SE(3)-Equivariant Diffusion Policy in Spherical Fourier Space
Authors: Xupeng Zhu
Fan Wang
Robin Walters
Jane Shi
Venue: ICML 2025 poster (2025)
Keywords: Imitation Learning
Equivariance
Closed-Loop Policy
Manipulation
Primary area: applications->robotics
TL;DR: An SE(3) equivariant visuomotor policy based on spherical Fourier representation that is sample efficient and generalize to novel 3D arrangement of the scene.

Abstract: Diffusion Policies are effective at learning closed-loop manipulation policies from human demonstrations but generalize poorly to novel arrangements of objects in 3D space, hurting real-world performance. To address this issue, we propose Spherical Diffusion Policy (SDP), an SE(3) equivariant diffusion policy that adapts trajectories according to 3D transformations of the scene. Such equivariance is achieved by embedding the states, actions, and the denoising process in spherical Fourier space. Additionally, we employ novel spherical FiLM layers to condition the action denoising process equivariantly on the scene embeddings. Lastly, we propose a spherical denoising temporal U-net that achieves spatiotemporal equivariance with computational efficiency. In the end, SDP is end-to-end SE(3) equivariant, allowing robust generalization across transformed 3D scenes. SDP demonstrates a large performance improvement over strong baselines in 20 simulation tasks and 5 physical robot tasks including single-arm and bi-manual embodiments. Code is available at https://github.com/amazon-science/Spherical_Diffusion_Policy.

SB(3)-Equivvriam) Diffusion Policy in Spherical Fourier Space
Xupeng Zhu 12 Fan Wang Robin Walters? 2 Jane Shi
Abstract
Diffusion Policies are effective at learning closed-loop manipulation policies from human demonstrations but generalize poorly to novel arrangements of objects in 3D space, hurting real-world performance. To address this issue, we propose Spherical Diffusion Policy (SDP), an SE(3) equivariant diffusion policy that adapts trajectories according to 3D transformations of the scene. Such equivariance is achieved by embedding the states, actions, and the denoising process in spherical Fourier space. Additionally, employ novel spherical FiLM layers to condi- tion the action denoising process equivariantly on the scene embeddings. Lastly, we propose a spherical denoising temporal U-net that achieves spatiotemporal equivariance with computational efficiency. In the end, SDP is end-to-end (F) equivariant, allowing robust generalization across transformed 3D scenes. SDP demonstrates large performance improvement over strong baselines in 20 simulation tasks and 5 physical robot tasks including single-arm and bi-manual embodiments. Code is available at https: wttp:hhww..coo/mmaaoooccccee sccieeeneecccicee////cc Spperrical_DiiffisssoooPP.L
ene
Ef
3D rotation
3D rotation
3D rotation
enc
Co
Figure I. SDP enforces that the policy is SO(3) equivariant. Specifically, in the second row, an SO(3) rotation that is applicd to the seene leads to equivalent rotation the latent spherical Fourier features in the neural networks enc, ca, and on the gen- erated trajectory (blue dots). Fourier features visualized spherical signal.
Consider, for example, grasping a dish that is randomly placed in the sink, threading a nut onto bolt with random orientation, or wiping the curved surface of a car. Diffu- sion Policy may struggle to attain robust 3D generalization without training on a large amount of costly human demon- strations to exhaust the possible 3D arrangements of the scene.
1. Introduction
Diffusion Policy (Chi et al., 2023) has emerged as an effec- tive method for learning closed-loop policies from human demonstration. This success is based on the ability of Diffu- sion models (Ho et al., 2020) approximate multi-modal human demonstrations (Mandlekar al., 2021). A particu- larly challenging aspect of real-world robotic manipulation, which is often underrepresented in synthetic benchmarks, is that objects may be found in a wide range of 3D poses.
We propose Spherical Diffusion Policy (SDP), Fourier space SE(3) equivariant method that automatically adapts to changes in the scene. SDP improves on recent works in equivariant diffusion policy learning which are limited to (2))))qquivariancc (Wang et al., 2024b), equivariant to only single-object transformations (Yang et al., 2024a; Tie et al., 2024), or computationally heavy (Tie et al.. 2024). In contrast, our method is light and SE(3) equivariant across complicated
'Work done an intern at Amazon Robotics Khoury Col- lege of Computer Science, Boston, Massachuselts, USA Amazon Robotics, Boston, Massachusetts, USA. Correspondence Xu- peng Zhu x northeastern.edu>.
multiple objects, allowing it perform tasks with less engineering. SDP achieves translational in- variance by formulating states and actions in the gripper frame (Chi et al., 2024). Figure/ illustrates the SO(3) equivariance the proposed method. If the scene is trans- formed by 3D rotation, then the denoised action trajectory
Proceedings of the ndnd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Spherical Diffusion Policy
will be rotated by the same rotation. Since this equivari- ance is embedded in the neural network, it does not rely on additional data to train and thus achieves high sample efficiency. The equivariance constraints lead to provable SE(3) generalization to transformed scenes.
G: (o))(1))(x)) I (P)))) for all g E G. Here, the ps denote group representations, mapping each group ele- ment to invertible matrix (Serre et al., 1977). The 2D special orthogonal group SO(2) describes planar rotations and its subgroup C. discretizes SO(2) into n rotations. Similarly, S0(3) describes 3D rotations. We denote the group of 3D translations T(3). The Special Euclidian group SE(3) = SO(3) x T(3) includes both 3D rotations and translations. For any group, the trivial representation çu assigns the identity matrix (o = I to each group element. This makes invariance a special case of equivariance where the output representation Pout = Po- For SO(3), there are higher-dimensione representations pr, P2.... that will be introduced later. Representations can be combined by di- rect sum p(g) = p' (gg) @ p"(g). where p' p'(g) and p" (g) are diagonal blocks in p(g).
The contributions of this work are:
1. novel method, Spherical Diffusion Policy, which is equivariant to 3D rotations and invariant to 3D transla- tions enabling generalization to unseen scenes,
2. novel spherical FiLM layer for SO(3) equivariant conditioning,
3. novel spherical denoising temporal U-net for denois- ing trajectories with petpottemmoraa-equvvariiiaa
4. theoretical validation that SDP is equivariant,
Equivariant Policy Learning assumes the policy is equiv- ariant π (gS) 9a. g E G. where G could be SO(2) group or SE(3) group. One way to achieve equivariance is by recognizing and modeling an equivariant function using equivariant neural networks. (Ryu et al., 2024) states that for Brownian Diffusion on the SE(3) manifold, if the target function (SS) is equivariant, then the denoising func- tion Ep is also equivariant: Eg gg,, k) ge A, k). EquiDiff (Wang et al., 2024b) extends this open-loop equiv- ariance (Ryu et al., 2024) into closed-loop setups, but it is limited to SO) equivariance. For an additional introduc- tion, see Appendix C.I.
5. empirical validation of SDP through extensive experi- ments that include 20 simulation and physical tasks including single-arm and bi-manual embodiments.
2. Background
Diffusion Policy is closed-loop imitation learning method that learns policy ((s) that maps states to action trajectories from expert demonstrations. The states 8 con- sist of camera observation o, e.g. images or voxels or point clouds, and the end-effector's 6DoF pose (3D trans- lation and 3D rotation) ET, ER and aperture Egrip- The ac- tion a specifies the 6DoF pose ar, aR and gripper aper- ture agrip- The policy takes an input history of l states S, = [se, st-1..... st, st]n+1] The output is an action se- quence of actions A,- = (de, 0t+11 (t+r-i]. i].
Another way to achieve equivariance is by canonicalizing the input S and output of neural network (Zeng et al., 2022; Wang et al., 2021; Jia et al., 2023; Chi et al., 2024). For example, if a is 3D translation, then canonicalizing S involves translating it inversely so that the action is at the origin: S = S - a, a = a a, a E T(3). Intuitively, canonicalization eliminates the transformation applied to the state and action by always evaluating the state in the canonicalized view. Refer to Appendix C.2 for proof.
Diffusion Policy (Chi et al., 2023) leverages Diffusion Mod- els (Ho et al., 2020; Song et al., 2021) to learn from multi- modal human demonstrations (Mandlekar et al., 2021). Dif- fusion policy infers actions by sampling AK from uniform Gaussian noise, then performing K iterations of denoising, producing AK, AK-1 ..., Ap. The final iterate AB is the The denoising process is defined by:
Spherical Harmonies (SH) are functions on the sphere YI": S2 + E which give an orthonormal basis for the function space (S R). They are indexed by degree I e Zzo and order -1 < m & I, m E Z. A spherical function in spatial space can be transformed into the frequency domain by a spherical Fourier transform: F: f + (c""), where ci" are Fourier coefficients. Inversely, the inverse spherical Fourier transform F-1 converts the Fourier coefficients to the spatial value: f() Ei= Σm cJ" y/m (u). Spherical functions and SH are SO(3) steerable and thus suitable for -equuuvaaaa networks. Essentially, a rotation of f in spatial space is equivalent to a rotation of of in the frequency domain by the Wigner D-matrices D which is orthogonal. That is, I' g S SO(3) is equivalent to qf' =m D (0) eT" where of' are Fourier coefficients
output action.
(AA 100---eosssi
k)) +2), (5,
11( (1)
where (a,S, k) is a learnable denoising function, pa- rameterized by 8, that estimates the noise ch based the state S,, the noisy action A8, and the step k. The parame- ters a, 7,0 a define the noise schedule and functions of the denoising step k. Finally, the denoising function is trained to predict the noise added to the expert action:
L (St, A + e, k) - =||2_
(2)
Equivariance describes the property of a function which commutes with the transformations of a symmetry group
2

