Diffusion Generative Modeling on Lie Group Representations

We thank the reviewer PLvS for their time for reading and reviewing our manuscript and for their insightful questions and comments.

Weaknesses

• Paper's contribution and novelty. TL;DR.: We propose a novel diffusion process governed by SDEs, which operates not directly on the Lie group itself, but on its space of representations. Specifically, the diffusion takes place in the (flat) space of linear operators (i.e., matrices) induced by the group action rather than on the (curved) group manifold.

  Formally, given $x\in X$ it can be written as $x=\rho_X(g)(x_0)$ for some fixed $x_0\in X$ and $g\in G$ (in the manuscript we denoted $x_0=0$), where the above map is a matrix map, i.e., $x=\rho_X(g_x)(x_0) = \Omega(g) x_0$ where $\Omega(g)$ is a $n\times n$ matrix. For instance, for $G=SO(2)$, $g$ is the angle and $\Omega(g)=[[\cos g, \sin g], [-\sin g, \cos g]]$ the known rotation matrix in 2d. Given this setup we can explain the three approaches:

  • Lie group diffusion extracts $g$ from $x$ and performs diffusion on (the generally curved) group $G$;
  • We propose to perform diffusion in the space of $\Omega$'s for any group $G$. This matrix-valued diffusion process induces a stochastic flow on $X$ by applying the evolving matrices to points in $X$, $x_t \rightarrow \Omega_t(g) x_{t-1}$. In this way, the entire process remains within Euclidean space, as the representation elements are matrices acting on a vector space.
  • Standard diffusion consists in taking $\Omega$ to be the identity matrix (+ constant vector). This corresponds to the representation of the translation group $T(n)$, which makes it explicitly a subcase of our general formalism.

  Compared to diffusion on the Lie group itself, our method is computationally simpler and more general. It avoids the geometric and optimization challenges of curved manifolds, does not require projections, and avoids the difficulty of sampling forward SDE trajectories for non-Abelian groups, where no closed-form solutions exists (but they do in our formalism!). We plan to clarify these points in the revised manuscript following the rebuttal. Please let us know if the above explanation is helpful for the reviewer to understand our work and its novelty in the context of standard Lie group diffusion.

• Further experiments and comparison. We performed extra experiments to further increase the benchmarking evaluation of our strategy:

  • Synthetic datasets (for table of results please refer to the answer to reviewer d94q): We performed a quantitative evaluation using the Wasserstein-2 (W2) distance on the synthetic 2D/3D datasets comparing standard (Fisher) score matching ($G=T(N)$) to our proposed approach with Lie group $G=SO(N)\times\mathbb{R}^+$. One strong bias of such experiments is due to how close the prior distribution is to the target one and this affect the performance of the generative process. To mitigate this effect we report a normalized W2 metric, where we divide the W2 distance for the sampled points with the W2 distance of the priors. We noticed that GSM performs on par or better in most datasets, especially the ones where there is a clear inductive bias given by symmetry. In the MoG datasets the standard score matching (with $G=T(N)$) does perform better, which is to be expected since there is no rotation symmetry to be exploited and translation symmetry is a good inductive bias since it helps identify the center of the various Gaussian modes. We also notice that outperformance of GSM is stronger in 3d than in 2d. We postulate that as the dimension grows, the model has a harder time to just "memorize" the distribution, and the symmetry-awareness advantage also increases.

  • Further CrossDocked benchmarking (for implementation details please refer to the answer to reviewer d94q): We conducted additional experiments on CrossDocked2020 using a BBDM model, finding that while BBDM and our Lie algebra-based method achieve comparable RMSD (Lie: $2.91\pm1.05$, BBDM: $2.92\pm1.57$), BBDM produces unphysical poses due to linear interpolation in Euclidean space. Unlike our method and RSGM, BBDM fails to preserve global $SO(3)$ rotational structure, resulting in a higher rotation $\text{MAE}(D(x_0, x_0), D(\hat{x}_0, \hat{x}_0)) = 0.43 \pm 0.21$ between the two distance matrices, compared to $0.0$ for both our method and RSGM. These results demonstrate that our method combines the best of both worlds: it maintains the symmetry structure of the data without sacrificing performance, matching unconstrained models in accuracy while outperforming Lie group diffusion approaches.

