1. Bibliographic Information

1.1. Title

Diffusion-EDFs: Bi-equivariant Denoising Generative Modeling on SE(3) for Visual Robotic Manipulation

1.2. Authors

The paper is authored by a team of researchers from several institutions, indicating a significant collaborative effort:

• Yonsei University: Hyunwoo Ryu, Jiwoo Kim, Hyunseok An, Junwoo Chang, Chaewon Hwang, Jongeun Choi.

• University of California, Berkeley: Joohwan Seo, Roberto Horowitz, Jongeun Choi.

• Samsung Research: Taehan Kim.

• Massachusetts Institute of Technology (MIT): Yubin Kim.

• Ewha Womans University: Chaewon Hwang.

  The authors have strong backgrounds in robotics, control theory, and machine learning, particularly in the area of geometric deep learning and equivariant models. Jongeun Choi and Roberto Horowitz are established figures in control systems and robotics.

1.3. Journal/Conference

The paper was published as a preprint on arXiv. While the paper itself does not state a conference venue, its predecessor, EDFs, was published at ICLR 2023. The quality and topic are well-suited for top-tier robotics and machine learning conferences such as the Conference on Robot Learning (CoRL), Robotics: Science and Systems (RSS), or the International Conference on Learning Representations (ICLR).

1.4. Publication Year

2023

1.5. Abstract

The abstract introduces Diffusion-EDFs, a novel approach for visual robotic manipulation that combines diffusion generative models with $SE(3)$ equivariance. The authors highlight the model's key advantages:

• Method: It is a diffusion-based generative model that operates on the $SE(3)$ manifold, designed to be bi-equivariant.
• Data Efficiency: It achieves impressive data efficiency, requiring only 5 to 10 human demonstrations for end-to-end training.
• Training Speed: The training process is very fast, taking less than an hour.
• Performance: Benchmark experiments show that the method has superior generalizability and robustness compared to state-of-the-art approaches.
• Validation: The proposed methods are validated through real-world hardware experiments.

1.6. Original Source Link

• Original Source: https://arxiv.org/abs/2309.02685

• Publication Status: This is a preprint available on arXiv.

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2. Executive Summary

2.1. Background & Motivation

• Core Problem: The central challenge addressed is teaching robots complex 6-Degree-of-Freedom (6-DoF) manipulation tasks, like picking up an object and placing it precisely, from a small number of human demonstrations. Humans can learn new physical tasks from just a few examples, but it is notoriously difficult for robots to achieve this level of data efficiency and generalization.

• Existing Gaps: Prior research faced a trade-off:

  1. Diffusion Models for Robotics (e.g., SE(3)-Diffusion Fields): These models are powerful for capturing the stochastic and multi-modal nature of human demonstrations (i.e., there can be multiple valid ways to perform a task). However, they typically require a large number of demonstrations to train and do not generalize well to new object positions or orientations not seen during training.
  2. Equivariant Models for Robotics (e.g., EDFs, R-NDFs): These models build in physical symmetries, specifically $SE(3)$ equivariance (invariance to 3D rotation and translation), which dramatically improves data efficiency and generalization. A model that understands that rotating the entire scene doesn't change the fundamental task can learn from far fewer examples. However, a leading equivariant method, Equivariant Descriptor Fields (EDFs), relied on energy-based models (EBMs), which are extremely slow to train (over 10 hours for a few demonstrations).

• Innovative Idea: The paper's key insight is to synthesize the best of both worlds: combine the generative power of diffusion models with the data efficiency and generalization of $SE(3)$-equivariant models. They propose Diffusion-EDFs as a diffusion-based successor to the energy-based EDFs, aiming to preserve the high performance of EDFs while drastically reducing the training time.

2.2. Main Contributions / Findings

The paper makes several key contributions to the field of robotic learning:

1. A Novel Bi-equivariant Diffusion Model on SE(3): This is the first work to develop a fully $SE(3)$ bi-equivariant diffusion model for visual robotic manipulation. The authors provide the necessary theoretical framework to design a diffusion process and a score-matching model that respects the complex symmetries of robot manipulation tasks (bi-equivariance, which accounts for both scene and end-effector transformations).