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Spherical Diffusion Policy
of f". For example, degree 0 (po) Fourier coefficients co E RR scalars that are invariant to rotation, and degree (p1) Fourier coefficients C1 E IR are 3D vectors with Wigner D- matrix given by a standard 3D rotation matrix. A Spherical Fourier signal up to degree I. has (T. + 1)² coefficients (Cohen et al., 2018; Bonev et al., 2023). SDP leverages this compact representation. Convolving two sets of Spherical Fourier signals (Cohen et al., 2018; Klee al., 2023) leads to signal over (3), which has Σ/ (21 1)² coefficients, as adopted in ET-SEED (Tie et al., 2024).
investigated equivariance in the grasp learning. (Wang et al., 2021; Huang al., 2024c; Simeonov al., 2022; Zhao et al., 2023a; Ryu et al., 2024; Huang et al., 2024a; Gao et al., 2024; Zhu et al., 2025b) developed equivariant open- loop policies. (Van der Pol et al., 2020; Wang et al., 2022b;a; Jia et al., 2023; Liu et al., 2023; Wang et al., 2024c;b; Yang et al., 2024b;a) verified effectiveness of equivariance in closed-loop agent. Among these works, (Zhu et al., 2022a; Zhao al., 2023a; Jia al., 2023; Liu et al., 2023; Huang et al., 2024c; Wang et al., 2024b; Zhu et al., 2025a) utilize discrete equivariance that suffers from discretization error. On the other hand, (Ryu et al., 2024; Huang et al., 2024a; Gao et al., 2024; Hu et al.,, 2024; Zhu et al., 2025b) leverages continuous equivariance but is limited to open-loop settings. Moreover, EquiBot (Yang et al., 2024a) are limited to degree = 1 representation that suppresses rich information, and ET-SEED(Tie al., 2024) uses heavy SO(3) irreducible representation that needs two-stage inference to alleviate computation burden. Furthermore, (Yang et al., 2024a;b; Tie et al., 2024) requires a segmentation pipeline engineered for each task to exclude everything but one object in the workspace. In contrast, our work the first to leverage con- tinuous and compact spherical Fourier features achieve SE(3) equivariant, end-to-end and computationally effi- cient closed-loop policy.
Equiformer (Liao & Smidt, 2023) and EquiformerV2 (Liao et al., 2024) are SE(3) equivariant graph neural net- works (GNN) (Passaro & Zitnick, 2023). In contrast to conventional GNNs that treat each node the graph as a scalar, Equiformer attaches spherical features to each node. These features are compactly approximated by truncated Fourier coefficients, up to degree 1 < L. Messages are aggregated from neighbor nodes in the graph through the edges by equivariant graph attention. This is followed by an equivariant spherical linear and activation layer. The spheri- cal linear layer treats degree / Fourier coefficients as high dimensional vectors to perform a linear mapping in each degree separately. The spherical activation layer (Geiger & Smidt, 2022) performs inverse Fourier transform, then per- forms conventional activation point-wise on the sphere, and lastly converts the activation back to Fourier coefficients.
Diffusion Models and Equivariant Diffusion Models Diffusion Models (Sohl-Dickstein et al., 2015; Ho et al., 2020; Song et al., 2021) are probabilistic generative models that demonstrated a strong capability modeling multi-modal distribution. Such capability is achieved by iteratively re- moving noise from an initial sample randomly drawn from an underlying distribution. Equivariance is introduced dif- fusion models in (Xu et al., 2022; Hoogeboom et al., 2022; Yim et al., 2023) in the context of molecule generation. Dif- fusion Models are applied to robotics in open-loop settings (Ke et al., 2024; Ryu et al., 2024; Jiang et al., 2023; Urain et al., 2023; Huang et al., 2024b) and closed-loop settings (Janner et al., 2022: Pearce et al., 2023; Chi et al., 2023; 2024: Ze et al., 2024; Wang et al., 2024a; Liu al., 2024; Brehmer et al., 2024). The most relevant works on equivari- ant Diffusion Policy include (Wang et al., 2024b; Zhao et al., 2025; Yang et al., 2024a; Tie et al., 2024), where (Wang et al., 2021; Zhao et al., 2025; Hu et al., 2025) is limited to discretized SO(2) equivariance. (Yang et al., 2024a; Tie et al., 2024) is SO(3) equivariant thus requiring engineer- ing effort to segment everything but the target object, even though, these methods are designed handle a single object in the scene. Moreover, (Tie et al., 2024) is based on heavy SO(3) irreducible representations and needs 2 stage diffu- sion process. In contrast, our method is SE(3) equivariant, based on compact yet expressive spherical representations that can end-to-end learning without task specifc engineer- ing effort and generalize to multi-ohject tasks.
3. Related Works
Closed-loop Robot Policy Imitation Learning learns robot skills from human demonstrations through machine learning. Though it is a straightforward and general frame- work, facing multiple challenges. One challenge is the error compounding effect where the action prediction er- ror causes future states to diverge from the training states and further exacerbate the next action prediction (Ke et al., 2021). To combat this, action chunking (Lai et al., 2022: Mandlekar et al., 2021; Chi et al., 2023; Zhao et al., 2023b) proposes predicting and executing a trajectory of actions instead of one step of action. Another challenge is to learn from multi-modal human demonstrations. Multiple methods are proposed to fit a modll policy, including Gaussian Mixture Model (Mandlekar et al., 2021; Zhu et al., 2022b), Variational Auto Encoder (Zhao et al., 2023b; Mousavian et al., 2019), Energy-Based Models (Implicit Models) (Flo- rence et al., 2022), and Diffusion Models (Janner et al., 2022; Pearce et al., 2023; Chi et al., 2023). Based on (Chi et al., 2023), this work leverages additional inductive bias equivariance, to achieve significantly better performance.
Equivariance on Robot Learning Robotic policies op- erate in the 3D world, sharing rich symmetries. (Zhu et al., 2022a; Huang et al., 2022; Zhu et al., 2023; Hu et al., 2024)
3

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Spherical Diffusion Policy
A
A
A
step
A
A
S:
A
Point Cloud
enc
state
C₁ spherical Fourier feature
encoder
A
Spherical Denoising Temporal U-net
A
Proprioceptive
nolsy action
noise
Encoding state into spherical Fourler feature
Iterating K' denolsing steps with SDTU
Figure Method overview. During inference, SDP embeds state Sy into a spherical scene feature C, by the encoder enc. Then, SDTU estimates the noise based on the noisy actions Ař, step kt and the scene feature C.. Later, the noise is subtracted from the noisy actions, generating cleaner actions A A This denoising process is performed for K iterations, generating clean trajectory A.
4. Method
{0, 1 and we canonicalize the state and action to the left
= 0 and the right = gripper's position.
4.1. Method Overview
The Spherical Diffusion Policy model maps observations to actions =SS) = A. We assume the optimal policy is SE(3) equivariant and enforce this assumption in the model. Specifically, we enforce rotation equivariance (gS)) gA.g g S3(3), and translation invariance π (1S) = A, E T(3). The model is thus -equivarra where T(3) acts trivially on the actions.
4.2. Representing State and Action by Spherical Signal
In this section, we propose spherical representation of the state and action for the policy. There are several advantages of using spherical Fourier features as latent features. First, the truncated spherical Fourier coefficients provide a com- pact approximation of spherical features and compatible with SO(3) rotations, rather than computationally heavy SO(3) irreps used in ET-SEED (Tie et al., 2024). Further- more, higher degree coefficients can represent finer details than EquiBot( Yang et al., 2024a) which adopted Vector Neu- ron (Deng et al., 2021) that only supports up to 3D vectors (analogous to type-l 1), suppressing rich higher degree information in the latent features. For example, vector rep- resentations cannot capture spherical distributions with two distinct modes. Lastly, spherical features support equivari- ance to continuous group SO(3) (continuous rotation), in contrast to discretized group C's (discretized rotation) in EiuiDiff (Wang et al., 2024b) which suffers from discretiza- tion emror.
The rotational equivariance of is enforced by equivari- ant denoising function eø, as proven in (Ryu et al., 2024; Wang et al., 2024b). Specifically, we use an equivariant conditional denoising function (S, A + k)) to estimate the noise for a noisy action + * the step k, and state S. We model 6g using three components as shown in Figure 2: the spherical encoder embeds the state into multichannel spherical scene feature enc(S) C, and then ii) spher- ical denoising temporal Unet (SDTU) estimates the noise from the noisy action and step, conditioned on the scene feature eg (C, A + k) using iii) spherical FiLM (SFiLM) layers to achieve this equivariant conditioning. Since these three components are equivariant, the denoising function is equivariant by composition.
The end-effector state e, the action a,, and the noise have the same geometric structure consisting of 3D position, 3D rotation, and ID gripper aperture information. We de- compose the end-effector data 3D position vector, 3 x 3 rotation matrix, and a ID scalar. The rotation matrix can be viewed as 3 column vectors. We represent the po- sition vector by degree vector, the rotation matrix by degree vectors, and the aperture representation. That e, 54, E Pec p Po- Intuitively, the position and rotation matrix are rotated by the rotation matrix corresponding to the rotation of the state, while the gripper aperture stays unchanged.
Translational invariance of π is achieved using relative ac- tion formulation (Chi et al., 2024), which canonicalizes the state-action (Zeng et al., 2022) with respect to translations by centering the observation on the gripper and defining action positions relative to the gripper:
scalar in the trivial
Scan,i (0 ex, ex e C sppp))
c = (Ag eT, Al.. A Agrip).
See Appendix C.2 for proof. For the single-arm setting, d E {0} denotes the gripper. Additionally, we propose bi-manual relative action representation. In this case,