  • Quantitative MNIST evaluation(please refer to the answer to reviewer d94q for table of results): We evaluated the FID as well as the classification accuracy on the generated images from both our model and the BBDM one. While our model is only slightly better on the FID, it is far superior in the classification accuracy (93% vs 80%) (the classifier is a simple network trained on the original (unrotated) MNIST dataset, whose embeddings we also extract for the FID computation). This happens because BBDM sometimes generate bridges between different classes, as it cannot enforce a strict "rigid" rotation. We showed one instance of this occurence in Figure 6 of the manuscript.

Questions

1. The reviewer is correct that our use of notation is a bit sloppy here. In line 35 it is the usual definition of a linear representation on a vector space. In line 86 we generalized this to be a general group action (thus not needing the structure of a vector space for $X$), but more importantly the map in 86 is the map in 35 applied at a specific point $x\in X$, using the fact that $GL(X) : X\rightarrow X$. To reconcile the two, these definitions relate via $\rho_X(g, x) := \rho_X(g)(x)$. We will revise the notation and clarify the text accordingly.
2. This simply follows from the fact that $GL(X) : X\rightarrow X$, and the differential of this has indeed $TX$ as the image.
3. In Figure 1 we wish to emphasize how elements in $X$ are obtained through the representation of the group action. Here $0$ is not necessarily the classical origin $0=(0,0,0,...)$ of $\mathbb{R}^n$, but it is any point in the image of the identity element $e\in G$, $0=\rho_X(e)$. $g_1$ and $g_0$ are obviously swapped in the bottom side of the figure, we thank the reviewer for catching that!
4. We appreciate the suggestion of the reviewer to make the notation clearer!
5. Thank you for catching that!
6. We did not compare our results with the works of Zhu et al. and Kong & Tao. While it is fair to cite these papers as they also approach group-aware diffusion using the structure of the Lie algebra, these methods are quite new and not been yet applied to real-world scenarios like molecule generations.
7. $\xi_A$ is defined in line 93, and as Figure 2 depicts, it corresponds to the flow on $X$ induced by the map $\rho_X$, given a flow (geodesic) on $G$ corresponding to a Lie algebra element $A$. When the conditions of section 2.2 are satisfied, $\tau$ is well-defined. It can still be that there might be an ambiguity, for instance for $SO(2)$, $\tau=\theta$ is the angle parameter, which is $2\pi$ periodic. In such cases, we establish an interval for allowed values.
8. Thank you!
9. If $X$ is not homogeneous for $G$, then $G$ cannot connect all points in $X$, and the orbits partition $X$. In such cases, our diffusion process will be restricted to the orbit of the prior initial condition, which is not the standard case where each (prior) point can be moved by the dynamics to any target point. However, the framework (score estimation, Langevin dynamics, etc.) still applies on each orbit separately.
10. Equivariant models restrict the score function to a fixed representation of the $G$, limiting its expressivity. In contrast, our method does not enforce equivariance on the score function, but it leverages the group action to guide the generative dynamics while allowing the score to be freely learned in $\mathbb{R}^n$. For instance, in $X = \mathbb{R}^2$, a strictly $SO(2)$-equivariant model cannot capture angular variation, as equivariance dictates how the score must transform under rotations. Our approach overcomes this by modeling dynamics under a product group (e.g., $SO(2) \times \mathbb{R}^+$), allowing us to capture both angular and radial dependencies. Crucially, the score function remains unrestricted: the group governs the direction of the Langevin dynamics but does not constrain the form of the learned function.
11. This is a very good question! We address it already in lines 283-284. In short, our results for Generalized Score Matching hold also for a generic curved manifold. Our Theorem 3.1 requires the flatness of $X$ since it is the ultimate goal of our work to derive a diffusion process following Lie group trajectories but all in Euclidean space. Nonetheless, it would be a very nice theoretical extension to extend our Theorem 3.1 to more generic spaces.

Limitations

In Appendix G, we discussed several limitations of our approach. Another potential challenge can lie in the flexibility of our framework. While its flexibility is actually a strength, as it enables accurate inductive bias modeling of the data structure, it can also require more extensive parameter tuning, particularly when working with novel datasets or symmetry groups as compared to standard score matching methods. We will make a separate section about the limitations.