2. Dramatic Improvement in Training Efficiency: Diffusion-EDFs are shown to be significantly faster to train than their predecessor, EDFs. The paper claims a 15x speedup, reducing training time from over 10 hours to under an hour on a standard GPU, making the approach far more practical.

3. Advanced Hierarchical Architecture for Scene Understanding: They propose a new U-Net-like, multi-scale architecture for their equivariant networks. This gives the model a wider "receptive field," allowing it to understand scene-level context (e.g., relationships between multiple objects) instead of being limited to object-centric reasoning. This is a crucial step towards solving more complex, sequential tasks.

4. State-of-the-Art Performance: Through extensive simulation and real-world experiments, Diffusion-EDFs are shown to outperform existing state-of-the-art equivariant (R-NDFs) and diffusion-based (SE(3)-Diffusion Fields) methods. The model demonstrates remarkable generalization to unseen objects, poses, and cluttered environments, even when trained on only 5-10 demonstrations and without needing object segmentation.

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3. Prerequisite Knowledge & Related Work

3.1. Foundational Concepts

3.1.1. Diffusion Models (Score-Based Generative Modeling)

Diffusion models are a class of generative models that learn to create data by reversing a gradual noising process.

• Forward Process: You start with a real data sample (e.g., an image or, in this paper, a robot pose $g_0$). Over a series of timesteps $t$, you gradually add a small amount of Gaussian noise. After many steps, the original data is transformed into pure, structureless noise. This process is mathematically fixed and does not involve learning.
• Reverse Process (Denoising): The core of the model is a neural network trained to reverse this process. At any timestep $t$, the network takes the noised data $g_t$ as input and tries to predict the small amount of noise that was added. By repeatedly subtracting this predicted noise, the model can start from pure noise and gradually "denoise" it back into a realistic data sample.
• Score Matching: The denoising process is often framed as "score matching." The score of a data distribution at a point is the gradient of the log-probability of the data at that point, i.e., $\nabla_x \log p(x)$. The neural network is trained to predict this score for the noised data distribution at every timestep $t$.
• Sampling via Langevin Dynamics: Once the score function $\mathbf{s}_t(g) = \nabla \log P_t(g)$ is learned, you can generate new samples using an iterative process like Langevin dynamics. This involves taking small steps in the direction of the score (the direction of highest probability increase) while adding a small amount of noise to encourage exploration.

3.1.2. The Special Euclidean Group SE(3)

$SE(3)$ is the mathematical group that describes all possible rigid-body motions (orientations and positions) in 3D space.

• Elements: An element $g \in SE(3)$ represents a transformation and can be described by a pair $(R, \mathbf{p})$, where $R \in SO(3)$ is a 3x3 rotation matrix and $\mathbf{p} \in \mathbb{R}^3$ is a 3D translation vector.
• Application in Robotics: $SE(3)$ is fundamental to robotics, as it precisely describes the pose (position and orientation) of a robot's end-effector or any object in the world relative to a reference frame.
• Group Operation: The composition of two transformations $g_1 = (R_1, \mathbf{p}_1)$ and $g_2 = (R_2, \mathbf{p}_2)$ is another transformation $g_1 g_2 = (R_1 R_2, R_1 \mathbf{p}_2 + \mathbf{p}_1)$.

3.1.3. Equivariance

Equivariance is a powerful property for neural networks that deals with symmetries.

• Definition: A function $f$ is equivariant with respect to a group of transformations $\mathcal{G}$ if transforming the input and then applying the function is equivalent to applying the function first and then transforming the output. Formally, for a transformation $T_g$ from the group, $f(T_g(x)) = S_g(f(x))$, where $S_g$ is the corresponding transformation on the output space.
• $SE(3)$ Equivariance: In the context of 3D data like point clouds, an $SE(3)$-equivariant network ensures that if you rotate or translate the input point cloud, the output features will be rotated or translated in the exact same way. This is extremely useful because the network doesn't have to re-learn how to recognize an object in every possible orientation. It inherently understands the geometry of 3D space.

3.1.4. Bi-equivariance

This is a specialized form of equivariance crucial for manipulation tasks, as defined in this paper and its predecessor, EDFs. It considers two simultaneous frames of reference: the world/scene frame and the robot's end-effector frame.