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Spherical Diffusion Policy
conditions on sphere
conditions
space
skip connection
on sphere
Spherical
Spherical down conv
Spherical conv
up corv
Spherical
A noisy actions
Spherical Fourier features
mix channel
SILU
Spherical Fourier features
skip
naise
temporal down/up
connections
convolution
Spherical Denaising Temporal U-net Eg
Spherical downlup convolution
Figure 3. Spherical denoising temporal U-net (SDTU). Left: The SDTU éo estimates the noise e, based on the noisy actions A denoising step index and the encoded scene C. The SDTU has U-net architecture, with 4 spherical down up convolution blocks. Right: details of a spherical down or up convolution block.
We adopt point clouds as an observation and treat color information as degree 0 spherical coefficients (same as (Ryu et al., 2024)), as the color invariant to the point cloud rotation. The point cloud encoded to latent vector by 5- layer ResNet (He al., 2016) encoder enc(-). The encoder implemented by EquiformerV2 (Liao al., 2024) which extracts high-degree spherical signal from the point cloud, see Appendix E for details. The robot state is concatenated to the output of the encoder yielding C.
in Equation. 3 is SO(3) equivariant:
(
ΣΣ JET KEin
to mi (4)
For proof see Appendix A.1, the proof essentially follows Schur's lemma (Schur, 1905), which states that any linear combination of SO(3) Fourier features is equivariant. This convolution is followed by spherical activation (Cohen et al., 2018; Geiger & Smidt, 2022) for expressiveness. Stride and transposed convolution are used for down- and up- sampling in the U-net, as in Diffuser (Janner et al., 2022). Spherical FiLM layers are adopted, allowing for equivariant condi- tioning, and are described the next section. Figure summarizes the SDTU,
4.3. Spherical Denoising Temporal U-net
The spherical denoising temporal U-net Eg infers the noise from the noisy action A + c², denoising step index k, and state embedding C' as AA + k)). The vector C' encodes the state spherical Fourier space up to degree L. The input A + c and the output are spherical signals in Pee: introduced in Section 4.2. The denoising step index is encoded using sinusoidal embeddings, treated degree O features.
4.4. Spherical FILM Conditioning Layer
We propose equivariant spherical FiLM (SFiLM) layers to extend the Feature-wise Linear Modulation (FiLM) layer (Perez al., 2018) used by Diffuser (Janner al., 2022) into the spherical Fourier domain. The condition sphere C' is projected into a scaling condition and an offset condition 3 by equivariant linear layers (Geiger & Smidt, 2022): 7 = T(C), 3 = BCC). Then, SFiLM conditions each degree separately. Specifically, SFiLM treats ",,,, 3r as 2/ + 1 dimensional vectors, to modulate the hidden feature h4. by projecting " h scaling condition and adding St offset condition:
The SDTU is ID U-net with spherical Fourier features that are spatiotemporal equivariant. The temporal equivariance achieved using convolution along the time dimension as proposed in Diffuser (Janner al., 2022). We incorporate an additional spherical Fourier dimension in the latent fea- tures to achieve spatial equivariance (SO(3) equivariance). This equivariance is enforced by mixing channel tempo- ral convolution on each degree 1 of the spherical Fourier coefficient:
h |
ΣΣ hi.m. -
(3)
SFiLM 3:)
+ Br
(5)
j= VEAn
where i, indexing the input and the output channels respec- tively, I, m is the degree and order of the spherical Fourier coefficient h, in denotes all input channels. Subscript j indexes the time the prediction trajectories.
SFiLM supports high degree Fourier coefficients for expres- siveness, which differs from EquiBot (Yang et al., 2024a) that only supports degree which drops rich information in the latent features.
Proposition 4.1. The mixing channel temporal convolution
Proposition 4.2. The SFiLM layer Equation. 5 is SO(3)
5

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Spherical Diffusion Policy
equivariant:
diffusion policy based on point-cloud representations. 5) ACT (Zhao et al., 2023b) (Action Chunking Transformer) model capturing multi-modality via Variational Au- toencoder (VAE). 6) BC-RNN (Mandlekar et al., 2021) a behavioral cloning approach that captures multi-modality using a Gaussian Mixture Model (GMM) and accounts for non-Markovian dynamics via Recurrent Neural Network (RNN), relevant baseline, ET-SEED (Tie al., 2024), is not included because the code was unavailable before the initial submission. Following (Chi et al., 2023; Wang et al., 2024b), we train all baselines using DDPM (Ho et al., 2020) with 100 denoising steps. For details on hyperparameters, see Appendix D. We report the maximum test success rate throughout training, averaging results over 50 rollouts for each of the three seeds.
D())- (FiLM
(()))
SO(3)
(6)
The proposition is proved by the orthogonal property of the Wigner D-matrices D and Schur's lemma (Schur, 1905), please see Appendix A.2 for details.
5. Experiments
5.1. Simulation Experiments
Experimental Settings We conduct simulation experi- ments using the MimicGen (Mandlekar et al., 2023) environ- ment, built on the Mujoco simulator (Todorov et al., 2012), which features diverse tasks that are contact-rich, precise, and long-horizon (see Figure 4). Unlike scripted demon- strations, which are unimodal, or Reinforcement Learning (RL) agent-generated demonstrations, which are Markovian, MimicGen generates multi-modal, non- Mon-Markov trajec- tories from a few human demonstrations (Mandlekar et al., 2021), making it well-suited for benchmarking learning from human demonstrations.
Results on Tasks with SE(3) Initialization Table I shows that SDP outperforms all baselines across all tilt- ing ranges, except for the Coffee 0' task, demonstrating superior sample efficiency. Notably, the tilting range increases, SDP achieves a more significant relative perfor- mance improvement over the baselines. This highlights SDP's strong SE(3) generalization, enabled by its contin- uous SE(3) equívariance property. However, performance declines for all methods, including SDP, as the tilting range increases. We hypothesize that this drop is caused by point- cloud occlusion and object instability due to gravity, both of which disrupt SF(3) equivariance.
MimicGen provides observations in the form of RGBD im- ages from both front view and an in-hand view, along with a 7-DoF robot state. The RGB images have reso- lution of 84 x 84 x 3, while RGBD data can be used to reconstruct either 3D colored voxels (849) or colored point clouds (PCD) with 1024 points. For PCD, we exclude table points, following DP3 (Ze et al., 2024). The action space in MimicGen consists of 6-DoF gripper pose and a 1-DoF gripper aperture. Three control modes are used: Absolute Control, which defines the gripper trajectory in the robot frame; Relative Control, which defines it in the current grip- per frame; and Velocity Control, which determines the next gripper pose relative to the previous one (Chi et al., 2024).
Results on Tasks with SE(2) Initialization Table 2 shows that SDP outperforms all baselines across 10 tasks, except for Coffee and Coffee_Preparationn Despite the varia- tions in SE(2), SDP still demonstrates a notable advantage, suggesting that its continuous SE(2) equivariance bene- fits learning more effectively than the discrete Cg equiv- ariance in EquiDiff. The lower performance of SDP on Coffee and Coffee.Preparate low-resolution point clouds, which struggle to capture fine details-such as the slack between the coffee pod and its -eeeptale-potentiallly hindering precise manipulation.
may be attributed to the
To evaluate robustness, we modify four MimicGen tasks with SE(3) initialization by randomly tilting the table within defined range and randomly placing objects on the table- top while keeping the robot base upright. Benchmarking is conducted across three difficulty levels with progressively increasing tilt ranges: [0]. 15", 15%, and ,- Ad- ditionally, we also compare various baselines across all 12 original MimicGen tasks.
5.2. Physical Experiments
Experimental Settings We further evaluate the perfor- mance of SDP across 5 physical tasks in Figure 5, using a robot station shown in Figure A1. Turn involves ma- nipulating an articulated object, while Push Eraser requires pushing small eraser. Grasp_Bor challenges the policy to maintain closed kinematic chain. Flip Book involves rich contact between the end-effector, the tabletop. and the book. Pack Package is long horizon task. The observations are captured by two stationary RGBD cameras positioned above the workspace to minimize occlusion. Point clouds with 1024 points are reconstructed from the RGBD images (for
We compare several baselines in our experiments: 1) EquiD- iff (Wang et al., 2024b)- an (2()))equivariant diffusion policy using either voxel or RGB image observations. 2) DiffPo (Chi et al., 2023) the original diffusion policy, employing either convolutional (-C) or transformer (-T) backbone in the diffusion network. 3) EquiBot (Yang et al., 2024a)- - an SO(3) equiivarrt diffusion policy with up to degree representations. 4) DP3 (Ze et al., 2024) -
6

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Spherical Diffusion Policy
Coffee 0
-5 15
(c) -30 30
Thr. Pc. D2
(c) Square D2
(f) Thread. D2
Figure 4. MimicGen tasks with (3) initialization ((a)-cc), showing of tasks) and SE(2) initialization ((d)(f), showing 3 of 12 tasks).
Table 1. Evaluation success rate 4 MimicGen tasks with 3 levels of SE(3) initialization. We train all the baselines progressively tilted environments with 100 demonstrations. As the degress (E(() initialization increases, SDP maintains reasonable performance while the performance of other baselines drop severely. Results averaged over three seeds.
Coffee
Three Pc. Assembly
Square
Threaling
Average
Method
0
15
30"
0
15a
30
0"
15"
30
0"
15"
30"
0
15
30"
8B(33
54
33
67
37
62
38
53
39
63
35
EquiDiff (Wang
(O(2)
DiiPoo (Chi et al,, 2023)
NA
Abs
RGB
44
23
16
4
8
0
17
10
8
18
EquiBor (Yang ol., 20246)
SO(3)
Abs
PCD
0
0
1
0
1
6
4
D
2
3
Table 2. Evaluation success rate 12 MimicGen tasks with SE(2) initialization. We train all the baselines with 100 demonstrations. SDP demonstrates the best performance on 10 tasks. Results averaged over three seeds.
Stack
Three Pc: Hammer
Mug Cleanup
Nit Assembly
Pick
Coffee Preparation
Avenge
Square
Threading
Assembly
Cleanup
Method
DI
D2
DI
Rate
EquiDiff (Wang 20124))
Vonel
65
64
EquiDir (Wing 20246)
DVTPP-CC (Chi at al., 2023)
DilPo-T (Chi et 2023)
Abs
RGB
RGB
76
51
11
17
44
47
52
30
55
35
65
38
29
42
(Ze 2024)
69
al., 20336)
EquiDie (Wang
20128))
EquiDii (Wang al, 20240)
DulPPo (Chi et al., 2023)
Vol
RGB
RGB
$1
75
26
11
13
41
43
43
25
29
42
42
35
42
49
33
33
BC RNN (Mandickar
RGB
(a) Single-arm Turn Lever
(b) Single-arm arm Push Eraser
(c) Bi-manual Grasp Box
(d) Bi-manual Multi-step Flop_Book
(c) Bi-manual Multi-step Pack Pack_Packan Package
Figure 5. Five physical robotic manipulation tasks.
Eraser we also tested 2048 points). The actions are the DoF' gripper poses for single-arm tasks 12 DoF gripper poses for bi-manual tasks.
Training Dataset from Human Demonstrations use Gello (Wu al., 2023) to collect demonstrations with ob- jects initialized in random SE(3) poses. For the single- arm tasks, collect 30 successful human demonstrations. Additionally, we record an extra 10% demos as the recov-

