt . 2 Now we deduce the explicit expressions for the case when G = SO(3).
The expression for the projection operator Px (v) for G = SO(3). Recall from Appx. C.1, the tangent space Tx M of M at x can be explicitly expressed as (Eqs. (21, 22) ): • The vertical tangent space Vx : Vx = {(⃗a × ⃗x(1) , ⃗a × ⃗x(2) , · · · , ⃗a × ⃗x(N ) ) | ⃗a ∈ R3 }. • The horizontal space Hx , which is the orthogonal complement of the vertical space: N N n o X X Hx = v = (⃗v (1) , · · · , ⃗v (N ) ) ∈ R3N ⃗v (n) = ⃗0, ⃗x(n) × ⃗v (n) = ⃗0 . n=1
29
n=1
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The horizontal projection mapping is defined by Px (v) = vH = v − vV , ∀v ∈ Tx M, and we can PN H find an explicit expression of it. By definition, n=1 ⃗x(n) × ⃗v (n) = ⃗0, then N X
⃗x(n) × ⃗v (n) =
n=1
N X
V
⃗x(n) × ⃗v (n) .
n=1
Assume vV = (⃗a × ⃗x(1) , ⃗a × ⃗x(2) , · · · , ⃗a × ⃗x(N ) ), then N X
⃗x(n) × ⃗v (n) =
n=1
N X
V
⃗x(n) × ⃗v (n) =
n=1
N X
⃗x(n) × (⃗a × ⃗x(n) )
n=1
N D E D E X = ⃗x(n) , ⃗x(n) ⃗a − ⃗x(n) , ⃗a ⃗x(n) = n=1
N X
∥⃗x
(n) 2
∥ I−
n=1
N X
⃗x
(n) (n) ⊤
⃗x
! ⃗a,
n=1
where we have used the identity ⃗x(n) × (⃗a × ⃗x(n) ) = ⃗x(n) , ⃗x(n) ⃗a − ⃗x(n) , ⃗a ⃗x(n) . By denoting K :=
N X
∥⃗x(n) ∥2 I −
n=1
N X
⊤
⃗x(n) ⃗x(n) ,
(25)
n=1
PN we have ⃗a = K−1 ( n=1 ⃗x(n) × ⃗v (n) ), and V
v = (⃗a × ⃗x
(n)
)n =
−1
K
N X
! ⃗x
(n′ )
× ⃗v
(n′ )
! × ⃗x
n′ =1
(n)
, n
where the last cross-product is applied on each 3-dimensional coordinate vector ⃗x(n) . Henceforth, we have: ! ! N X V (n) −1 (n′ ) (n′ ) (n) Px (v) = v − v = ⃗v − K ⃗x × ⃗v × ⃗x , ∀v ∈ Tx M. n′ =1
n
The expression for the horizontal-lifted mean curvature vector field h̃(x) for G = SO(3). (a) We start with a characterization of the horizontal-lifted mean curvature vector field h̃(x) in the general case. As we have mentioned, h̃(x) reflects the effect of the change of the volume of the equivalence class, which helps us derive an explicit expression of it. The volume can be defined by a Riemannian structure. As each equivalence class can be seen to be induced by the Lie group, we first establish a relation between the two spaces. When the Lie group G acts smoothly, freely, properly, and isometrically on M, we can define a mapping Φx : G → M, g 7→ g · x which identifies the Lie group to the equivalence class π(x) that x is in. This Φx is a bijection since freeness indicates that if g · x = g ′ · x, then g = g ′ . Using this bijection, we can induce structures on G from those on M. Firstly, a Riemannian structure G can be defined. Let {g i }G i=1 be a coordinate chart of G, and {gi }i=1 be the induced frame on G, i.e., G {gi }i=1 is a basis set of Tg G. Then a Riemannian metric can be defined by: G,x
Gxij (g) := ⟨gi , gj ⟩g
:= ⟨Φx∗g gi , Φx∗g gj ⟩M Φx (g) .
(26)
The Riemannian structure defines a measure on G in the form of the volume form (i.e., top-ranked (rank-G) exterior form on G), which can be explicitly expressed by: p Volx (g) := det(Gx (g)) dg 1 ∧ · · · ∧ dg G , ∗ i i i where {dg i }G i=1 is a basis set of Tg G and dg (gj ) = gj (g ) = δj . Through the embedding mapping x Φ , the volume form can also be seen as a measure on the equivalence class π(x) through x.