• Left Equivariance (Scene Equivariance): If the entire scene (including the target placement location) is transformed by $\Delta g$, the correct final end-effector pose should also be transformed by $\Delta g$. The transformation is applied on the left of the pose: $g' = \Delta g \cdot g$. This is intuitive: if you move the table, the robot should move its hand accordingly. The following diagram from the paper (Figure 6a) illustrates this.

  该图像是示意图，展示了场景等变性和抓取等变性的概念。左侧（图a）展示了场景等变性，公式为 $g_w{e}' = g_{w'w}g_{we} = \Delta g g_{we}$。右侧（图b）展示了抓取等变性，公式为 $\Delta g = g_{we}{e}' = g_{we}\Delta g^{-1}$。图中使用了三维坐标系表示机器人抓取操作的变换关系。

• Right Equivariance (Grasp Equivariance): If the object is re-grasped differently, or if we change the coordinate system of the end-effector itself by a transformation $\Delta g$, the final world pose must be adjusted to compensate. The transformation is applied on the right: $g' = g \cdot \Delta g^{-1}$. For example, if you grasp a mug by a different part of its handle, you'll need to adjust your final arm pose to place the mug correctly on the hanger. The above diagram (Figure 6b) also illustrates this concept.

  A policy is bi-equivariant if it satisfies both left and right equivariance.

3.1.5. SO(3) Group Representation Theory

To build equivariant neural networks, we use representation theory.

• Representation: A representation of a group (like the rotation group $SO(3)$) is a way to map each group element to a linear operator (like a matrix) such that the group structure is preserved.
• Irreducible Representations (irreps): These are the fundamental "building blocks" of all representations. For $SO(3)$, irreps are classified by a non-negative integer $l$, called the type or spin.

  • Type-0 ($l=0$): These are scalars. They are invariant to rotation (e.g., distance, mass). The representation is just the number 1.

  • Type-1 ($l=1$): These are standard 3D vectors. They transform under rotation via the rotation matrix $R$ itself.

  • Higher Types ($l \ge 2$): These are higher-order tensors that capture more complex, higher-frequency geometric information.

    Equivariant networks operate on features that are composed of these different irrep types, ensuring that each part of the feature transforms correctly under rotation.

3.2. Previous Works

3.2.1. Equivariant Descriptor Fields (EDFs) [61]

Diffusion-EDFs is a direct evolution of EDFs.

• Core Idea: EDFs are bi-equivariant energy-based models (EBMs). They learn an energy function $E(g | O_s, O_e)$ over the space of possible end-effector poses $g \in SE(3)$, conditioned on the scene point cloud $O_s$ and grasped object point cloud $O_e$. Successful poses correspond to low-energy regions.
• Methodology: They construct an equivariant network that outputs a descriptor field. The energy is calculated based on the similarity between descriptors from the scene and the grasped object at a given pose.
• Strengths: Excellent data efficiency and generalization due to built-in bi-equivariance.
• Weakness: EBMs are notoriously difficult and slow to train. Training requires MCMC-based sampling (like Langevin dynamics) within the training loop, which is computationally expensive. The paper states this took over 10 hours for a simple task.

3.2.2. SE(3)-Diffusion Fields [75]

This is a key diffusion-based baseline.

• Core Idea: Applies a diffusion model directly on the $SE(3)$ manifold to generate end-effector poses. It learns a score function conditioned on visual input.
• Methodology: It uses a standard (left-invariant) Brownian diffusion process on $SE(3)$ and a score network that is not explicitly equivariant.
• Weakness: Because it lacks built-in $SE(3)$ equivariance, it relies on extensive data augmentation (e.g., showing the robot many different rotations of the same scene) to learn the symmetry. This leads to poor data efficiency and weaker generalization compared to truly equivariant models.

3.2.3. R-NDFs (Relational Neural Descriptor Fields) [68]

This is a key equivariant baseline.

• Core Idea: Extends Neural Descriptor Fields (NDFs) to handle relational tasks between multiple objects. It learns equivariant descriptors for objects and a relational module reasons about their interactions to predict a transformation.
• Weakness: The authors of Diffusion-EDFs argue that R-NDFs lack "locality." Their architecture, which may involve global operations like centroid subtraction, makes them brittle. Specifically, they perform poorly when presented with raw, unsegmented point clouds containing clutter, as they cannot easily distinguish the target object from its surroundings.