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Spherical Diffusion Policy
lation, which defines actions in the current gripper frame. 3) DP3-canonical DP3 (Ke et al., 2024) with canonicalized observation-action space, by transforming the point cloud and trajectory to the gripper frame. This achieves SE(3)- invariance. 4) EquiBot Rel. removes SDP's spherícal Fourier features and replaces its model with EquiBot (Yang et al., 2024a), while keeping the relative action formulation. 5) DP3 A Absolute Action (Ke et al., 2024) the original DP3 policy, this baseline ablates relative action, spherical repre- sentation, and equivariance. 6) EquiBot Absolute Action- removes both the spherical Fourier features and the relative action formulation, adopting EquiBot (Yang et al., 2024a) in the absolute action formulation.
Table Success rate (%) of 5 physical experiments over 20 evalu- ation episodes. The action space and number of training demon- strations are listed under each task. Overall, SDP is 61% better than EquiDiff' and 71% better than D-f Results from one seed. using point clouds with 2048 points.
Tum Lever
Push kinser Dolf Demus
Grasp Box
Hip Book
Pack Package
Arg.
Dof
Dof 66 Demus
Succ.
Method
35 Demos
Demos
66 Demos
Rate
SDP
35/ 90*
65
70
Equitirr
20
30
35
0
B
17
DiftFo-C
10
10
15
0
D
ery demos specific poses. Similarly, for the bi-manual Grasp_Box task, we collect 33 demos at random SE(3) poses. For more challenging bi-manual tasks like Flip Book and Pack Tack_Package we collect 66 demos. Figure A3 visu- alizes the SE(3) training pose distribution for all five tasks. Further details on the tasks are provided in Appendix B.2,
The results are shown in Table 4. Discrete SDP trivially modifying (Wang al., 2024b) to be SE(3) equivariant, sufferers from discretization error thus underperforms SDP, The relative action formulation that achieves 3D transla- tional equivariance, also plays key role, as its removal in SDP Abs. causes major performance drops, particularly in the Coffee and Square tasks. DP3-canonical leverages SE(3) invariant features, outperforms DP3 that did not lever- age, but still underperforms SDP by a large margin, demon- strating the advantage of equivariant features. This matches the finding in (Miller et al., 2020). The EquiBot Rel. abla- tions result in significant performance drops across all four tasks, indicating that the spherical Fourier representation is the most critical factor for SDP's performance. Finally, DP3 Abs., EquiBot Abs. further amplifies the performance degradation, demonstrating that removing both the relative action formulation and spherical Fourier representation is more detrimental than removing either one alone.
Results and Discussion We benchmark SDP against the top two baselines, EquiDiff' (Wang et al., 2024b) and DiffPo- C (Chi et al., 2023), from the simulation experiment (Table 1). We train all three models using DDIM (Song et al., 2021) and inference with 8 denoising steps for all tasks. Each base- line is evaluated on 20 rollouts per physical task, with each rollout initialized using object poses in novel SE(3) poses unseen in the training set. The success rates are summarized in Table 3, with detailed breakdown of the success rates provided in Table A1. For the Push Eraser task, increasing the PCD resolution to 2048 points enables accurate localiza- tion of the eraser, resulting in a performance improvement of over 50%.
SDP significantly outperforms all baseline methods across every task and embodiments, achieving a 61% higher suc- cess rate than EquiDiff and 71% improvement over DiffPo- C. These significant gains in sample efficiency and spatial generalization are largely attributed to its inherent SFF(3) equivariance. For instance, in the Turn- task, SDP successfully locates and rotates a lever that is randomly clamped in 3D space. In comparison, Eiui frequently misdirects the gripper to the workspace center, entirely miss- ing the lever, while DifiPo approaches the lever but only hovers nearby without engaging it. Further discussion of common failures can be found in Appendix B.3.
Table 4. Ablation study. The relative action space and the spherical representation are critical for the SE(3) generalization, while the latter one more important. Results from one seed.
Rel.
Spher. Rep.
Equi- variance
Coffee 15
Thr. Pt: As. 15
Square 15"
Thread. 15"
Avg
Method
Act.
SR
SDP
/ (E(33)
54
49
38
53
49
Discrete SUP
/
Octabedron
42
16
48
35
J SO(3) equ.
18
42
44
26
Dan-canoniaal
X SE(() law.
40
0
8
12
15
Equi Rel.
K
SE((3
18
2
6
8
DP3 Abs.
20
4
Equi
SO(3) equ.
5.3. Ablation Study
5.4. Additional Studies
Table 4 presents six ablations: 1) Discrete SDP- replaces SDTU that has continuous equivariance with discrete equiv- ariant denoising U-net with Octahedron (cubical) discretiza- tion, this is a SE(3) adaption of (Wang et al., 2024b; Zhao et al., 2025). 2) SDP Absolute Action - defines actions in the workspace frame instead of SDP's relative action formu-
Performance VS Degree / in spherical Fourier features As shown in Table 5, increasing the degree from 1 to 2 improves performance across all tasks. However, beyond degree 2, performance saturates while computational cost increases. Since SDP is designed for real-time robot control, we select 1 2.
8

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Spherical Diffusion Policy
Inference Speed Table 6 presents the inference time com- paring SDP with four other baselines. DiffPo is the fastest (0.09s) while ETSEED is the slowest (29.4s), The inference time of SDP is on the same order of magnitude as that of the best Diselo (approximately 5x), while SDP achieves continuous SE(3) equivariance, significantly better performance than DiffPo and Equi in simulation and physical experiments, and does not require preprocessing.
Table Success rate VS degree of the spherical Fourier feature. Results from one seed.
Degree I Coffee 15"
Thr. Po: As 15"
Squar 15"
Threading 15"
Avg. SR
52
52
65
52
56
2
(SDP))
54
49
38
53
49
44
46
26
Sample Efficiency To assess the impact of training dataset size on performance, we evaluate SDP, EquiDiff, and Do on four MimicGen tasks with tilted angles in the range [0, 15%). using 100, 316, or 1000 demonstrations. Figure 6 summarizes the results, with each point representing the average success rate across all four tasks.
SDP leverages spherical features by using EquiformerV2 (Liao et al., 2024) and SDTU, which is more lightweight (66x x faster inference speed, 32x larger batch size) than the SE transforrre that ET-SEED is based on. Moreover, SDP supports high orders of spherical harmonics, which is more expressive than Vector Neuron. Lastly, SDP achieves continuous -equivuuuvariiance, where EquiDiff enforces discrete C8 C SO(2) equivariance.
SDP achieves a success rate of 48.5% using only 100 demon- strations, surpassing EquiDiff by 4% while utilizing just one-tenth of the data, indicating a 10x improvement in data efficiency. Similarly, EquiDiff attains a 20% success
Table 6. Comparison of inference time. At the costs of 5x solver than Dof SDP achieves continuous SE(3) equivariance and does not need preprocessing. The inference time and the training batch size are tested on a commercial GPU with 24GB RAM.
rate with 100 demonstrations, matching the performance of DifiPo, which requires approximately 300 demonstrations, supporting a 3 gain in data efficiency.
Inference Time (4)
Pro- processing ig
Batch Size 1
Equivariance
Neuma
Data Scaling Curves
Method
Netwoek
61-
SDP
0.44
No
32
SB(3)
Equiformrr22 SDTUU
Diffoo
0.09
No
64
None
Convolation
EquiDiff
0.14
No
64
C8 (O
ESCNN
10
Equittr
0.18
Segmestation
64
8E(3)
Vector Neuron
ST-SEE
29.4
Segmentation
SE(3)
SE(3) Transfommrr
32.5
6. Conclusion and Limitations
20.25
3 X
This paper introduces the Spherical Diffusion Policy (SDP), an -equivaauiaai policy that generalizes to 3D scene ar- rangements using only a few demonstrations. SDP achieves this through three key components: 1) spherical Fourier features, providing compact and precise representations for continuous SO(3) equivariance; 2) spherical FiLM, enfore- ing equivariant conditioning; and 3) spherical denoising temporal U-net, ensuring spatiotemporal equivariant denois- ing. SDP significantly outperforms strong baselines across simulation and real-world experiments, demonstrating ef- fectiveness in both single-arm and bi-manual embodiments.
EquiDVT
DEfiP
31.6
1000
Training Demo
Figure 6. Impact of training dataset size on task success rates: in- creasing the number of demonstrations from 100 to 316 (a 3x
increase) yields an average success rate improvement of 9-12%
across four tasks, with initial success rates of 48.5% (SDP), 20.3%
(EquiDiff), and 8.8% (DiffPo). At 1, 1,000 demonstrations (10x),
the average success rate saturates for SDP at 60%, while EquiDiff and DifiPo continue to improve, reaching 44.5% and 32.5%, re- spectively.
One limitation of the proposed method is that it operates in position control, ignoring contact forces, which leads to protective stops in the Flip Book task. An important future direction is to address this by learning compliant, force-aware policy (Kohler et al., 2024; Hou et al., 2024) in an equivariant manner. Another limitation is the low- resolution point cloud processing in the observation encoder, which struggles to capture fine details, such as these in the Push Eraser task. Using a more efficient Graph Neural Network (Zhao al., 2021; Luo et al., 2024) could help mitigate this issue.
The observed performance saturation of SDP beyond 300 demonstrations may be attributed to scene occlusions, as re- lying solely on agent-view and in-hand cameras can obscure critical areas of the workspace. Additionally, all demon- strations are generated from 10 raw human demonstrations (Mandlekar et al., 2023), so increasing the number of gener- ated demonstrations may not increase diversity in the data. Furthermore, the kinematic constraints of the robot may limit its ability to access certain regions, thereby impacting overall task success.
9

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Spherical Diffusion Policy
Impact Statement
work for (3)equvvari networks. In Proceedings of the IEEE/CVF International Conference Computer Vision, pp. 12200-12209, 2021.
On the bright side, the proposed method enhances spatial generalization for manipulation policies, making poten- tially deployable in real-world scenarios significantly reduce human workload. On the dark side, the method lacks awareness of common scenes, which could result in risky actions such as harming individuals, causing fires, or damaging objects.
Florence, P., Lynch, C., Zeng, A., Ramirez, O. A., Wahid, A., Downs, L.,, Wong, A., Lee, J., Mordatch, I., and Tompson, J. Implicit behavioral cloning. In Conference on Robot Learning, pp. 158-168. PMLR, 2022.
Gao, C., Xue, Z., Deng, S., Liang, T., Yang. S., Shao, L.., and Xu, H. Riemann: Near real-time se (3)-equivariant robot manipulation without point cloud segmentation. arXiv preprint arXiv:2403. 2024.
Acknowledgments
The authors would like to thank Michael Schultz and Nathan Gere for the help on the physical experiments, Haojie Huang for the help on the simulation environments, Pranay Thangeda and Erica Aduh for the help on the robot platform setups.
Geiger, M. and Smidt, T. e3nn: Euclidean neural networks. arXiv preprini arXiv:2207.099:: 220.092277. 2022.
He, K., Zhang, X., Ren, S., and Sun, J. Deep residual learn- ing for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 770-778, 2016.
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Zhu, X., Qi. Y., Zhu, Y., Walters, R., and Platt, R. Equact: An se(33)-eqivvaria) multi-task transformer for open-loop robotic manipulation, 2025b. URL / https:///axxiv arxiv. hrg/abs/2a 2505. 21351.
Zhu, Y., Joshi, A., Stone, P., and Zhu, Y. Viola: Object- centric imitation learning for vision-based robot manip- ulation. In 6th Annual Conference on Robot Learning. 2022b.
13