The relation between the horizontal-lifted mean curvature vector field and the volume of equivalence class lies in the effect under time variation. For this, we first define how a spatial function can be extended to vary in time (unnecessarily uniquely). Definition 37. Let Φ : G → M be an immersion. A smooth variation of Φ is a smooth mapping Φ· (·) : (−ϵ, ϵ) × G → M satisfying: • For any t ∈ (−ϵ, ϵ), Φt (·) is an immersion; 30
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• Φ0 (·) = Φ(·). The relation is explicitly given by the following conclusion: Proposition 38. (First variation of volume (Chavel, 1995, Exercise. III.14)) Let Φxt be a smooth variation of Φx . Define the corresponding time-dependent Riemannian metric tensor Gxt,ij (g) := p x ⟨(Φxt )∗g gi , (Φxt )∗g gj ⟩M det(Gxt (g)) dg 1 ∧ · · · ∧ dg G the corresponding Φx (g) , and Volt (g) := t
volume form. Then the mean curvature vector h̃(x) satisfies the following formula: Z Z d Volxt (g) = − ⟨h̃(Φx0 (g)), vΦx0 (g) ⟩Volx0 (g), dt t=0 G G
(27)
where vΦx0 (g) := γ̇0 for which (γt )t is the curve γt = Φxt (g). This proposition formalizes the intuition that the mean curvature vector represents the R effect of volume change of equivalence class. The left hand side represents the change rate of G Volxt (g), i.e., the volume of the equivalence class through x, and the equality on the right hand side indicates that this change is given by the projection of the equivalence-class movement v in the direction of the mean curvature vector h̃(x). Particularly, as the mean curvature vector h̃(x) is horizontal everywhere, if the movement v is vertical (almost) everywhere, then the volume does not change, which matches the intuition. (b) Before diving into the G = SO(3) case, we first derive a general explicit expression for h̃(x) under a common construction that common Lie groups have, including SO(3). Let the coordinate system of G be defined in a left-invariant way, i.e., gi |g = (Lg )∗e gi |e . x ′ Then consider the mapping ψg (g ) := (gg ′ ) · x, which leads to two equivalent expressions from its differential map. The first leverages ψgx (g ′ ) = Φx (Lg (g ′ )) and gives (ψgx )∗g′ g = Φx∗gg′ (Lg )∗g′ g for any g ∈ Tg′ G. The second leverages ψgx (g ′ ) = g · (g ′ · x) = Lg (Φx (g ′ )) and gives (ψgx )∗g′ g = (Lg )∗(g′ ·x) Φx∗g′ g, which indicate that: Φx∗ (Lg )∗ = (Lg )∗ Φx∗ . By further noting the isometry of the group action on M, the metric defined in Eq. (26) can be simplified as: M
x x Gxij (g) =⟨Φx∗g gi , Φx∗g gj ⟩M Φx (g) = Φ∗g (Lg )∗e gi |e , Φ∗g (Lg )∗e gj |e g·x M
M
=⟨(Lg )∗x Φx∗e gi |e , (Lg )∗x Φx∗e gj |e ⟩g·x = ⟨Φx∗e gi |e , Φx∗e gj |e ⟩x = Gxij (e),
(28)
which turns out to be constant over G. This turns the left-hand-side of Eq. (27) simplified to: Z p ∂t=0 det(Gxt ) dg 1 ∧ · · · ∧ dg G ZG q p = ∂t=0 log det(Gxt ) det(Gx0 ) dg 1 ∧ · · · ∧ dg G G Z p =∂t=0 log det(Gxt ) Volx0 (g) G p =V0x ∂t=0 log det(Gxt ), R where Vtx := G Volxt (g) is the volume of the equivalence class π(x) induced from Φxt . For the time-dependent embedding Φxt (g), we consider a special type that is induced from an leftinvariant vector field v on the total space M: Φxt (g) := Expg·x (tvg·x ), for t around zero. This definition indicates that γ̇0 in Eq. (27) is: ∂t=0 Φxt (g) = vΦx0 (g) = vg·x = (Lg )∗x vx ,
(29)
which is left-invariant by definition. Moreover, as a horizontal-lifted vector field, h̃(x) is naturally left-invariant (see Eq. (17)). Together with the isometry of the group action on M, this indicates
31
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that the right-hand-side of Eq. (27) can be simplified to: Z − ⟨h̃(Φx0 (g)), vΦx0 (g) ⟩Φx0 (g) Volx0 (g) ZG Z = − ⟨(Lg )∗x h̃(x), (Lg )∗x vx ⟩g·x Volx0 (g) = − ⟨h̃(x), vx ⟩x Volx0 (g) G G Z x x = − ⟨h̃(x), vx ⟩x Vol0 (g) = −V0 ⟨h̃(x), vx ⟩x . G