We hope that our responses have adequately addressed the reviewer’s questions and concerns. We would be grateful if the reviewer could acknowledge the efforts we have made, particularly in generating new benchmarks, by considering to increase the overall score when updating their review. Should any issues remain unresolved, we would be more than happy to provide further clarification.

We thank the reviewer d94q for their time reading and reviewing our manuscript for their and insightful questions and comments. We are particularly pleased that the reviewer found the goal of the paper clear, theoretically sound and that the experiments demonstrate our claims.

Weaknesses

• The suggestion of the reviewer regarding the structure of Section 3 makes absolutely sense and we will implement it in the final version of the manuscript.

• Additional experiments and quantitative benchmarks

  • We include a quantitative evaluation using the Wasserstein-2 (W2) distance on the synthetic 2D/3D datasets comparing standard (Fisher) score matching ($G=T(N)$) to our proposed approach with Lie group $G=SO(N)\times\mathbb{R}^+$. One strong bias of these distributions is that all the datasets considered are in fact symmetric with respect to the origin in $\mathbb{R}^{2,3}$, as is the standard Gaussian which is the prior for the Fisher score matching models. The similarity of the prior distribution to the target one affects decisively the performance of the generating process. Indeed, both methods produce visually indistinguishable samples across all manifolds tested, as they only differ to the relative weighting given to the different modes (in the updated manuscript we will provide images of all the generated distributions). To try to at least partially mitigate this effect, we report a normalized W2 metric, obtained by dividing the W2 distance for the sampled points with the W2 distance of the priors. We noticed that GSM performs on par or better in most datasets, especially the ones where there is a clear inductive bias given by symmetry. In the MoG datasets standard score matching (GSM with $G=T(N)$) does perform better than the group $G=SO(N)\times \mathbb{R}^+$, (which is to be expected since there is no rotation symmetry to be exploited and translation symmetry is a good inductive bias since it helps identify the center of the various Gaussian modes. We also notice that outperformance of GSM is stronger in 3d than in 2d. We postulate that as the dimension grows, the model has a harder time to just "memorize" the distribution, and the symmetry-awareness advantage also increases.

  Table: Comparison of GSM ($G=SO(N)\times\mathbb{R}^+$) and standard Fisher Score matching ($G=T(N)$) on 2D and 3D synthetic datasets. Best results are in bold. When numbers are too close, we consider them on par.

  Dataset	Group	W2
  MoG (2D)	$SO(2)\times\mathbb{R}^+$ (GSM)	0.34
  MoG (2D)	$T(2)$ (standard)	$\boldsymbol{0.15}$
  Concentric Circles (2D)	$SO(2)\times\mathbb{R}^+$	$\boldsymbol{0.19}$
  Concentric Circles (2D)	$T(2)$	$\boldsymbol{0.17}$
  Line (2D)	$SO(2)\times\mathbb{R}^+$	$\boldsymbol{0.33}$
  Line (2D)	$T(2)$	0.56
  MoG (3D)	$SO(3)\times\mathbb{R}^+$	$0.40$
  MoG (3D)	$T(3)$	$\boldsymbol{0.24}$
  Torus (3D)	$SO(3)\times\mathbb{R}^+$	$\mathbf{0.14}$
  Torus (3D)	$T(3)$	0.35
  Möbius Strip (3D)	$SO(3)\times\mathbb{R}^+$	$\boldsymbol{0.06}$
  Möbius Strip (3D)	$T(3)$	0.16