3.3. Technological Evolution

The field of robotic manipulation learning has evolved through several key stages:

1. Imitation Learning: Early methods focused on directly cloning human demonstrations, often using standard neural networks (e.g., CNNs, MLPs). These methods required massive datasets to generalize.
2. Introducing Geometric Priors (SE(2) Equivariance): Methods like Transporter Networks [86] introduced $SE(2)$ equivariance for 2D planar manipulation tasks (top-down picking and placing). This dramatically improved data efficiency by baking in translational and in-plane rotational symmetries.
3. Moving to 3D (SE(3) Equivariance): For full 6-DoF manipulation, researchers began developing $SE(3)$-equivariant models. Works like NDFs, R-NDFs, and EDFs built networks that could reason about 3D geometry and poses in a data-efficient manner.
4. Generative Models for Policies: Concurrently, generative models like VAEs, GANs, and more recently, diffusion models, were explored for learning policies. Diffusion models proved particularly effective for modeling complex, multi-modal distributions of successful actions.
5. Synthesis (This Paper): Diffusion-EDFs represents a synthesis of the two most promising recent trends: it combines the geometric power of $SE(3)$-equivariant models with the generative modeling capabilities of diffusion models, addressing the key weaknesses of both parent approaches (slow training for equivariant EBMs, poor data efficiency for non-equivariant diffusion models).

3.4. Differentiation Analysis

• vs. EDFs: The core innovation is replacing the slow energy-based model with a fast-to-train score-based diffusion model. This changes the learning objective from contrastive divergence (EBMs) to score matching (diffusion), leading to a ~15x training speedup.

• vs. SE(3)-Diffusion Fields: The key difference is the explicit enforcement of bi-equivariance. Diffusion-EDFs design a novel bi-equivariant diffusion process and score network, whereas SE(3)-Diffusion Fields use a standard non-equivariant approach that relies on data augmentation. This makes Diffusion-EDFs vastly more data-efficient and robust to out-of-distribution poses.

• vs. R-NDFs: Diffusion-EDFs are designed for locality and raw-scene processing. Thanks to their EDF-based architecture and contact-based diffusion origin, they can operate directly on cluttered, unsegmented point clouds where R-NDFs fail. Additionally, their new hierarchical architecture provides a larger receptive field for better scene-level context.

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4. Methodology

4.1. Principles

The core principle of Diffusion-EDFs is to learn a generative model of successful robot end-effector poses $g \in SE(3)$. Instead of directly modeling the complex, often multi-modal distribution of correct poses $P_0(g)$, they use a diffusion model. This involves two steps:

1. Training: A known diffusion process gradually adds noise to successful poses from human demonstrations. A neural network, called the score model, is trained to predict the "score" (gradient of the log-probability) of this noised pose distribution. The critical innovation is designing both the diffusion process and the score model to be bi-equivariant, ensuring data efficiency and generalization.

2. Inference: To generate a new successful pose, the model starts with a random pose (pure noise) and iteratively refines it by taking small steps in the direction of the learned score function, effectively reversing the diffusion process. This sampling is done via annealed Langevin dynamics.

   The following figure provides an overview of the process.

   该图像是示意图，展示了Diffusion-EDFs的核心概念与流程。左侧部分(a)描述了通过双等变扩散进行端到端训练，其中目标末端执行器姿态 $g_0$ 的拓扑变化。右侧部分(b)则展示了通过双等变去噪生成末端执行器位置的过程，强调了相对放置姿态的不变性。

4.2. Core Methodology In-depth (Layer by Layer)

4.2.1. Problem Formulation and Bi-Equivariance

The goal is to learn the policy distribution $P_0(g_0 | O_s, O_e)$, where $g_0 \in SE(3)$ is the target end-effector pose, $O_s$ is the point cloud of the scene, and $O_e$ is the point cloud of the object held by the end-effector. This policy must be bi-equivariant. This property is formally expressed as: $ P_0(g | O_s, O_e) = P_0(\Delta g g | \Delta g \cdot O_s, O_e) \quad \text{(Left/Scene Equivariance)} $ $ P_0(g | O_s, O_e) = P_0(g \Delta g^{-1} | O_s, \Delta g \cdot O_e) \quad \text{(Right/Grasp Equivariance)} $ for any transformation $\Delta g \in SE(3)$.