p In this case, Eq. (27) can be simplified to V0x ∂t=0 log det(Gxt ) = −V0x ⟨h̃(x), vx ⟩x , which indicates: 1 ⟨h̃(x), vx ⟩x = − ∂t=0 log det(Gxt ), ∀x ∈ M. 2 To further simplify the expression, next we will prove the following conversion from the time derivative to spatial derivative, i.e., ∂t=0 log det(Gxt ) = ⟨∇x log det(Gx0 ), vx ⟩x , henceforth h̃(x) = − 12 ∇x log det(Gx ). Firstly, using Jacobi’s formula in matrix calculus, we have ∂t=0 log det(Gxt ) = tr Gxt −1 ∂t=0 Gxt , and by Eq. (28), we have: M
M
∂t=0 (Gxt )ij = ⟨∂t=0 (Φxt )∗e gi , (Φx0 )∗e gj ⟩x + ⟨(Φx0 )∗e gi , ∂t=0 (Φxt )∗e gj ⟩x . We next convert the time derivative ∂t=0 (Φxt )∗e gi into spatial derivative by leveraging the leftinvariance of v that defines Φxt . Consider a general g ∈ G and g ∈ Tg G. From the definition of a tangent vector, for any test function f around Φx0 (g) = g · x, we have: (Φxt )∗g g (f ) = g f (Φxt (·))|g = gi ∂gi f (Φxt (·))|g = gi (∂xα f |g·x )∂gi Φxt (g)α = gi ∂gi Φxt (g)α ∂xα (f ), where we have used Einstein’s summation convention where summations over indices repeated in the superscript and subscript are implied. Taking the derivative w.r.t t and by leveraging Eq. (29), we have: ∂t=0 (Φxt )∗g g = gi ∂gi (∂t=0 Φxt (g)α )∂xα α α = gi ∂gi vg·x ∂xα = gi ∂gi (Lg )∗x vx ∂xα ∂Lα ∂Lα g g = gi ∂gi vxβ ∂xα = gi ∂xβ vβ ∂xα β ∂x x ∂g i x x α ∂Lα g = vxβ ∂xβ gi ∂xα = vxβ ∂xβ (Φx0 )∗g g ∂xα , ∂g i x which converts the time derivative to spatial derivatives by exchanging the order of differentiating g · x = Lg (x) = Φx0 (g) w.r.t x and g. By leveraging the linearity of inner product, we know M M ⟨∂t=0 (Φxt )∗e gi , (Φx0 )∗e gj ⟩x = vxβ ∂xβ (Φx0 )∗e gi , (Φx0 )∗e gj x , so: M
M
∂t=0 (Gxt )ij = vxβ ∂xβ (Φx0 )∗e gi , (Φx0 )∗e gj x + (Φx0 )∗e gi , vxβ ∂xβ (Φx0 )∗e gj x
= vxβ ∂xβ ⟨Φx∗e gi , Φx∗e gj ⟩x = vxβ ∂xβ (Gx0 )ij , hence ∂t=0 log det(Gxt ) = tr Gx0 −1 ∂t=0 Gxt = tr Gx0 −1 vxβ ∂xβ Gx0 = vxβ tr Gx0 −1 ∂xβ Gx0 = vxβ ∂xβ log det(Gx0 ) = ⟨∇x log det(Gx0 ), vx ⟩x . Therefore, the mentioned expression indeed holds: 1 (30) h̃(x) = − ∇x log det(Gx ). 2 (c) Finally, we derive the expression for the specific case of G = SO(3). Its embedding into M through x ∈ M is given by Φx (g) := g · x = gx, where in the last expression, g is meant to be its 3 × 3 rotation-matrix form, and gx := (g⃗x(n) )n denotes a set of matrix-vector products. Its push-forward mapping is given explicitly as Φx∗g g = gx, where g ∈ Tg G also takes its 3 × 3 matrix form. As indicated by Eq. (19), {J1 , J2 , J3 } as defined in Eq. (20) forms a basis for so(3), i.e., the tangent space at e = I, which also induces a basis {gJ1 , gJ2 , gJ3 } for Tg G, and this basis frame is left invariant, satisfying the construction above. Therefore, the asked basis frame {gi }i at g can be constructed by gi = √12 gJi . Then the Riemannian metric tensor G is given explicitly by: 32
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M
Gxij (g) = ⟨gi x, gj x⟩gx =
N 1 1 1 X (n) ⊤ (gJi x)⊤ (gJj x) = x⊤ J⊤ J x = ⃗x Ji Jj ⃗x(n) . i j 2 2 2 n=1
⃗ ⃗⊤ ⃗ To proceed, one can easily verify that J⊤ i Jj = δij I − 1j 1i , where 1i denotes the 3-dimensional one-hot vector where 1 is placed at the i-th dimension. Leveraging this expression, we have: 1 Gx (g) = K(x) =: Gx , 2 PN PN ⊤ where K(x) := n=1 ∥⃗x(n) ∥2 I − n=1 ⃗x(n) ⃗x(n) is the same as defined in Eq. (25). Note that this expression indeed does not depend on g. From Eq. (30), we know: 1 1 h̃(x) = − ∇x log det Gx = − ∇x log det K(x). 2 2 Using Jacobi’s formula d log det G = tr(K−1 dK) again, we have: N N X X ⊤ ∂K (n) i ⃗1i ⃗x(n) ⊤ + ⃗x(n)⃗1i ⊤ ) , K := ∥⃗x(n) ∥2 I − ⃗x(n) ⃗x(n) , I − ( = 2⃗ x i ∂⃗x(n) n=1 n=1 where ⃗1i ∈ R3 is a one-hot vector at i. Therefore, i ⊤ −1 ∂K tr K = 2 tr(K−1 )⃗x(n) − 2⃗1i K−1 ⃗x(n) , i (n) ∂⃗x and finally we get the expression for the mean curvature vector field for SO(3): 1 = −(tr(K−1 )I − K−1 )⃗x(n) . h̃(x) = − ∇⃗x(n) log det G n 2 n