• We performed additional experiments on CrossDocked2020 comparing our Lie-algebra induced generalized score matching against standard equivariant Euclidean diffusion (BBDM) using identical network architectures (please see Reviewer 2 DmbC response for implementation details). While RMSD metrics are comparable, GSM generates 3D poses that faithfully preserve $SE(3)$ transformations, whereas BBDM produces implausible final ligand poses (with deformed geometries, e.g. non-planar aromatic rings, or wrongly streched bonds) that furthermore deviate from proper global rotations. In particular, to measure this effect we computed the mean absolute error on the distance matrices, i,e., $\text{MAE}(D(x_0, x_0), D(\hat{x}_0, \hat{x}_0)) = 0.43 \pm 0.21$, while RSGM and our method achieve $0.0$ by design. Thus, our approach offers a clear advantage over other strategies, as it keeps the RMSD of unconstrained diffusion while keeping the perfect structure due to the symmetry inductive bias, and outperforming Lie group diffusion. We will include the new experiment in the final version of the manuscript. Also note that the group is $G=SE(3)$ with $\dim G = 6$ (so it is of high dimensionality in a way), and the representation on which it acts has dimensionality $3N$ where $N$ is the number of atoms in the ligand, also of very high dimension, showing scalability of the method.

• We also perform additional benchmarking in the MNIST experiment, as the reviewer suggested. In particular we evaluted the FID as well as the classification accuracy on the generated images from both our model and the BBDM one. While our model is only slightly better on the FID, it is far superior in the classification accuracy (the classifier is a simple network trained on the original (unrotated) MNIST dataset). This happens because BBDM sometimes generate bridges between different classes, as it cannot enforce a strict "rigid" rotation. We showed one instance of this occurrance in Figure 6 of the manuscript.

  Model	Average Accuracy ($\uparrow$)	Average FID ($\downarrow$)
  GSM	$\boldsymbol{0.93 \pm 0.04}$	$\boldsymbol{130.8 \pm 22.0}$
  BBDM	$0.80 \pm 0.10$	$133.4 \pm 19.0$

Questions

• In the manuscript we already presented a synthetic dataset in 4D with group $G=SO(4)\times \mathbb{R}^+$, showing that the method straightforwardly applies to higher dimensional groups. The results can be found in Figure 5h of the manuscript. In appendix A.6 we provided the necessary mathematical derivation for applying our method to $SO(N)$. We did search in the short time we had at our disposal for the rebuttal if there is some easy to implement dataset from material science, but unfortunately we did not find a standard one on which to run our method (material science does not lie unfortunately within our area of competence, and it seems it would require some further knowledge about the data and experience to find "the right questions to ask").

• Unfortunately, most of the continuous groups actions relevant for common data type are (product of) rotations ($SO(N)$), translations $T(N)$, dilations $\mathbb{R}^+$. These are also the groups that are tackled in virtually all the papers on Lie group diffusions. Other groups like the unitary groups $U(N)$ are relevant for quantum mechanics applications, but they are complex groups and the diffusion process needs to take place on a complex manifold, which has not been tackled yet systematically in the literature. This is however a very interesting direction for future work, but would be out of scope for the current manuscript.

• We thank the reviewer for raising this important point. While the formal conditions on the space $X$ and the fundamental fields $\Pi$ may initially appear technical, they are in fact natural and routinely satisfied in common settings involving Lie group actions. We explicitly verified these conditions in every experiment presented in the paper, including those involving $SO(2), SO(3), SO(4), T(N)$, and $SE(3)$.

  The results of the conditions can be formulated in a practical series of steps that the practitioner must follow/check when setting up the theoretical framework for learning on a new use case:

  • Identify the space $X$ where diffusion is performed. This is not necessarily the space where the data resides, but rather the space where the network outputs. For instance: $X$ has dimension 1 for the MNIST experiment, while dim data-space $=28 \times 28$. (Pre-condition)
  • Dimension matching of the group action: Ensure that the dimension of the image of the group action satisfies $\dim \rho_X(G) \geq \dim X$. This step is often straightforward, as it involves computing the kernel of the group action (i.e., the group elements whose action is the identity element on $X$) and ensuring the dimension of its complement matches or exceeds $\dim X$. (Condition 1)
  • Group coverage of the space: Verify that the group "covers" the entire space . For example, $T(N)$ generates all of $\mathbb{R}^N$, and $SO(2)$ generates all rotations of an image. This property is usually so obvious that it doesn’t require explicit mention. (Conditions 1 and 2)
  • Lie algebra differential operator commutators: Compute the action of the Lie algebra differential operators and check that the commutators vanish. While this calculation can sometimes be lengthy, it follows systematically from the representation of the group action. (Condition 3)

  We hope that this clarifies the intuition behind the conditions and makes it clear that they are natural and mostly very easy to satisfy and check.