4.2.2. Bi-equivariant Score Function

The score function, $\mathbf{s}(g | O_s, O_e) = \nabla \log P(g | O_s, O_e)$, must also obey specific transformation rules if the underlying probability distribution $P$ is bi-equivariant. Proposition 1 in the paper derives these rules:

1. Left Invariance: The score remains unchanged when the scene and pose are transformed together. $ \mathbf{s}(\Delta g g | \Delta g \cdot O_s, O_e) = \mathbf{s}(g | O_s, O_e) $
2. Right Equivariance: The score transforms in a specific way when the end-effector frame and grasped object are transformed. $ \mathbf{s}(g \Delta g^{-1} | O_s, \Delta g \cdot O_e) = [\mathrm{Ad}_{\Delta g}]^{-T} \mathbf{s}(g | O_s, O_e) $
   • $\mathrm{Ad}_{\Delta g}$ is the adjoint representation of the group element $\Delta g = (\Delta R, \Delta \mathbf{p})$. It describes how velocities or forces in the Lie algebra transform when the reference frame is changed. Its matrix form is: $ \mathrm{Ad}_{g} = \begin{bmatrix} R & [\mathbf{p}]^{\wedge} R \ \mathbf{0} & R \end{bmatrix} $ where $[\mathbf{p}]^{\wedge}$ is the skew-symmetric matrix corresponding to the cross-product with vector $\mathbf{p}$. The transformation rule $[\mathrm{Ad}_{\Delta g}]^{-T}$ is crucial for designing the equivariant score model.

4.2.3. The Bi-equivariant Diffusion Process

A standard diffusion process on $SE(3)$ is not bi-equivariant. The authors design a novel process to overcome this.

1. The Challenge: Standard Brownian diffusion on a Lie group is defined by $g_{t+dt} = g_t \exp[dW]$, where dW is a noise term in the Lie algebra $\mathfrak{se}(3)$. This process is left-invariant but not right-invariant, so it cannot produce a bi-equivariant distribution from a bi-equivariant one. The paper proves that no non-trivial bi-invariant diffusion kernel exists on $SE(3)$.

2. The Solution: Diffusion in an Equivariant Frame. The key idea is to apply the diffusion noise not in the end-effector frame or the world frame, but in a dynamically chosen diffusion frame $d$.

The diffusion procedure is as follows:

• D1. Sample a target pose from the demonstrations: $g_0 \sim P_0(g_0 | O_s, O_e)$.

• D2. Sample a diffusion frame $g_{ed}$ relative to the end-effector frame $e$, based on the scene and grasped object: $g_{ed} \sim P(g_{ed} | g_0^{-1} \cdot O_s, O_e)$. This conditioning is crucial.

• D3. Sample a standard diffusion displacement $\Delta g_{t|0}$ from a simple, left-invariant kernel $K_t$.

• D4. Apply the displacement to $g_0$ in the diffusion frame $d$ to get the noised pose $g_t$. The transformation is $g_t = g_0 g_{ed} \Delta g_{t|0} g_{ed}^{-1}$.

  The overall conditional probability (the diffusion kernel) is given by integrating over all possible diffusion frames: $ P_{t|0}(g | g_0, O_s, O_e) = \int_{SE(3)} d g_{ed} P(g_{ed} | g_0^{-1} \cdot O_s, O_e) K_t(g_{ed}^{-1} g_0^{-1} g g_{ed}) $

3. Making it Practical (Proposition 4). The paper shows a remarkable simplification: if the base kernel $K_t$ is the standard Brownian motion kernel $\mathcal{B}_t$, then for the overall process to be bi-equivariant, we only need to select the origin (translational part) of the diffusion frame in an equivariant way. The rotational part can be ignored.