E
T RAINING AND S AMPLING M ETHOD IN G ENERAL C ASE
Training Objective The diffusion model on the total space M is trained by the denoising score matching objective. Since the vertical components of the velocity are not strictly needed, we propose to supervise the model only on the horizontal components and allow arbitrary vertical output of the model. Recall that the horizontal projection operator Px projects a vector to its horizontal component, i.e., Px (v) = vH . Thus the improved training objective is given by 2 L(θ) := Ep(t) w(t)E(x0 ,x1 )∼pjoint ,ϵ∼N (0,I) Pxt vθ (xt , t) − (α̇t x0 + β̇t x1 + γ̇t ϵ) .
ODE Sampler After the training stage, Pxt (vθ (xt , t)) is an approximation of the ground truth vector field in the horizontal subspace. For the deterministic sampler, we need to simulate the horizontal lift of the projected ODE, which is given by dxt = Pxt v(xt , t) dt. dt In practice, the ODE process is approximated by numerical solvers, e.g. the Euler method and Runge-Kutta methods.
SDE Sampler For the stochastic sampler, we need to simulate the horizontal lift of the projected original SDE in Eq. (3). According to Thm. 1 and Thm. 4, the lifted process is given by p dxt = Pxt (vθ (xt , t) + gt sθ (xt , t)) dt + γηt h(xt ) dt + 2γηt Pxt dwt , where we introduce the hyperparameter γ for protein generation following Geffner et al. (2025). The training and sampling algorithm is summarized in Algorithm 2 and 3. 33
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Algorithm 1 Training for pprior = N (0, I)
Algorithm 2 Training for general pprior
1: repeat 2: (x0 , x1 ) ∼ pjoint 3: t ∼ pt 4: xt = α̂t x0 + βt x1 5: Take a gradient descent step on ∇θ w(t) ∥Pxt (Dθ (xt , t) − x1 )∥2 6: until converged
1: repeat 2: (x0 , x1 ) ∼ pjoint , ϵ ∼ N (0, I) 3: t ∼ pt 4: xt = αt x0 + βt x1 + γt ϵ 5: vt = α̇t x0 + β̇t x1 + γ̇t ϵ 6: Take a gradient descent step on ∇θ w(t) ∥Pxt (vθ (xt , t) − vt )∥2 7: until converged
Algorithm 3 Sampling 1: x0 ∼ pprior 2: for i = 0 to K − 1 do 3: ∆ti = ti+1 − ti 4: if ODE sampling then 5: xti+1 = xti + Pxti vθ (xti , ti )∆ti 6: end if 7: if SDE sampling then 8: di = Pxti (vθ (xti , ti ) + ηti sθ (xti , ti )) + γgti h̃(xti ) 9: ϵ ∼ N (0, I) p 10: xti+1 = xti + di ∆ti + 2γηti ∆ti Pxti ϵ 11: end if 12: end for
F
A DDITIONAL E XPERIMENTAL R ESULTS
F.1
E FFICIENCY AND C OMPLEXITY A NALYSIS
Complexity analysis. In this subsection, we give a detailed discussion on the computational cost of our method. As mentioned in Thm. 4, we need to compute the inversion of the matrix K and the cross product for the horizontal projection operator Px and the mean curvature vector h̃(x). For the calculation of K−1 , notice that K is always a 3 × 3 matrix, so construction cost of K−1 is only linear O(N ), where N is the number of atoms (linear O(N ) cost for constructing K, and constant O(1) cost for inversion). The cross product is conducted atom-wise, so its computational cost is also linear O(N ). So we can conclude that the overall computational complexity is O(N ) for both Px and h̃(x). We would like to mention that the alignment operation adopted in the heuristic alignment-based diffusion strategies also has the same complexity. To see this, for aligning x ∈ R3×N towards y ∈ 1 R3×N , the Kabsch-Umeyama algorithm constructs the optimal rotation matrix as (H⊤ H) 2 H−1 , ⊤ 3×3 where H := yx ∈ R requires a linear O(N ) cost. In practice, the O(N ) computational cost is negligible compared to the cost of gradient back-propagation through the neural network. A comparison of practical training times is shown in the following table. Table 4: Training speed comparison on QM9. Methods training speed (iters/s)