We thank the reviewer for their thoughtful and constructive feedback, which has helped us improve the experimental section through the inclusion of new metrics. We hope that our additional work demonstrates our commitment to addressing the reviewer's concerns, and we would be grateful if this could be reflected in an updated review score.

Dear Reviewer d94q,

Thank you for the positive feedback and for reviewing our work. We're pleased that our responses addressed your concerns and that the new experiments demonstrated the strength of our approach.

We didn't mean to dismiss your material science suggestion, we find it quite interesting and hope other researchers will explore applying our framework to such applications.

Please feel free to ask if any other questions or remarks arise during the remaining rebuttal discussion period. Thank you again for your time and effort.

We thank the reviewer FJCA for their time reading and reviewing our paper and for their insightful questions and comments.

Weaknesses

• We performed several extra experiments to further increase the benchmarking evaluation of our strategy for three of our experiment setups. In summary:

  • Quantification of performance on the synthetic datasets (for the table of results please refer to the answer to reviewer d94q): We conducted a quantitative evaluation using the Wasserstein-2 (W2) distance on synthetic 2D and 3D datasets, comparing standard (Fisher) score matching ($G = T(2)$) with our proposed approach based on Lie groups ($G = SO(2)\times\mathbb{R}^+$ and $SO(3)\times\mathbb{R}^+$). A strong bias in such experiments arises from the similarity between the prior and target distributions. To account for this, we report a normalized W2 metric, dividing the W2 distance between samples and target by the W2 distance between target and the corresponding priors. We observe that GSM performs on par or better in most datasets, particularly where symmetry provides a clear inductive bias. In the MoG datasets, standard score matching ($G = T(N)$) outperforms the Lie group model ($G = SO(N)\times \mathbb{R}^+$), which is expected since no rotational symmetry is present, while translation symmetry effectively helps locate the Gaussian modes. The performance gap becomes even more pronounced in 3D, where GSM shows stronger advantages. We hypothesize that in higher dimensions, memorizing the target distribution becomes more difficult, and models that incorporate symmetry more explicitly benefit increasingly from this inductive bias.

  • We performed a new experiment on CrossDocked2020 (see Response to Reviewer DmbC below for details of the implementation): We evaluated a (equivariant) Brownian Bridge Diffusion Model (BBDM) on CrossDocked2020 and compared it to our Lie algebra-based method. Both achieved similar RMSD (Lie: $2.91\pm1.05$, BBDM: $2.92\pm1.57$), but BBDM suffers from unphysical poses and fails to preserve global SO(3) rotation (despite being an equivariant network), yielding a rotation $\text{MAE}(D(x_0, x_0), D(\hat{x}_0, \hat{x}_0)) = 0.43\pm0.21$ (computed between the pairwise internal distance matrices of the two point clouds) versus $0.0$ for our method, indicating that our Lie algebra-induced diffusion offers a clear advantage over standard Diffusion models in this bridging problem.

  • Quantitative MNIST evaluation: We evaluated the FID as well as the classification accuracy on the generated images from both our model and the BBDM one. While our model is only slightly better on the FID, it is far superior in the classification accuracy (the classifier is a simple network trained on the original (unrotated) MNIST dataset, whose embeddings we also extract for the FID computation). This happens because BBDM sometimes generate bridges between different classes, as it cannot enforce a strict "rigid" rotation. We showed one instance of this occurrence in Figure 6 of the manuscript. We report the evaluation in the following table (best results in bold):

    Model	Average Accuracy ($\uparrow$)	Average FID ($\downarrow$)
    GSM	$\boldsymbol{0.93 \pm 0.04}$	$\boldsymbol{130.8 \pm 22.0}$
    BBDM	$0.80 \pm 0.10$	$133.4 \pm 19.0$

• We acknowledge that Lie group and representation theory may be unfamiliar to many in the machine learning community. In response to the reviewer’s suggestion, we added a table in the appendix summarizing key notation, definitions, and intuitive explanations. The full version is available in the response to Reviewer DmbC.

• We understand the reviewer's point. We will just have one label per column to improve consistency of notation.