The diffusion origin selection mechanism $P(\mathbf{p}_{ed} | g_0^{-1} \cdot O_s, O_e)$ is implemented with a clever heuristic based on physical intuition: diffusion should originate from regions of likely contact. $ P(\mathbf{p}{ed} | g_0^{-1} \cdot O_s, O_e) \propto \sum{p \in O_e} n_r(p, g_0^{-1} \cdot O_s) \delta^{(3)}(\mathbf{p}_{ed} - p) $

• $n_r(p, g_0^{-1} \cdot O_s)$ counts the number of scene points within a radius $r$ of a point $p$ on the grasped object (when viewed from the object's frame).
• $\delta^{(3)}$ is the Dirac delta function.
• In essence, the diffusion origin $\mathbf{p}_{ed}$ is chosen to be one of the points on the grasped object, with a higher probability for points that are close to the scene at the target pose. This focuses the diffusion process on geometrically meaningful, contact-rich areas.

4.2.4. Score Matching Objective

The score model $\mathbf{s}_t(g | O_s, O_e)$ is trained to predict the score of the true (but intractable) diffused marginal distribution $P_t$. The paper shows that this can be achieved by minimizing a simple Mean Squared Error (MSE) loss, which does not require computing $P_t$ directly: $ \mathcal{J}t = \frac{1}{2} \left| \mathbf{s}t(g | O_s, O_e) - \nabla \log K_t(g{ed}^{-1} g_0^{-1} g g{ed}) \right|^2 $ The training objective is the expectation of this loss over all data and diffusion samples: $\mathcal{I}_t = \mathbb{E}_{g, g_0, g_{ed}, O_s, O_e}[\mathcal{J}_t]$. The term $\nabla \log K_t(\dots)$ is the score of the simple base kernel, which can be computed analytically.

4.2.5. The Bi-equivariant Score Model Architecture

This section details the neural network architecture for the score model $\mathbf{s}_t$, which must satisfy the bi-equivariance properties from Proposition 1.

1. Decomposition: The score $\mathbf{s}_t \in \mathfrak{se}(3)$ is split into a translational part $\mathbf{s}_{\nu;t} \in \mathbb{R}^3$ and a rotational part $\mathbf{s}_{\omega;t} \in \mathbb{R}^3$.

2. Model Formulation: The scores are computed as an integral (in practice, a sum) over query points on the grasped object $O_e$. $ \mathbf{s}{\nu;t}(g | O_s, O_e) = \int{\mathbb{R}^3} d^3x \rho_{\nu;t}(\mathbf{x}|O_e) \tilde{\mathbf{s}}{\nu;t}(g, \mathbf{x} | O_s, O_e) $ $ \mathbf{s}{\omega;t}(g | O_s, O_e) = \int_{\mathbb{R}^3} d^3x \rho_{\omega;t}(\mathbf{x}|O_e) \tilde{\mathbf{s}}{\omega;t}(g, \mathbf{x} | O_s, O_e) + \int{\mathbb{R}^3} d^3x \rho_{\nu;t}(\mathbf{x}|O_e) \mathbf{x} \wedge \tilde{\mathbf{s}}_{\nu;t}(g, \mathbf{x} | O_s, O_e) $

• $\rho_{\square;t}(\mathbf{x}|O_e)$: An equivariant density field (a scalar field) that gives a weight to each query point $\mathbf{x}$.

• $\tilde{\mathbf{s}}_{\square;t}(g, \mathbf{x} | \dots)$: The core score field, a vector field that must have specific equivariance properties. The rotational score has an "orbital" component ($\mathbf{x} \wedge \tilde{\mathbf{s}}_{\nu;t}$) and a "spin" component ($\tilde{\mathbf{s}}_{\omega;t}$).

  3. The Equivariant Score Field $\tilde{\mathbf{s}}$: The key component, the score field, is modeled using two Equivariant Descriptor Fields (EDFs), $\psi$ and $\varphi$, and an equivariant tensor product. $ \tilde{\mathbf{s}}{\square;t}(g, \mathbf{x} | O_s, O_e) = \psi{\square;t}(\mathbf{x} | O_e) \otimes_{\square;t}^{(1)} \mathbf{D}(R^{-1}) \varphi_{\square;t}(g\mathbf{x} | O_s) $

• $\psi(\mathbf{x} | O_e)$: An EDF that generates a high-dimensional descriptor (a collection of different irrep types) for a query point $\mathbf{x}$ on the grasped object $O_e$.

• $\varphi(g\mathbf{x} | O_s)$: An EDF that generates a descriptor for the point $g\mathbf{x}$ (the query point transformed by the current pose $g$) within the scene $O_s$.