Original diffusion
GeoDiff alignment
Af3 alignment
Quotient-space diffusion
4.19
4.07
4.08
4.10
All the results are tested on a single Nvidia A100 GPU. From the results, we can see that the additional computational cost brought by the alignment and projection is negligible. Numerical stability. In our quotient-space diffusion model framework, we need to calculate the matrix inversion of K, which may have numerical issues for near-collinear systems of points. In 34
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Figure 4: Training loss vs. training epochs. We find that our training is stable in practice. practice, we add an ϵI term before conducting matrix inversion, that is, we calculate (ϵI + K)−1 in practice, where I is the 3 × 3 identity matrix. This treatment is widely adopted in algorithms facing similar situations, e.g., the practical implementation of the Kabsch-Umeyama algorithm for alignment. Our typical choice of ϵ is 1e-8, and we found that the training process is stable under this setting. We have shown the training curve of the model on the protein backbone generation task in Fig. 4, which indicates no numerical issues arise during the training process. F.2
T HE I MPLEMENTATION OF G-E QUIVARIANT V ECTOR F IELD
In Thm. 4, we require that the vector field is SO(3)-equivariant. In practice, this can be implemented by using a SO(3)-equivariant network architecture or applying data augmentation. In this subsection, we justify that both of these choices are valid, such that the diffusion model can generate a SO(3)-invariant distribution.
Diffusion model with data augmentation. The optimal solution of the Euclidean diffusion model is given by D∗θ (xt ) = E[x1 |xt ] (Song et al., 2021; Karras et al., 2022). When the data distribution is augmented by random rotation, the data distribution becomes SO(3)-invariant. Thus, the optimal diffusion model can recover the SO(3)-invariant data distribution. When the transition density p(xt |x1 ) is SO(3)-equivariant, i.e. p(xt |x1 ) = p(g · xt |g · x1 ), ∀g ∈ SO(3), the optimal network is SO(3)-equivariant. To see this, let g ∈ SO(3) be an arbitrary rotation matrix. Since D∗θ (g · xt ) = E[x1 |g · xt ], by the Bayes formula, E[x1 |g · xt ] = =
−1 Ep · x1 )] Eptarget (x1 ) [x1 p(g · xt |x1 )] (x ) [x1 p(xt |g = target 1 −1 Eptarget (x1 ) [p(g · xt |x1 )] Eptarget (x1 ) [p(xt |g x1 )]
g · Eptarget (g−1 x1 ) [g −1 x1 p(xt |g −1 · x1 )] = g · E[x1 |xt ], Eptarget (g−1 x1 ) [p(xt |g −1 x1 )]
where we use the equivariance property of the transition density to get the second equality and the invariance property of ptarget to get the third equality. Thus, we can conclude that the optimal solution under these conditions is SO(3)-equivariant. Geffner et al. (2025) also gives an empirical validation that a well-trained neural network becomes nearly equivariant even if its architecture is not equivariant.
Equivariant architecture. When the model is required to be SO(3)-equivariant, the optimal solution of the diffusion model is not E[x1 |xt ]. To figure out the optimal solution, we consider the training loss at time t. The loss function at t is given by Lt (θ) = E∥Dθ (xt , t) − x1 ∥2 Z Z = d3N x1 d3N xt p(x1 , xt ) ∥Dθ (xt , t)∥2 + ∥x1 ∥2 − 2⟨Dθ (xt , t), x1 ⟩ . The optimal solution satisfies D∗θ (xt , t) =
argmin Dθ is SO(3)-equivariant
35
Lt (θ).