• We understand the reviewer’s concern regarding the real-world significance of our method, and to this regard we wish to share with the reviewer our point of view on the matter and our current/future plans. When preparing this paper, we faced a strategic choice. One option was to focus on a single high-impact application, such as molecular design or protein docking, and demonstrate competitive (out)performance in that specific context. The other was to present a more methodological contribution: a general framework grounded in rigorous theory, validated across multiple data types to illustrate its versatility and correctness. We deliberately chose the second path. Our motivation is that the relevance of symmetry and Lie group structure in data extends far beyond individual applications such as molecular conformer generation or protein docking. By establishing the theoretical foundations and practical applicability of our method in a variety of controlled settings, we aim to provide the machine learning community with a robust and general-purpose tool. While specific applications often require extensive domain-specific tuning, these can obscure the contribution of the underlying method itself. Nonetheless, we agree that a focused, real-world application would further underscore the utility of our approach. We are currently working on a follow-up study dedicated to Lie group-aware molecular generation in protein binding pockets.

Questions

• The blurriness of the images is due to a interpolation layer we added to the sampling process due to the fact that the pixels do lie on a integer grid, but the diffusion process is in continuous angle space. Unless the angle is a multiple of 90 degrees, the transformed pixels will lie outside the grid, and we necessitate an interpolation step to assign values on the original grid. Performing this procedure for each step of the diffusion process (T=10) results in this smoothness effect. To address the blurriness in visualization, we initialize the rotation angle change $\delta_\theta = 0$ and iteratively update it during each diffusion sampling step by adding the score network output (properly scaled by the diffusion scheduler) and Brownian noise. At the final timestep, we apply the accumulated rotation change $\delta_\theta$ to the original (perturbed) MNIST image. The trajectory figures will be updated in the final manuscript.

• That's a very good question. Given the technical and methodological nature of the paper, the main purpose of the experiment is to show the validity and the effectiveness of the method. A full implementation of the method for semi-flexible docking would go beyond the scope of this work, but it is indeed the topic of a follow-up focused on a very similar use-case as discussed here. With the QM9 experiment, we aim to validate our formalism (as outlined in Sections 2 and 3) by demonstrating that it holds even in the absence of an exploitable symmetry structure. We compare two group actions, $G_1 = T(3N)$ (standard diffusion) and $G_2 = (\mathbb{R} \times \mathrm{SO}(3))^N$ (generalized score matching), and show that both models solve the task equally well, with no significant differences in performance or energy. This supports our claim that, as long as the setup $(G, X, \rho)$ satisfies the conditions in Section 2.2, it can be used to learn any unconstrained, unconditional distribution, regardless of whether it aligns with specific degrees of freedom. So we choose on purpose two groups with no data-related symmetry to demonstrate the universality and correctness of the approach, rather than claiming some particular advantage in this case.

• For the Casimir, we begin by making a small clarification. We did not add such term explicitely by hand. The term arises naturally in both backward and forward SDEs when imposing the condition that they are paired, namely, that they generate the same time-marginal probability distributions $p_t(x)$ at all times $t$. In the manuscript, we described the intuitive meaning of such term, namely, that it compensates for the orbit deviation due to the curvature, since it is not obvious from just looking at the formulas. To better demonstrate its effect, we have sampled from the trained model with and without the Casimir term in the backward process for the $SO(2)$ components in the 2d synthetic datasets. We then report the average radial distribution $r=\sqrt{x_1^2 + x_2^2}$ of the sampled points and of the original dataset

  Dataset	With Casimir	Without Casimir	Original data
  GMM (2D)	8.3	9.6	8.3
  Concentric Circles (2D)	6.4	7.3	6.4
  Lines (2D)	2.0	2.5	2.0

  As we can see from the table, with Casimir the model captures perfectly (at least the mean) of the true radial distribution. The sampling with no Casimir always leads to a systematic overshooting of the radial component (we recall that the $\mathrm{SO}(2)$ Casimir is purely radial, as depicted in Figure 3 of the manuscript).

We hope that our responses have adequately addressed the reviewer’s questions and concerns. We would be grateful if the reviewer could acknowledge the efforts we have made, particularly in generating new benchmarks, by considering to increase the overall score when updating their review. Should any issues remain unresolved, we would be more than happy to provide further clarification.