• $\mathbf{D}(R^{-1})$: Rotates the scene descriptor $\varphi$ into the end-effector's frame.

• $\otimes^{(1)}$: A time-conditioned equivariant tensor product. This is a learnable operation that combines the two high-dimensional descriptors ($\psi$ and $\varphi$) and contracts them down to a single type-1 vector (a 3D vector), which is the score field value.

  The paper proves in Proposition 6 that this construction for $\tilde{\mathbf{s}}$ has the exact left-invariance and right-equivariance properties required to make the entire score model $\mathbf{s}_t$ bi-equivariant.

4.2.6. Multi-Scale EDF Architecture

To improve the model's ability to understand both fine-grained local geometry and coarse-grained global context, the paper introduces a new hierarchical, U-Net-like architecture for the EDFs.

该图像是示意图，展示了Diffusion-EDFs方法中多尺度EDF（子图a）和场模型（子图b）的结构。图中包括输入点云、查询点、时间嵌入及各层的对应模型，如等变U-Net下行和上行块，以及边缘编码器。该模型集成了多种特征以增强在三维空间（SE(3)）的处理能力，支持机器人视觉操控任务的优化与训练。

• Encoder (Feature Extractor): The input point cloud is processed through multiple scales. At each coarser scale, the points are downsampled using Farthest Point Sampling (FPS), and the graph convolutions (Equiformer layers) use a larger message-passing radius. This allows the model to build up features that cover a wide receptive field.

• Decoder (Field Model): The field model takes the multi-scale features from the encoder and a query point. It upsamples the features and combines them using skip connections, similar to a U-Net, to compute the final field value. This structure allows the model to leverage both global context and local detail.

• Efficiency: The heavy encoder part is run only once per scene, while the much lighter field model is queried repeatedly during the iterative denoising process, making inference efficient.

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5. Experimental Setup

5.1. Datasets

The experiments do not use standard large-scale datasets but are instead based on a small number of human demonstrations collected for each specific task, highlighting the method's data efficiency.

• Data Source: Human demonstrations for pick-and-place tasks. 10 demonstrations were used for each simulation task, and 4-10 for each real-world task.

• Simulation Environment: The SAPIEN simulator [83] was used. The scenes consist of objects from PartNet-Mobility and ShapeNet datasets.

• Tasks:

  • Simulation (Figure 3):

    1. Mug-on-a-Hanger: Pick a mug by its rim and place it on a hanger by its handle. This requires precise 6-DoF control.

    2. Bottle-on-a-Tray: Pick a bottle by its cap and place it on a tray.

       该图像是图示，展示了仿真实验中的两个任务：在(a) Mug-on-a-Hanger任务中，红色杯子需要通过边缘提起并放在绿色挂钩上；在(b) Bottle-on-a-Tray任务中，红色瓶子需通过瓶盖提起并放置在绿色托盘上。

  • Real World (Figure 5):

    1. Mug-on-a-hanger: Similar to the simulation task.

    2. Bowls-on-dishes: A sequential task requiring the robot to place red, green, and blue bowls on dishes of matching colors in the correct order. This tests scene-level understanding.

    3. Bottles-on-a-shelf: A multi-modal task where the robot must pick up multiple identical bottles and place them on a shelf.

       该图像是图表，展示了真实硬件实验中的多个任务场景，包括（a）杯子放置任务、（b）碗放置任务（按红绿蓝顺序）及（c）瓶子放置任务。每个任务展示了抓取点云的场景信息和机器人操作流程。

• Data Example: The input data consists of two point clouds: $O_s$ (the scene, observed by external cameras) and $O_e$ (the object held in the gripper, observed by a wrist-mounted camera). The label is the demonstrated successful final pose $g_0$.

5.2. Evaluation Metrics

The primary evaluation metric used across all experiments is Success Rate.

1. Conceptual Definition: This metric measures the percentage of trials in which the robot successfully completes the entire pick-and-place task. A trial is considered successful if the object is correctly picked and placed in a stable goal configuration.
2. Mathematical Formula: $ \text{Success Rate} = \frac{\text{Number of Successful Trials}}{\text{Total Number of Trials}} $
3. Symbol Explanation:

   • Number of Successful Trials: The count of experimental runs where the task objective was met.

   • Total Number of Trials: The total number of runs performed for a given experimental condition.