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The training loss can be simplified using the equivariant constraint: Z Z 3N Lt (θ) = d x1 d3N xt p(x1 , xt ) ∥Dθ (xt )∥2 + ∥x1 ∥2 − 2⟨Dθ (xt ), x1 ⟩ Z Z Z = drt dg d3N x1 p(x1 , g · rt ) ∥Dθ (g · rt )∥2 + ∥x1 ∥2 − 2⟨Dθ (g · rt ), x1 ⟩ , M/SO(3)
SO(3)
where M is defined in Appx. C.1, and rt ∈ R3N is a representation of a quotient-space element in the original Euclidean space. Since Dθ is SO(3)-equivariant, Dθ (g · rt ) = g · Dθ (rt ), then we have Z Z Z Lt (θ) = drt dg d3N x1 p(x1 , g · rt ) ∥Dθ (rt )∥2 + ∥x1 ∥2 − 2⟨g · D1θ (rt ), x1 ⟩ . M/SO(3)
Define p(rt ) = Z Lt (θ) =
SO(3)
R SO(3)
1 ,g·rt ) . Then we have d3N x1 p(x1 , g · rt ), and p(x1 , g | rt ) = p(xp(r t) # Z Z p(rt )∥Dθ (rt )∥2 − 2⟨Dθ (rt ), dg d3N x1 p(x1 , g · rt )g −1 · x1 ⟩
dg "
drt
R
M/SO(3)
SO(3)
Z
Z
Z drt
+ M/SO(3)
dg
d3N x1 p(x1 , grt )∥x1 ∥2 .
SO(3)
So we can conclude that D∗θ (rt , t) =
Z
D∗θ (g ′ · rt ) =
Z
Z
d3N x1 p(x1 , g | rt ) g −1 · x1 ,
Z
d3N x1 p(x1 , g | rt ) g ′ · g −1 · x1 , ∀g ∈ SO(3).
dg SO(3)
dg SO(3)
Notice that D∗θ (rt ) =
Z
Z dg
d3N x1 p(x1 , g | rt ) g −1 · x1
SO(3)
d3N x1 p(g · x1 )p(g · rt | g · x1 )x1 R dg d3N x1 p(g · x1 )p(g · rt | g · x1 ) SO(3) R R dg d3N x1 p(g · x1 )p(rt | x1 )x1 SO(3) R , = R dg d3N x1 p(g · x1 )p(rt | x1 ) SO(3) R which is equivalent to the case pdata = SO(3) dg p(g · x1 ), i.e. using the augmentation by random SO(3) rotation. R
SO(3)
dg
R
= R
F.3
T RAINING AND S AMPLING ACCELERATION
In this subsection, we study the training and sampling convergence speed of different methods. For the training convergence speed comparison, we plot the generation performance measured by the precision AMR median metric with respect to the training epochs for previous heuristic alignment methods and our quotient-space diffusion model in Fig. 5(Left). We only focus on the first 100 epochs for all the methods. These models are trained with the same architecture ET-Flow (SO(3)) and training configurations on the GEOM-DRUGS dataset. The results indicate that our method achieves a similar convergence speed to the AF3 heuristic method, because both methods reduce the learning difficulty of the model, as shown in Table 1. This theoretical benefit leads to faster convergence than the GeoDiff alignment method. We also notice that the AF3 alignment method starts to get worse generation performance after 80 training epochs. This happens due to the incompatibility between the training loss and the generation performance metric, as the AF3 method is originally designed for the protein structure prediction task, which is not evaluated by distributional metrics. For the sampling convergence speed comparison, we plot the generation performance measured by the precision AMR median metric with respect to the number of function evaluations (NFE) for the sampling process in Fig. 5(Right). For all these methods trained on the GEOM-DRUGS dataset, we use the Flow Matching ODE sampler (Lipman et al., 2023) with Euler discretization. From the results, we can observe that models trained with different strategies exhibit similar convergence
36
Published as a conference paper at ICLR 2026
Figure 5: Training and sampling convergence speed comparison on GEOM-DRUGS. (Left) The relationship between training epochs and generation performance measured by the precision AMR median metric. (Right) The relationship between the number of function evaluations (NFE) for sampling and generation performance measured by the precision AMR median metric.
trends (performance gradually degrades as NFE decreases), our quotient-space diffusion framework consistently outperforms all baselines across every NFE setting. F.4
Q UOTIENT S PACE BEYOND R3N /SE(3)
Our framework can generalize to quotient spaces generated by symmetry groups beyond the special Euclidean group SE(3). Possible examples include the U(1) symmetry in quantum wavefunctions, the SU(2) symmetry in particle physics, and the SO(3) symmetry in higher (> 3) representation spaces for tasks including the mean-field electron Hamiltonian matrix prediction. In this work, we focus on the SE(3) case for its significant relevance to scientific research (Abramson et al., 2024). Applications of our framework on the mentioned more diverse systems above are left as future work.