Dear Reviewer FJCA, thank you again for the time and effort during this rebuttal period, and we are very pleased that you are satisfied with our new benchmarks!

For the second question, we would like to recall that the (MNIST-image) representation we are dealing with is $x_{i,j}\in \mathbb{R}^{28\times28}$ describing pixel values in a 2d grid. So $\rho_X:G \rightarrow \mathrm{GL}(\mathbb{R}^{28\times28})$, defines a $784\times 784$- dimensional matrix. This matrix is block diagonal where each $2\times 2$-dimensional diagonal box is identical, since it implies an identical rotation for all the pixels. In pixel space, we can interpret the forward SDE applying a shared Brownian noise term on all $(i,j)$ pixel values through the flow coordinate which represents the rotation angle, see Eq. (54) in the Appendix of the submitted manuscript. We evaluate three GSM variants during the diffusion sampling process. We denote $x_1$ as MNIST image that comes from the rotated prior distribution.

GSM-v0 rotates intermediate images using the (infinitesimal) change in angle at each step: $x_{k-1} = \rho_X(g_{k})x_k = \Omega(g_k)x_k$, where $\Omega(g) \in \mathrm{SO}(2)^{28\times 28} \subset GL(\mathbb{R}^{28\times28})$ describes the (matrix) representation of the rotation. This is the matrix (as mentioned before, obtained as a direct sum of identital $2\times2$ matrices applied to the individual pixels) and this matrix is applied to the image $x_t$ for each $t$. Applying consecutive group actions leads to cumulative interpolation artifacts (due to interpolation in torch.nn.functional.rotate using the BILINEAR interpolation) as each intermediate image gets rotated, yielding FID: $333.5 \pm 59.7$ and accuracy: $0.92 \pm 0.04$.

GSM-v1 accumulates angle changes in form of generated representations and applies the total rotation only at the final predicted step: $x_0 = \Omega(g_{N})\Omega(g_{N-1})\cdot \cdots \cdot \Omega(g_0)x_1 \equiv \Omega(g_{N, N-1, N-2, ..., N-k, ..,0})x_1$ where $\Omega(g_{N, N-1, N-2, ..., N-k, ..,0})$ is the product of single (infinitesimal) rotations. While this reduces interpolation artifacts compared to GSM-v0, the intermediate trajectory still involves rotated images during the diffusion sampling process (as explained above in GSM-v0), and it achieves FID: $130.8 \pm 22.0$ and accuracy: $0.93 \pm 0.04$. This version was reported in our first response in the rebuttal period.

GSM-v2 rotates intermediate images (at each timestep) relative to the prior (perturbed) image: $x_k = \Omega(g_{k-1, k-2, .., 0}) x_1$. Note that each of the single matrices $\Omega(g_k)$ are still generated from the previous state $x_{k+1}$. This approach minimizes interpolation artifacts by always rotating from the perturbed prior image $x_1$ rather than from previously rotated intermediate results, achieving the best performance with FID: $85.77 \pm 15.74$ and accuracy: $0.96 \pm 0.02$. This result will be used in our final updated manuscript.

Note that in all methods the training is completely identical. All methods involve rotating intermediate images during sampling, but differ in their rotation strategies: incremental rotations from previous states (GSM-v0), accumulated rotation applied once (GSM-v1), or progressive rotation from the prior perturbed image (GSM-v2). The progression shows clear improvement as interpolation artifacts are minimized.

In other words, as our method can be seen as a matrix-valued diffusion, where we obtain a stochastic path on $X$ by applying the diffusion representation matrix to points in $X$ (see also response to Reviwer PLvS), the different methods differ by how we perform this operation. In GSM-v0 we apply this to every step. In GSM-v2 we apply only the combined (diffused to timestep t) matrix to the point in $x$. GSM-v1 is equivalent to GSM-v0, but for visualization purposes we apply the strategy of GSM-v2 to the final image.

Updated table for Accuracy and FID scores comparing GSM against BBDM

We thank the reviewer DmbC for their time reading our manuscript and for their insightful comments, which we will address below.