---

Spherical Diffusion Policy
A. Proofs
A.1. Proof Proposition 4.1
Proof. Focusing on the right-hand side of the Equation
ΣΣ D (g) 5 whi- JET
D D' (g) JET =D' (i,
ΣΣ
M.m.
(8)
the line is because of Schur's lemma (Schur, 1905), which states that any linear operation of SO(3) irreps acts as on each irreducible subspace is equivariant.
A.2. Proof of Proposition 4.2
Proof. Focusing on the right-hand side of Equation
SFiLM(P(G))))))))) 3)
(10)
D(g)h + D(g)3
= (D(g)a)) h,
(11)
[g
P(g)
= = D()h)) [g)
+ D(g)3
(12)
(13)
Because the Wigner D-matrices are orthogonal, we have:
(FLLM(D()M ( D(g)8))
(14)
-7Th D(g)h
D(g)3
(15)
D(a)( Iu by + Bi) |
(16)
D(g) (FiLM(
(17)
the line 16 is based on Schur's lemma (Schur, 1905).
B. Physical Experiments and Results
B.1. Physical Experiment Workstation
Our physical robotic workstation, as shown in Figure A1, composed of two collaborative UR5e manipulators, each equipped with a compliant ray-fin finger(Crooks al., Two scene cameras (RealSense D415) are positioned on either side of the workspace. GELLO controllers (Wu et al., 2023) are utilized to collect human demonstrations for physical robotic manipulation tasks.
B.2. Physical Manipulation Tasks and Training Datasets
We experiment with SDP five physical tasks, shown in Figure A2: (a) Turn_Lever involves manipulating articulated object and (b) Push_Erased requires pushing small object with single manipulator; (c) Bi-manual Grasp_ challenges the policy to maintain a closed kinematic chain; (d) Flip Book involves rich contact between the end-effector and the book while transforming book's ' pose with dexterous complex coordinated manipulation; (c) Pack_ Package is long horizon task.
Lever: An expert demonstrator moves one -an-fin finger to make flush contact with the edge of the lever and then turn the Lever counter-clockwise least 60 degree around the fulcrum. Otherwise the task has failed. The lever, initialized with
14

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Spherical Diffusion Policy
D415 RGB-D Cameras
URSe with 6DoF
GELLO Teleoperation Arm
Compliant TPU Fingers
Figure AI. Overview of the manipulation experimental setup: two URSe manipulators, each with compliant ray-fin fingers,
stationary overhcad cameras, and GELLO teleop controllers.
SE(3) pose, is flexibly positioned with clamp, using combination of pitch and yaw angles within the 3D workspace of the manipulator. A total of 33 human demonstrations have been collected, composed of 30 successful demo and 3 (10%) recovery demonstrations where failed states were corrected to reach to the successful goal state.
Praseer An expert demonstrator moves one ray-fin finger to make contact with the Eraser, which is initialized with an SE(3) pose within the marked boundary on the whiteboard. Once the contact secure, the demonstrator will move the ray-fin finger push the Eraser, in straight line, towards the closest edge, until the eraser outside the marked rectangle, achieving successful goal state. Otherwise the task has failed. The whiteboard is positioned, flexibly with clamp, with approximate pitch angles of -15, 0, 30, 45, 60, and 90 degrees and approximate yaw angles of -15, O, and 15 degrees within the 3D workspace of the manipulator. In total, 33 human demonstrations were collected, consisting of 30 successful demonstrations and 3 (10%) recovery demonstrations.
For the bi-manual manipulation tasks, we design a set of 432 distinct SE(3) poses, consisting of regions on the x-y plane, 3 discrete pitch angles (0%, 8%, or 169) provided by 8 wedge, and 16 discrete yaw angles with 15 or 30 increments. We randomly sample an SE(3) pose from this set position box, book, container for collecting human demonstrations.
Grasp Box: an expert demonstrator moves two ray-fin fingers to pinch grasp the box sampled SE(3) pose, then lifts the pinched box minimally 40cm above the flat surface. Otherwise the task has failed. Similarly, we collect 33 human demonstrations including 3 recovery demos.
Flip_ Book: an expert demonstrator moves two ray-fin fingers to pinch grasp the book along its medium dimension at a sampled SE(3) pose, then rotates the pinched book in-hand so that the book is pinched along its smallest dimension, and then lifts the book minimally 40em above the flat surface. Otherwise the task has failed. Due the precision required for coordinated finger movements, we collect 66 human demonstrations including 6 recovery demos.
An expert demonstrator moves two ray-fin fingers to pinch-grasp the box, transports it to the pre-pack pose above the container, places the box inside, moves to the pre-lid-close pose, and finally closes the lid. The final goal state of the task is the box inside the container with the lid closed. any step fails, the task is considered a failure. Due to the precision required for coordinated finger movements, we collect 66 human demonstrations, including recovery demonstrations.
Pack Package:
15

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Spherical Diffusion Policy
Goal State
Goal State
Goal State
(a) Single-arm Teve_
(b) Single-arm
(c) Bi-manual Brx
Goal State
(d) Bi-manual Flip_ Blok
(c) Bi-manua Pack_ Packaae
Figure A2. Demonstrated robot actions for each task: a) Turn b) Eraser and c) Grasp_Box, each task with trajectory segments and their respective goal states; d) Flip_Book task with three trajectory segments and goal states, where both manipulators must perform coordinated movement after the initial pinch e) Pack Pack Package with five trajectory segments and goal states.
B.3. Detailed Evaluation Results
evaluate the baseline performance with 20 rollouts, using object poses randomly sampled from the SE(3) pose set in training. All poses are annotated, and evaluation conducted on novel, unseen poses.
For all five physical tasks, we report the success rate for each intermediate goal state in Table A1. We compare SDP with EquiDiff (Wang et al., 2024b) and DifiPo (Chi et al., 2023). SDP achieves strong performance with minimum 90% success rate on the first step, where both EquiDiff and DifiPo perform poorly. For subsequent steps, the success rate drops by up to 20% in the Blok task, where precise coordination both fingers is required for the in-hand pose rotation.
Figure A4 provides examples of task successes and failures. The most common failure cases SDP occur during book flip step. Lack of precise coordination between the two fingers may cause the book to either drop or be pinched too tightly, leading robot fault. Other failures include invalid pinching during the pick step or object drops due loose grip. For the Pack_Bor task, collisions with the container may occur during the transfer of the pinched box, and misalignment or incorrect placement can lead to collisions during the packing step. Interestingly, when positioning the lid to close, the robot may mistakenly identify the object as the lid. For the Turn Lever task, the finger may drift away from the lever while attempting to complete the required rotation. For the Push_Eraser Eraser task, the robot pushes the wrong direction, failing to push the eraser across the boundary. For the Grasp_ Box task, SDP struggles to pinch the box when its long dimension is parallel to the robot's front (i.e., when the box has yaw angle of 0 or 180 degrees).
16

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Spherical Diffusion Policy
(a) Single-arm Turn Lever
(b) Single-arm Push Eraser
(c) Bi-manual Grasp Box
(d) Bi-manual Flip Book
(e) Bi-manual Pack Package
Figure A3. Visualization of SE(3) Pose Distribution for Five Physical Tasks. The initial state out 20 episodes are visualized.
C. Additional Background
C.1. Equivariant Diffusion
The theory of Equivariant Diffusion has been extensively investigated in (Köhler et al., 2020; Brehmer et al., 2024; Ryu et al., 2024; Wang et al., 2024b). Based on these works, we summarize the equivariance property of the policy and the denoising function of the policy for completeness. There are two scenarios: diffuse equivariance and denoise equivariance. Proposition C.1 (Diffuse equivariance). If the policy is equivariant to group G, i.e., x(gS) = (T()), and the distribution D from which to sample the noise, invariant to group G. i.e., D = =gD, and the the denoising function satisfies
co n (S) c, k) = c, C,
(18)
then the denoising function equivariant to group G, iee c(gS, gA A k) = gc R).
Proof. We assume the denoising function satisfies Equation 18. Since the equation holds for all D and G-invariant,
17

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Spherical Diffusion Policy
Table AI. Breakdown of rates each step for five physical experiments over 20 evaluation episodes. The action space and number of training demonstrations are same as in Table 3.
Turn Lever
Push Eraser
Grasp Box
Flip Book
Pack Box
Method
Contact
Rotate
Contact
Push
Grasp
Lift
Grasp
Flip
Lift
Pick
Transport
Pack
Locate
Close
SDP
90
80
90
90
90
85
100
80
65
95
85
75
70
70
EDP
35
20
40
30
45
35
0
0
45
35
10
0
0
DP
30
10
15
10
30
15
35
70
35
5
0
Contact Mainta heed Buring Rotation
Contact Lest During Rotation
Tomerl Push Ishrection
Incorredd Fuxb Direction
(a) Single-arm Tum_Lever
(h) Single-arm Push_ Fraherr
Ni05
ATgnod Finsh: Bhretie
Miseligroa Flizhh Direatren
Certensi HTP
Inserreet Fúp Recation
Ronfanbaan
(c) Bi Banual Gra
(d) Bi manual Flip_Book
Aigned Plasemant Elrestion
Misallaredd Flare mene stion
(c) Bi-manual Pack Package
Figure A4. Examples of successes and failures each task, with green indicating successful behaviors and red indicating failures.
we can evaluate at g and ge,
ce (((, + ge, ge.
Using the equivariance of and substituting in Eqn. 18 for gives
co(gS,ga(S) + ge, k) geo(S, (5) c.
18