G
E XPERIMENTS
G.1
M OLECULAR S TRUCTURE G ENERATION
This appendix summarizes our experimental setup, which strictly follows that of ET-Flow (Hassan et al., 2024). We detail the datasets, model architecture, training, sampling, and evaluation. For a more comprehensive discussion of each component, we refer the reader to the appendices of their original paper. Dataset. First, we evaluate our framework on the molecule structure generation task. In this scenario, our goal is to generate the 3D coordinates of a molecule given the graph structure of the molecule. We conduct the experiments on the GEOM datasets (Axelrod & Gomez-Bombarelli, 2022), which provide structure ensembles generated by metadynamics in CREST (Pracht et al., 2024), and we focus on the GEOM-QM9 and GEOM-DRUGS datasets. Following the data processing and splits from (Hassan et al., 2024), we use the random splits with train/validation/test of 243473/30433/1000 for GEOM-DRUGS and 106586/13323/1000 for GEOM-QM9. In addition, data with disconnected molecule graphs are removed for GEOM-DRUGS (Hassan et al., 2024). Our reproduction is based on the modified data-processing pipeline following the released configs thus different from the results reported in the original paper. Settings. We primarily follow the setting in (Hassan et al., 2024). We set the Gaussian distribution as the prior distribution on GEOM-QM9 and use the harmonic prior for GEOM-DRUGS (Volk et al., 2023). Following (Jing et al., 2022; Xu et al., 2022), we report the RMSD-based metrics, e.g. Coverage and Average Minimum RMSD (AMR) between generated and ground truth structure ensembles. We parameterize vθ by using equivariant graph transformer architectures from ETFlow (Hassan et al., 2024), including the O(3) and SO(3) equivariant variants, which also serves as a verification that our framework is compatible with different backbone models. For training, we use AdamW as the optimizer, and set the hyper-parameter ϵ to 1e-8 and (β1 , β2 ) to (0.9,0.999). We use 37
Published as a conference paper at ICLR 2026
the dynamic gradient clipping as (Hassan et al., 2024; Hoogeboom et al., 2022b). The peak learning rate is set to 5e-4 for GEOM-DRUGS and 7e-4 for GEOM-QM9. The batch size is set to 48 for GEOM-DRUGS and 128 for GEOM-QM9. The weight decay is set to 1e-8. The model is trained for 1000 epochs for both datasets. The noise scale σ is set to 0.1. We also use 50 time steps with the Euler solver for sampling. All models are trained on 8 NVIDIA A100 GPUs. Baselines. Following (Hassan et al., 2024), we choose strong baselines trained on GEOM-DRUGS and GEOM-QM9 for a challenging comparison. We report the performance of GeoMol (Ganea et al., 2021), GeoDiff (Xu et al., 2022), Torsional Diffusion (Jing et al., 2022), and MCF (Wang et al., 2023). G.2
P ROTEIN
This appendix summarizes our experimental setup, which strictly follows that of Proteı́na (Geffner et al., 2025). We detail the datasets, model architecture, training, sampling, and evaluation. For a more comprehensive discussion of each component, we refer the reader to the appendices of their original paper. G.2.1
DATASET
For training, we utilize the Foldseek AFDB clusters (DFS ) dataset as curated and described in the Proteı́na. This dataset is a high-quality, non-redundant subset of the AlphaFold Database (AFDB), containing 588,318 cluster-representative protein structures with lengths between 32 and 256 residues. The dataset is annotated with hierarchical CATH labels, which are leveraged during training. Our data processing and handling strictly follow the pipeline detailed in Appendix M of (Geffner et al., 2025). G.2.2
M ODEL A RCHITECTURE AND T RAINING
Our model architecture is the same as the efficient, non-equivariant transformer proposed by (Geffner et al., 2025). Specifically, we adopt the variant that forgoes the use of computationally expensive triangle update layers. The model is trained using the conditional flow matching (CFM) objective. Key aspects of the training protocol from Proteı́na are preserved, including their novel Beta-Uniform mixture for the time-sampling distribution p(t), the use of self-conditioning, and data augmentation with random rotations. All model and training hyperparameters, such as embedding dimensions, number of layers, attention heads, and optimizer settings, are kept consistent with hysmall perparameters saved in their released checkpoint Msmall FS . The hyperparameters for the MFS model are detailed in Table 5, in comparison with the larger models from the original Proteı́na paper. G.2.3
S AMPLING