• We understand that Lie group and representation theory is very technical and not common knowledge amount the machine learning community. We appreciate the reviewer's suggestion and we added a table in the appendix with the notation of the main concepts used in this work, together with their definition and a more intuitive explanation of the ideas behind them. The full table is presented here

• Additional experiments/benchmarks:

  • We perform additional experiments on CrossDocked2020 using a Brownian Bridge diffusion model (BBDM) similar to the MNIST experiment in Section 5. We sample a rotated ligand endpoint $x_T$ using the Riemannian Score-Based Generative Models (RSGM) scheduler, with the original ligand as $x_0$, and sample intermediates as $x_t= \alpha_t x_0 + (1 - \alpha_t) x_T + \sigma_t \epsilon$, where $\epsilon \sim N(0, I)$ and $m_t = t/T, ~~ \sigma_t = 2(m_t - m_t^2)$ using $T=100$ diffusion steps. As the Euclidean baseline, we train an equivariant Fisher score network with $3N$ degrees of freedom to predict the ground-truth pose $\hat{x}_0$. Similar to MNIST, intermediate BBDM samples display unphysical 3D poses due to linear interpolation and noising. We use the same network architecture as in the GSM and RSGM experiments to learn the correct $SO(3)$ rotation. Unlike existing experiments (our method and RSGM), the Euclidean BBDM in this setting attempts to learn only global $SO(3)$ rotation, neglecting translation. Since the problem is implicitly 3-dimensional but the equivariant score network predicts all $3N$ ligand atom coordinates, final samples with implausible coordinate trajectories tend to have higher energies due to unphysical poses including bond stretching, non-planar aromatic rings, and deformed rings. In terms of mean/std RMSD on the CrossDocked2020 testset, our method (Lie algebra: $2.91\pm 1.05$) is comparable with BBDM ($2.92\pm1.57$). However, since BBDM models all $3N$ atomic coordinates, the overall dynamics does not follow global SO(3) rotation, achieving $\text{MAE}(D(x_0, x_0), D(\hat{x}_0, \hat{x}_0)) = 0.43\pm0.21$, while RSGM and our method achieve $0.0$ by design. This indicates that Lie algebra induced diffusion offers a clear advantage over standard Diffusion models in that particular bridging problem. We will include the new experiment in the final version of the manuscript.

  • Synthetic datasets (please refer to the response to reviewer d94q for the full table of results): We ran a quantitative evaluation using the Wasserstein-2 (W2) distance on synthetic 2D and 3D datasets, comparing standard (Fisher) score matching ($G = T(2)$) with our proposed method based on Lie groups ($G = \mathrm{SO}(2)\times\mathbb{R}^+$ and $\mathrm{SO}(3)\times\mathbb{R}^+$). One challenge in these experiments is that the similarity between the prior and target distributions can affect the results. To reduce this bias, we report a normalized W2 metric, which divides the W2 distance between the samples and the target by the W2 distance between the prior and the target. We find that GSM performs as well or better in most datasets, especially in cases where symmetry gives a useful inductive bias. In the MoG datasets, standard score matching ($G = T(N)$) performs better than the Lie group model ($G = \mathrm{SO}(N)\times \mathbb{R}^+$), which makes sense since these data don’t have rotational symmetry, and translation symmetry helps the model find the centers of the Gaussian modes. The difference becomes even clearer in 3D, where GSM shows stronger performance. We believe that in higher dimensions, it is harder for a model to memorize the distribution, and the benefits of using symmetry-aware biases become more important.

  • Quantitative MNIST evaluation (table with numerical results can be found in the response to Reviewer d94q): We evaluated the FID as well as the classification accuracy on the generated images from both our model and the BBDM one. While our model is only slightly better on the FID, it is far superior in the classification accuracy (93% vs 80%) (the classifier is a simple network trained on the original (unrotated) MNIST dataset, whose embeddings we also extract for the FID computation). This happens because BBDM sometimes generate bridges between different classes, as it cannot enforce a strict "rigid" rotation. We showed one instance of this occurrance in Figure 6 of the manuscript.

We hope our responses have effectively addressed the reviewer’s questions and concerns. We would sincerely appreciate it if the reviewer could take into account the additional efforts made, particularly in generating new benchmarks, by updating the overall score in their review. If any issues remain unclear or need further discussion, we would be happy to provide additional clarification.