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Spherical Diffusion Policy
desired.
Proposition C.2 (Denoise equivariance). The action prediction is equivariant to group G. ie., x(gS) = gT((S), when the denoising function is equivariant to group G, i.e,, <(S,, gA k) = ge(S, AK k), and the distribution 'D from which to sample the noise, is invariant to group G. i.e., = gD.
Proof. Simplifying Equation. by dropping have:
A = a(A - A + 2), D
(19)
When & = K, the denoise is sampled form D: co A, K) d1, d1 ~ D, and the denoised action A A is:
A a(O ydl d2),
d1 ~D
(20)
a(22 yd1)
(21)
algdz 7ghh,, )
d,
(22)
gga(d's 7d()
(23)
9-
(24)
When &s = K - 1, the denoise is applied to the noisy action, to generate the cleaner action. Transforming the input the denoising function in Equation. 19 by g:
K 1) 2) a 11011-0oog8,_h
(25)
(S, A-1 K 1) + gz) = =a(gAK-1 -a(AAK-1 - 7E0 K-1 AK11 - -1)+2)
(26)
(27)
-
(28)
Equation. 28 holds for k K --1,1,1 1,K .., thus by applying it iteratively, have gA²,
Proposition. C.I. C.2 specify the prerequisites of an equivariant diffusion policy. Empirically we find that the group invariant distribution D can be relaxed to a Gaussian distribution M,, which is simple and achieves good performance.
C.2. Translation Invariance by Canonicalization
Translation invariance is achieved using a relative action formulation (Chi et al., 2024) and state-action canonicalization (Zeng et al., 2022; Wang et al., 2021; Zhu et al., 2022a; Jia et al., 2023). We summarize and proof this property. Proposition C.3. The relative state-action formulation is T(3) (Iranslational) invariant.
Proof. Translating both the state and the action by g E T(3) we have:
g ((0 + g) (er + g), (ET + g) (eT + g):
eR, Cgrip)
(0 ET, eT - ET, eR; Egrip)
Sean
9A can
((AT + g) - (ET + g), AR. Agrip)
(Ar - eT, AR. Agrip)
A can
Therefore (Soon) = π (caa))) = gA con
19

---

Spherical Diffusion Policy
D. Hyperparameters
Hyperparameters for diffusion-based baseline methods are listed Table A2. SDP generally adopts Diffusion Policy's hyperparameters, except for batch size, because SDP heavier than other baselines.
SDP
EquiDiff
EquiBot
DifiPo
DP3
DP3 paper
Batch Size
32
64
64
64
128
128
Prediction Horizon
16
16
16
16
16
Action Horizon
8
8
8
8
Learning Rate
le-4
le-4
le-4
le-4
le-4
le-4
Epochs
500
500
500
500
500
3000
Learning Rate Scheduler
cosine
cosine
cosine
cosine
cosine
cosine
Noise Scheduler
DDPM
DDPM
DDPM
DDPM
DDIM
DDIM
Diffusion Train/Test Step
100
100
100
100
100/10
100/10
Encoded Scene Dimension
128
128
128
128
64
64
Table A2. Hyperparameters for baselines.
E. Architecture of the Point Cloud Encoder
The point cloud encoder is 5-layer ResNet (He al., 2016), consisting of EquiformerV2 (Liao al., 2024) graph convolution layers in the hidden layers and EquiformerV2 origin convolution layer in the last layer to aggregate all the features into a single point.
EqulformerV2 Graph
Equifamerv2 Graph Conv
EqulformerV2
Equifomerv2 Graph Conv
Equifomne'' Origion
Channno Wise Lineer
1024 pts, c
1024 pts,
258 pts, 64 c
pts,
pts.
pt 128
de (0. o, 0)
Equilormer22 Graph Convolation
EquifomerV2 Origion Convolution
Figure A5. Overview of the Point Cloud Encoder. Top: the point cloud encoder. Bottom: the details of each block in the encoder. "pts" stands for the number of points and "c" stands for the number of channels.
20