To facilitate a direct comparison with the publicly available Proteı́na checkpoints, we trained our model with an identical hierarchical fold class conditioning mechanism. However, to ensure a fair assessment of foundational generative capabilities, all experiments reported in our main text were performed in a strictly unconditional setting. We applied the same sampling protocol across all models, using 400 sampling steps and enabling self-conditioning, which consistently improved performance. No other guidance techniques, such as autoguidance, were utilized. We use deterministic ODE sampling to assess distributional fidelity and SDE sampling to explore the designabilitydiversity trade-off. We adapt the SDE formulation and its Euler-Maruyama numerical scheme, detailed in Appendix I of (Geffner et al., 2025), for our quotient space framework, while retaining all other configurations, such as the sampling scheduler and g(t), from the original paper. G.2.4
E VALUATION
We evaluate our models rigorously adheres to the metrics established and validated in the Proteı́na paper. We assess model performance using the standard suite of metrics in protein design:
• Designability. Quantified by the self-consistency RMSD (scRMSD) protocol, using ProteinMPNN for inverse folding and ESMFold for structure prediction, with a success threshold of scRMSD less than 2Å. 38
Published as a conference paper at ICLR 2026
Table 5: Hyperparameters for Proteı́na model. Hyperparameter
MFS
Mno-tri FS
Msmall FS
Proteı́na Architecture sequence repr dim # registers sequence cond dim t sinusoidal enc dim idx. sinusoidal enc dim fold emb dim pair repr dim seq separation dim pair distances dim (xt ) pair distances dim (x̃(xt )) pair distances min (Å) pair distances max (Å) # attention heads # tranformer layers # triangle layers # trainable parameters
768 10 512 256 128 256 512 128 64 128 1 30 12 15 5 200M
768 10 512 256 128 256 512 128 64 128 1 30 12 15 — 200M
512 10 128 196 196 196 196 128 64 128 1 30 12 12 — 60M
Proteı́na Training # steps batch size per GPU # GPUs # grad. acc. steps
200K 4 128 1
360K 10 96 1
150K 5 16 1
• Diversity. Measured in two ways: by the average pairwise TM-score among designable samples, and by the number of distinct structural clusters identified by Foldseek at a TMscore threshold of 0.5. • Novelty. Assessed by calculating the maximum TM-score of each designable sample against reference structures in the PDB and AFDB databases. We also adopt the novel probabilistic metrics introduced by (Geffner et al., 2025), to measure how well our model captures the true distribution of protein structures: • FPSD. Measured the distributional similarity between generated and reference structures in the feature space of a pre-trained fold class predictor. • fS. Evaluated both the quality and diversity of samples based on the confidence and entropy of fold class predictions. • fJSD. Quantified the similarity between the categorical fold class distributions of generated and reference sets. It is noteworthy that we have omitted the Diversity and Novelty metrics from our main text to avoid comparisons with potentially inaccurate results in the literature. This decision is based on a bug recently identified in the alntmscore output of FoldSeek versions prior to v10 (release 10-941cd33), which renders many previously reported TM-based metrics incorrect (also found in (Daras et al., 2025)). To provide a controlled and accurate benchmark, we conducted our own analysis using the FoldSeek v10 (release 10-941cd33). We limited this re-evaluation to the released small Proteı́na model and our corresponding model trained in the quotient space. The full results of this comparison are summarized in Table 6.
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Published as a conference paper at ICLR 2026
Table 6: Complete performance comparison of the released Proteı́na checkpoints against our version in the quotient space. Best results are marked in bold.
Model
Designability (%)
Diversity
Novelty vs.
FPSD vs.
fS
fJSD vs.
Cluster↑
TM-Sc.↓
PDB↓
AFDB↓
PDB↓
AFDB↓
(C/A/T)↑
PDB↓
AFDB↓
SDE Sampling Msmall FS , γ = 0.35 Msmall FS , γ = 0.35 + ours small MFS , γ = 0.45 Msmall FS , γ = 0.45 + ours Msmall FS , γ = 0.50 Msmall FS , γ = 0.50 + ours
96.0 97.6 92.2 92.6 89.2 90.2
0.44 (209) 0.40 (197) 0.55 (253) 0.51 (253) 0.57 (255) 0.51 (231)
0.50 0.48 0.49 0.47 0.48 0.47
0.86 0.86 0.84 0.85 0.83 0.84
0.91 0.91 0.90 0.90 0.89 0.90
386.5 274.7 332.9 244.5 306.2 228.0
378.2 277.1 320.4 246.3 290.8 228.7
1.77/4.97/17.78 2.24/6.69/20.99 1.83/5.01/20.22 2.24/6.68/23.47 1.86/4.92/21.15 2.25/6.59/25.24
2.17 1.68 1.93 1.43 1.81 1.32
1.73 1.55 1.49 1.28 1.36 1.17
ODE Sampling Msmall FS Msmall + ours FS
13.8 15.6
0.90 (62) 0.87 (68)
0.43 0.43
0.80 0.80
0.87 0.86
83.18 69.94
21.93 17.56
2.45/5.63/31.76 2.57/6.40/32.14
0.58 0.41
0.12 0.11
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