Reviews:
[{"review_id": "rKiX1W2hTw", "reviewer": "Reviewer_mcma", "summary": "This paper presents a novel SE(3)-equivariant diffusion policy, named Spherical Diffusion Policy (SDP), aimed at improving generalization for robotic manipulation tasks across varying 3D transformations. The key motivation stems from the assumption that embedding states, actions, and denoising processes in spherical Fourier space ensures SE(3)-equivariance, thereby enabling robust generalization across transformed scenes without extensive data collection. Specifically, the framework incorporates a spherical encoder to embed scene features, spherical FiLM layers for equivariant conditioning, and a spherical denoising temporal U-net for spatiotemporal equivariance. Additionally, theoretical analyses verify the equivariance of the proposed method. Extensive simulation and physical experiments demonstrate the effectiveness and superior performance of SDP over state-of-the-art baselines on multiple challenging robot manipulation tasks.\n\n**update after rebuttal**\n\nAfter reading the rebuttal, most of my concerns have been addressed, and I am inclined to keep my original score.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their thoughtful feedback. We respond below:\n\n> \"Some SE(3) Diffusion Models should be cited as the related work, such as: ...\"\n\nThank you for highlighting these relevant works. We will include citations to these papers in the revised related work section.\n\n>\"Although demonstrated in Sec. 4.2, the practical advantages of the proposed ... remain unclear. Specifically, what does it mean that \"Vector Neuron only supports up to l=1\"?\"\n\nBoth Vector Neurons (VN) and our method can be interpreted as representing features in spherical Fourier space, up to a specified maximum frequency $l$. While our method supports arbitrary spherical harmonic types (we use $l_{max}=2$ in our paper), VN is limited to using only scalars ($l=0$) and vectors ($l=1$) as features. The scalars in VN are mathematically equivalent to type-0 features, which are invariant to rotation. The 3D vectors $V = [V_x, V_y, V_z]$ used in VN correspond to type-1 features $c_1 = [c^{-1}_1, c^{0}_1, c^{1}_1]$, as both are three-dimensional and transform under rotation via a rotation matrix. Thus, VN only supports spherical features up to type-1. However, type-1 features have limited representational capacity—for example, they are incapable of capturing spherical distributions with two distinct modes.\n\n>\" could you clarify how the \"truncated spherical Fourier coefficients ...\" in contrast to the \"computationally heavy SO(3) irreps used in ET-SEED\"?\"\n\nBoth SDP and ET-SEED achieve SE(3) equivariance by assigning SO(3)-steerable features to each point in the point cloud. While both methods use irreducible representations (irreps), SDP employs lightweight, order-wise linear layers—each order $m$ only connects to itself. In contrast, ET-SEED (based on the SE(3)-Transformer) utilizes heavier and more redundant fully connected linear layers that connect all orders $m$ across all types $l$.\n\n> \"...comparisons regarding computational efficiency are currently missing.\"\n\nEmpirically, although ET-SEED employs a two-stage diffusion process, its inference time is 60× slower than that of SDP (29.4s vs. 0.44s; see Table below). Moreover, SDP's inference time is on the same order of magnitude as that of Diffusion Policy (SDP is approximately 5× slower than DP).\n\nTable R1. Practical statistics for SDP and SOTA baselines.\n|                            | Diffusion Policy | EquiDiff                 | EquiBot                    | SDP                        | ET-SEED                           |\n|----------------------------|------------------|--------------------------|----------------------------|----------------------------|-----------------------------------|\n| Inference Speed (Second) ↓ | 0.09             | 0.14                     | 0.18                       | 0.44                       | 29.4                              |\n| Training Batch Size ↑      | 64               | 64                       | 64                         | 32                         | 1                                 |\n\n\n> \"Table 1 indicates that as the degree of SE(3) initialization increases, SDP continues to exhibit significant performance degradation, .... Could you address this point explicitly?\"\n\nAuthors agree that SE(3) equivariant methods should perform consistently no matter how the task is rotated or translated in 3D, provided the transformations are of the full environment. However, in Table 1, these tasks are not perfectly SE(3) transformed, since the gravity direction is not transformed, the camera view (occlusion) is not transformed, and the robot (kinematics) is not transformed. All of these factors could add complexity to the task (e.g., greater table tilt leads to increased object instability), thus SDP continues to exhibit performance degradation. Despite the increasing complexity, we find that SDP outperforms all baselines on all levels of SE(3) initialization of the tilted table tasks in Table 1.\n\n> \"Please clarify the fundamental differences ... in Spherical Fourier Neural Operators.\"\n\nBoth methods leverage the spherical Fourier (SF) transform to process signals on the sphere. However, our method is purely based on SF, while SFNO combines an SF neural network for global convolution with a pointwise MLP for local non-linearity. Additionally, our proposed SDTU enables SO(3) and temporal equivariant convolution, and SFiLM provides equivariant spherical conditioning—capabilities not present in SFNO.\n\n> \"Why not directly consider SE(3)-invariant representations?\"\n\nLeveraging SE(3)-invariant representations is a valuable baseline suggestion, and we have included it as an ablation. Specifically, we implemented a baseline called DP3-cano in [Table R2](https://openreview.net/forum?id=U5nRMOs8Ed&noteId=IN1AESox8P), which achieves SE(3)-invariance by canonicalizing (i.e., normalizing or transforming) both the point cloud and the action into the gripper frame. However, experimental results show that DP3-cano significantly underperforms the equivariant method SDP.", "claims_and_evidence": "Yes.", "methods_and_evaluation_criteria": "Yes.", "theoretical_claims": "Yes.", "experimental_designs_or_analyses": "Yes.", "relation_to_broader_scientific_literature": "The paper advances prior works on equivariant diffusion policies by generalizing leveraging spherical Fourier representations for improved generalization in robotic manipulation tasks.", "essential_references_not_discussed": "Some SE(3) Diffusion Models should be cited as the related work, such as:\n[1] SE (3) diffusion model with application to protein backbone generation, ICML'2023\n[2] SE (3) diffusion model-based point cloud registration for robust 6d object pose estimation, NeurIPS'2023\n[3] SE(3)-DiffusionFields: Learning smooth cost functions for joint grasp and motion optimization through diffusion, ICRA'2023", "other_strengths_and_weaknesses": "1. Strengths\n- The paper introduces an innovative SE(3)-equivariant diffusion policy framework, effectively enabling model generalization across SE(3)-transformed environments;\n\n- The proposed spherical Fourier features, spherical FiLM layers, and spherical denoising temporal U-Net achieve SE(3)-equivariance with solid theoretical grounding;\n\n- Extensive empirical evaluations in both simulation and real-world robotic experiments demonstrate substantial improvements in performance and generalization compared to existing state-of-the-art methods.\n\n2. Weaknesses\nAlthough demonstrated in Sec. 4.2, the practical advantages of the proposed spherical Fourier representations over previous SE(3)-equivariant representations (e.g., Vector Neuron and ET-SEED) remain unclear. Specifically, what does it mean that \"Vector Neuron only supports up to l=1\"? Furthermore, could you clarify how the \"truncated spherical Fourier coefficients provide a compact approximation of spherical features and are compatible with SO(3) rotations,\" in contrast to the \"computationally heavy SO(3) irreps used in ET-SEED\"?\n\n- Table 1 indicates that as the degree of SE(3) initialization increases, SDP continues to exhibit significant performance degradation, raising concerns about the actual effectiveness and robustness of the proposed method. Could you address this point explicitly?\n\n- Please clarify the fundamental differences between the spherical Fourier representations employed in this paper and those used in Spherical Fourier Neural Operators.\n\n- Given the importance of inference speed in robotic control applications, comparisons regarding computational efficiency are currently missing. It would be beneficial if the authors could provide clarity on this aspect.", "other_comments_or_suggestions": "See weaknesses.", "questions_for_authors": "Why not directly consider SE(3)-invariant representations rather than focusing on designing SE(3)-equivariant representations? Both approaches yield similar results, and SE(3)-invariant representations might even be better designed than equivariant ones？", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "D6N4aY75si", "reviewer": "Reviewer_CbPJ", "summary": "This paper works on improving the equivariance of diffusion policy for manipulation tasks. To this end, the authors align the input point clouds and robot state into a canonical coordinate frame to achieve translational equivariance, and then project the encoded observations onto spherical harmonic basis to achieve SO(3) equivariance. To further handle the SO(3) equivariance within network structure itself, the authors upgrade the 1D convolution in vanilla diffusion policy to a mix channel temporal convolution, and the FiLM condition to a spherical FiLM layer. To demonstrate the effectiveness of the proposed method, especially its SE(3) equivariance, the authors augment the scene to manipulate with additional rotation and translation,  and achieved superior performance than existing baselines that use absolute control or velocity control mode.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their thoughtful feedback. We respond below ([link to figures](https://limewire.com/d/DAndu#qMh5UOCXI6)):\n\n>\"... the point clouds and robot arm states can be further aligned within gripper's local coordinate frame by applying a known rotation.\nIf the authors target to achieve SO(3) equivariance to object's absolute rotation, then it is equivariant when the scene is under a global rotation to a demonstration.\"\n\nLeveraging SE(3)-invariant representations is a valuable baseline suggestion, and we have included it as an ablation. Specifically, we implemented a baseline called DP3-cano, which achieves SE(3)-invariance by canonicalizing (i.e., normalizing or transforming) both the point cloud and the action into the gripper frame. However, experimental results show that DP3-cano significantly underperforms the equivariant method SDP.\n\nTable R2. Additional Ablations.\n|                 |                         | Coffee 15 | Three Pc. As. 15 | Square 15 | Threading 15 | Avg SR |\n|-----------------|-------------------------|-----------|------------------|--------|-----------|--------|\n| SDP             | SE(3) equivariance      | 54        | 49               | 38     | 53        | 49     |\n| Discrete U-net* | Octahedron equivariance | 42        | 16               | 34     | 48        | 35     |\n| DP3-cano*       | SE(3) equivariance        | 40        | 0                | 8      | 12        | 15     |\n| DP3*            | None                    | 20        | 0                | 0      | 4         | 6      |\n\n*Currently, these baselines have been trained for 300 out of the planned 500 epochs. We expect the final average success rate to improve by approximately 4%.\n\nWe hypothesize that local equivariance in SDP plays a crucial role. Furthermore, Benjamin et al. [1] demonstrate that using equivariant features results in significantly lower prediction errors compared to invariant features in molecular property prediction tasks.\n\n[1] Benjamin Kurt Miller, Mario Geiger, Tess E. Smidt, Frank Noé, Relevance of Rotationally Equivariant Convolutions for Predicting Molecular Properties, Machine Learning for Molecules Workshop at NeurIPS 2020\n\n> \"Second, the translation equivariance is achieved by explicit alignment of point clouds ... Normalization is a standard preprocessing in point cloud networks, ... I prefer to have SO(3) in the title instead of SE(3) to be more focused and reduce confusion.\"\n\nFor clarification, our method is SE(3)-equivariant. Specifically, SDP achieves SO(3) equivariance through the use of spherical Fourier representations, and T(3) (translation) equivariance via canonicalization. It is important to note that canonicalization differs from normalization in that the generated action is also transformed into the canonical frame.\n\n> \"it would be beneficial to discuss it and correspondingly validate the maximum degree of spherical harmonics in ablation studies.\"\n\nWe agree that truncated spherical harmonic coefficients approximate the underlying spherical function, and using a low maximum frequency can lead to a loss of important details. As shown in Table A3 of the paper, setting $l = 1$ results in a noticeable performance drop. However, performance saturates at $l = 2$ and $l = 3$, suggesting that higher frequencies provide diminishing returns. Due to space limitations, this table is currently located in the appendix, but we will strengthen the connections in the main text to highlight this finding.\n\n> \"... it worth to conduct experiments on more demonstrations, such as the settings in EquiDiff to see how the performance saturate.\"\n\nThis is a great question—we have plotted data scaling curves in Figure R1. Each point represents the average performance across four tilted-table tasks (with tilt angles in $[0, 15^\\circ]$) for SDP, EquiDiff (EDP), and DiffPo (DP). Notably, SDP trained with $10^2$ demonstrations outperforms EDP trained with $10^3$, while EDP with $10^2$ demonstrations achieves comparable performance to DP trained with approximately $10^{2.5}$ demonstrations.\n\n> \"... no further elaboration about the detailed settings, such as wether the control signal is directly from the policy network or converted from relative control signals.\"\n\nThe absolute position control does not have translational equivariance while relative position control does, as explained in Section 4.1 and proof in Appendix C.2.\n\nThe ablation study in Section 5.3, which uses absolute action control, highlights the importance of translational equivariance in our method.\n\n> \"The architecture of the encoder\"\n\nSee Figure R2.\n\n> \"Any motivation to have physical experiments with only one ray-fin finger ...\"\n\nWe use the shared robot in our lab and didn’t make any changes on the hardware. \n\n> \"Is that better to remove e_i^T−e_i^T\"\n\nYes, it's for the convenience of presentation in the paper.", "claims_and_evidence": "The major claim of this paper is its SE(3) equivariance and its better sample efficiency and generalization. These claims are in general well supported with straight forward proof of propositions as well as extensive experiments. \nHowever, I have two major concerns regarding to this claim:\n- First, do we really have to achieve SO(3) equivariance within the network architecture itself when the observation inputs are point clouds? \n  - If the author target to achieve SO(3) equivariance to relative rotation to the end-effector, the point clouds and robot arm states can be further aligned within gripper's local coordinate frame by applying a known rotation. \n  - If the authors target to achieve SO(3) equivariance to object's absolute rotation, then it is equivariant when the scene is under a global rotation to a demonstration. However, during the experiments, the proposed method demonstrate significantly better performance on manipulation tasks with multiple rigid objects, which needs more justification.\n- Second, the translation equivariance is achieved by explicit alignment of point clouds and proprioceptive states, therefore the diffusion policy network itself is SO(3) equivariant. Normalization is a standard preprocessing in point cloud networks, therefore it is better to explain how this normalization is distinct from others. In general, I prefer to have SO(3) in the title instead of SE(3) to be more focused and reduce confusion.\n\nThis paper needs more discussions and justifications about these two issues that will better validate the motivation of this paper.", "methods_and_evaluation_criteria": "The methods and evaluation criteria make sense in general, except for one minor issue:\n- Projection on to spherical harmonic basis will lose details of added noise and therefore potentially interferes the diffusion process. Therefore, it would be beneficial to discuss it and correspondingly validate the maximum degree of spherical harmonics in ablation studies.", "theoretical_claims": "I checked the propositions in this paper and they are generally correct.\nMy only concern is that theoretically the policy is only SO(3) equivariant to scene under a global rotation. However, the experiments demonstrate superior performance in scene with multiple randomly placed objects, such as Three Pc. Assembly, which requires more justification and explanation.", "experimental_designs_or_analyses": "The experiment designs are valid except for two minor issues:\n- First, the paper only compare with previous works with 100 demonstrations for training, it worth to conduct experiments on more demonstrations, such as the settings in EquiDiff to see how the performance saturate.\n- Second, in Sec.5.3 it is somewhat surprising that absolute position control is significantly worse. However, there is no further elaboration about the detailed settings, such as wether the control signal is directly from the policy network or converted from relative control signals. In addition, it is also unclear what are the conclusion we can draw from this performance gap.", "relation_to_broader_scientific_literature": "This paper is an application of SO(3) equivariance to diffusion policy tasks. Therefore, it has connection to previous works about equivariant networks. While the results demonstrated in this paper is domain-specific and does not expand the understanding of equivariant network architectures in general.", "essential_references_not_discussed": "Essential references have been discussed to understand this paper.", "other_strengths_and_weaknesses": "Strengths:\n - The performance is significantly better than existing work when number of demonstration is limited for training.\n\nWeaknessnes:\n- Some descriptions needs to be further elaborated: line-236~240: The architecture of the encoder is difficult to understand. Does the authors mean that a resnet is applied point-wisely first and then sent to EquiformerV2 for further feature extraction? It is better to illustrate the detail of encoder as well to reduce confusion.\n- Any motivation to have physical experiments with only one ray-fin finger as the end-effector on each arm? Which is less consistent to the simulation.", "other_comments_or_suggestions": "- Is that better to remove e_i^T−e_i^T since it is constantly zero, or is there any specific reason to keep it?", "questions_for_authors": "Please refer to the above comments and address major concerns about:\n- Claims And Evidence: what are the specific rotation this paper aims to be equivariant to, transformation between gripper coordinate frame, global transformation of the scene, or individual transformation of objects within a scene\n- Experiments: Discuss how relative position control and SE(3) equivariance contribute to the performance distinctly and their relation. Also discuss about how the performance varies when more demonstration is given. \n- Methods: Elaborate more about the structure of the encoder and the projection onto spherical harmonics's effect on the diffusion process.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "21ZnQTRY4X", "reviewer": "Reviewer_Sqpd", "summary": "This paper propose one new method called “Spherical Diffusion Policy (SDP)” for robot manipulation. The paper focus on 3D generalization, using SO(3) and T(3) equivariance (i.e., full SE(3) group) to handle random tilts and object placements. It design a special spherical encoder with spherical FiLM layer and “spherical denoising temporal U-net (SDTU).” The method is tested on many simulation tasks as well as real robot tasks (both single-arm and bimanual). The result show improved performance compared to baselines like EquiDiff or Diffusion Policy in environment with random initial poses.\n\n## After Rebuttal\nThough the overall results reported by authors are very good, in the authors' rebuttal, they are not able to reply my concern that the baseline is not tuned. The authors provide the param table while they deliberately miss the baseline (DP3) that is not tuned in the table. Also the reply from authors \"We use the DP3 results reported in the CoRL paper Equivariant Diffusion Policy\" show that the results are directly copied from previous works without justification. I am not sure if there is an issue on this but this looks suspicious to me. Considering the above mentioned factors, I would change my score from weak accept to weak reject. The other parts of this paper are all good.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their thoughtful feedback. We respond below:\n\n> \"Could authors tune some parameters of this baseline to see, like the longer prediction horizon? Besides, it is good to report the parameters for all baselines.\"\n\nWe use the DP3 results reported in the CoRL paper Equivariant Diffusion Policy. Additionally, we implemented a new baseline, DP3-canonicalization, which improves upon DP3 but still significantly underperforms SDP (see Table R2). Regarding DP3's poor performance on the MimicGen tasks, we hypothesize that this is due to its use of an MLP as the vision encoder. The MLP may struggle to capture information about multiple objects or their orientations—factors that are critical in the MimicGen tasks.\n\n> \"More ablation studies would be helpful to understand the method.\"\n\nIn Table R2, we add DP3 (no equivariance) and DP3-cano (global SE(3)-equivariance) as additional baselines for ablation. We find that SDP, which has local equivariance, outperforms the global SE(3)-equivariance used in DP3-cano.\nWe also introduce a new baseline, Discrete SDTU, which enforces discrete equivariance using the Octahedral or Cubic group, which discretize SO(3)—similar to EquiDiff, which uses the cyclic group C8 that discretize SO(2). Our results show that Discrete SDTU leads to an approximate 10% drop in performance, highlighting the advantage of SDP’s use of continuous spherical Fourier representations.\n\n> \"Detailed parameters for baselines for fair comparison.\"\n\nWe provide detailed parameters for baselines. SDP generally adopts Diffusion Policy’s hyperparameters, except for batch size, because SDP is heavier.\n\nTable R3. Hyperparameters for baselines.\n|                           | SDP    | EquiDiff | EquiBot | DP     |\n|---------------------------|--------|----------|---------|--------|\n| Batch Size                | 32     |    64    |    64   |   64   |\n| Prediction Horizon        |   16   |    16    |    16   |   16   |\n| Action Horizon            |    8   |     8    |    8    |    8   |\n| Learning Rate             |  1e-4  |   1e-4   |   1e-4  |  1e-4  |\n| Epochs                    |   500  |    500   |   500   |   500  |\n| Learning Rate Scheduler   | cosine |  cosine  |  cosine | cosine |\n| Noise Scheduler           |  DDPM  |   DDPM   |   DDPM  |  DDPM  |\n| Diffusion Train/Test Step |   100  |    100   |   100   |   100  |\n| Encoded Scene Dimension   |   128  |    128   |   128   |   128  |\n\n> \"report the failure mode of the baselines/their own method.\"\n\nDetailed failure modes are reported in Section B.3, Table A1, and Figure A4. The major failure mode is inaccurate action prediction when the end effector is about to make contact. We will strengthen the connection between the failure modes analysis section and the main text.", "claims_and_evidence": "The claims are supported by simulation and real-world experiments.", "methods_and_evaluation_criteria": "The proposed method and evaluation criteria make sense to me.", "theoretical_claims": "Yes.", "experimental_designs_or_analyses": "The experiment analysis overall looks good to me, with an extensive baseline comparison and a few ablation studies.\nHowever, there is a small problem regarding to baselines. I found that the results for DP3 (Ze et al., 2024) do not make sense to me, as it uses 3D information yet even worse than image-based methods, which does not align with my tuning experience. Could authors tune some parameters of this baseline to see, like the longer prediction horizon? Besides, it is good to report the parameters for all baselines.", "relation_to_broader_scientific_literature": "The contributions are very related to the literature.", "essential_references_not_discussed": "No", "other_strengths_and_weaknesses": "N/A", "other_comments_or_suggestions": "1. More ablation studies would be helpful to understand the method.\n2. Detailed parameters for baselines for fair comparison.", "questions_for_authors": "As mentioned above, the authors are suggested to tune the baseline methods or report the failure mode of the baselines/their own method. Besides, how the baseline is implemented and tuned should be reported.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "vhngnTnhum", "reviewer": "Reviewer_Kgae", "summary": "This paper introduces the Spherical Diffusion Policy (SDP), a novel method for robotic manipulation that enforces continuous SE(3) equivariance with spherical Fourier representations. The authors propose a spherical modification of the FiLM layer commonly used in diffusion-based policy networks, prove the invariance property of their proposed approach, and empirically verified their approach in both real and simulated domains.\n\n## update after rebuttal\n\nAccepted the original version of the paper. No updates to the review are necessary.", "questions": "", "limitations": "", "rating": 5, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their thoughtful feedback. We respond below:\n\n> \"However, it's not 100% clear how these experiments were performed from the description: \"We train all the baselines on progressively tilted environments with 100 demonstrations.\" Does tilt=30 degrees imply that demonstrations are on environments sampled between 0 and 30 degrees? Some clarification on this experiment would be helpful.\"\n\nWe are sorry for the confusion. Tilt=30 degrees imply the environment (the table top) is sampled between 0 and 30 degrees. We will add the clarification to the paper.\n\n> \"The paper is a bit densely written at several points, making it not very approachable\"\n\nCould the reviewer please point out which points can be improved?\n\n> \"No testing in cluttered scenes\"\n\nFor clarification, the (i) Pick Place task and (j) Kitchen tasks in MimicGen (see Figure R3 for visualization) involve multiple objects and can be viewed as slightly cluttered scenes. Unfortunately we don’t have time to test densely cluttered scenes.\n\n> \"No statistical analysis of the results or inference timing data reported\"\n> \"The authors claim that their work is more “computationally efficient” than competing approaches, but fail to provide any inference-time comparisons.\"\n\nThe step-wise success rate for physical experiments is reported in Table A.1. Moreover, in Table R1 we add the inference timing data.\n\n> \"Grammatical Issues\"\n\nThank you for pointing out the grammar issues and typos, we will fix these errors in the revision.\n\n> \"EquiBot (Yang et al., 2024a) are limited to degree l = 1 representation that suppress rich information. Can the authors elaborate on this point and give some examples of problems for which that would be a limiting constraint?\"\n\nPlease see the [response](https://openreview.net/forum?id=U5nRMOs8Ed&noteId=rKiX1W2hTw) to \"\"Although demonstrated in Sec. 4.2, the practical advantages of the proposed ... remain unclear. Specifically, what does it mean that \"Vector Neuron only supports up to l=1\"?\"\"\n\n> \"It was not clear to me why equation 5 was chosen in particular. It’s clear from the proof in A.2 that it preserves equivariance, but it seems other expressions (e.g. without normalization) would achieve the same goal.\"\n\nWe agree other expressions (without normalization or using tensor products, etc) could achieve the same equivariance. The proposed method stems from the nonlinearity operation in Vector Neurons. We find Equation 5 works well and we haven’t compared other expressions.", "claims_and_evidence": "The paper claims that enforcing SE(3) equivariance using spherical Fourier features leads to better generalization across varied 3D arrangements and improved sample efficiency. These claims are supported by detailed simulation results.", "methods_and_evaluation_criteria": "The authors evaluate their approach on a number of MimicGen tasks and real-world robotics tasks, finding that their method (SDP) outperforms other equivariant and non-equivariant baselines. While it seems the MimicGen environments only change the yaw dimension of the objects, the real world experiments use randomizations along other dimensions as well. \n\nOverall, I believe the set of tasks is sufficient for evaluation. Still, It would be nice to see how this method fares in more cluttered scenes.", "theoretical_claims": "The paper makes claims about the equivariance of the convolution and SFiLM, which I verified in the appendix.", "experimental_designs_or_analyses": "They additionally perform an analysis of how all baselines degrade with increasing initialization noise, showing that SDP degrades the most gracefully. However, it's not 100% clear how these experiments were performed from the description: \"We train all the baselines on progressively tilted environments with 100 demonstrations.\" Does tilt=30 degrees imply that demonstrations are on environments sampled between 0 and 30 degrees? Some clarification on this experiment would be helpful.", "relation_to_broader_scientific_literature": "This work extends and improves upon recent advances in diffusion-based policy learning and equivariant neural networks. It builds directly on prior works such as Diffusion Policy (Chi et al., 2023), EquiDiff, and EquiBot, addressing limitations related to discretized equivariance and computational inefficiencies.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "Strengths:\n- Novel algorithm for SE(3) invariant closed loop control for robot manipulation\n- Extensive evaluation and demonstration of the benefits of this approach\n- Detailed theoretical justification for the approach\n\nWeaknesses:\n- The paper is a bit densely written at several points, making it not very approachable \n- No testing in cluttered scenes\n- No statistical analysis of the results or inference timing data reported", "other_comments_or_suggestions": "Grammatical Issues:\n\n“… the proof essentially follows Schur’s lemma (Schur, 1905) that any linear operation of SO(3) irreps acts as on each irreducible subspace is equivariant.”\n\n“Diffser” → “Diffuser”\n\n“Diffusor” → “Diffuser”", "questions_for_authors": "EquiBot (Yang et al., 2024a) are limited to degree l = 1 representation that suppress rich information. Can the authors elaborate on this point and give some examples of problems for which that would be a limiting constraint?\n\nThe authors claim that their work is more “computationally efficient” than competing approaches, but fail to provide any inference-time comparisons. \n\nIt was not clear to me why equation 5 was chosen in particular. It’s clear from the proof in A.2 that it preserves equivariance, but it seems other expressions (e.g. without normalization) would achieve the same goal.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This paper proposes a novel method, Spherical Diffusion Policy, which is equivariant to 3D rotations and invariant to 3D translations enabling generalization to unseen scenes. Specifically, it combines a spherical encoder to embed scene features, spherical FiLM layers for equivariant conditioning, and a spherical denoising temporal U-net for spatiotemporal equivariance. The equivariance of the proposed method is verified both theoretically and empirically. While most of the reviewers appreciate the novelty of the paper, one reviewer raised concern on the parameters of one baseline. However, overall the experiments are sound, and I recommend an accept.
Decision: Accept (poster)