     The evaluation is performed under various out-of-distribution (OOD) scenarios to test generalization:

• Previously Unseen Instances: Using new object models from the same category that were not in the demonstrations.
• Previously Unseen Poses: Placing objects at novel starting positions and orientations.
• Previously Unseen Clutters: Adding distracting objects to the scene that were not present during training.
• All Combined: A combination of all three OOD conditions.

5.3. Baselines

The paper compares Diffusion-EDFs against two representative state-of-the-art methods.

• R-NDFs [68]: This is a leading $SE(3)$-equivariant method for relational manipulation. It is a representative baseline for data-efficient, geometry-aware learning. For this baseline, the authors use the officially provided pre-trained weights.

• SE(3)-Diffusion Fields [75]: This is a leading diffusion-based method for pose generation in manipulation. It is a representative baseline for generative policy learning. This model was trained from scratch on the same 10 demonstrations as Diffusion-EDFs. Since it is not equivariant, the authors also test it with rotational data augmentation.

  Crucially, the baselines were evaluated in two modes: (1) with ground-truth object segmentation (i.e., they are given a clean point cloud of only the target object) and (2) without segmentation (i.e., on the raw, cluttered scene), which is the same input Diffusion-EDFs receives.

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6. Results & Analysis

6.1. Core Results Analysis

The experimental results strongly validate the claims of Diffusion-EDFs. The primary results are presented in Table 1 for the simulation benchmarks.

The following are the results from Table 1 of the original paper:

(Note: The table structure has been simplified for clarity. $(n/a)$ indicates place success was not evaluated if pick failed. The original table contains some formatting errors and inconsistencies which have been regularized here based on the text's description.)

Analysis of Table 1:

• Superiority of Diffusion-EDFs: In every single scenario, Diffusion-EDFs (bolded rows) achieve the highest success rates, often by a very large margin. Even in the most challenging combined OOD setting, it maintains total success rates of 79% (Mug) and 87% (Bottle), whereas the baselines struggle.
• The Importance of Handling Raw Scenes: The most striking result is the complete failure of R-NDFs and SE(3)-DiffusionFields when operating without object segmentation (rows marked with ✗ in the "Without Obj. Seg." column). Their success rates drop to 0-1% in the trained setup and 0% in all OOD setups. This demonstrates that they are unable to function in realistic, cluttered scenes. In contrast, Diffusion-EDFs thrive in this setting, proving the effectiveness of their locality-aware, bi-equivariant design.
• Generalization Power: Diffusion-EDFs show minimal performance degradation when faced with unseen instances and poses. For example, in the Mug task, the total success rate only drops from 95% (Default) to 92% (Unseen Instances) and 96% (Unseen Poses). This confirms the powerful generalization endowed by $SE(3)$ bi-equivariance. SE(3)-Diffusion Fields, lacking this property, sees a more significant drop.
• Data Efficiency: All these high scores were achieved by training on only 10 demonstrations, without any large-scale pre-training, which is a testament to the model's data efficiency.

6.2. Ablation Studies / Parameter Analysis

While the paper does not have a formal "Ablation Study" section, the comparative experiments serve as powerful ablations:

• Ablation of Object Segmentation: The comparison between running baselines with and without segmentation is effectively an ablation study. The results (0% success without segmentation for baselines) prove that the ability to process raw scenes is not a minor feature but a critical component of Diffusion-EDFs' success, enabled by its local, equivariant architecture.
• Ablation of Equivariance: The comparison with SE(3)-Diffusion Fields serves as an ablation on the importance of $SE(3)$ bi-equivariance. Even with rotational augmentation, the non-equivariant model performs significantly worse, highlighting that explicitly building in the symmetry is superior to trying to learn it from augmented data.

6.3. Real Hardware Experiments

The real-world experiments further validate the model's capabilities in more complex and realistic scenarios.

The key challenges addressed by each task are summarized in Table 2: The following are the results from Table 2 of the original paper:

Key Takeaways from Hardware Experiments:

• Mug-on-a-hanger: The robot successfully performed this high-precision task, demonstrating accurate 6-DoF pose generation and generalization to new mugs and poses.

• Bowls-on-dishes: The robot learned the sequential and semantic nature of the task (e.g., place the red bowl first, match colors). This is a powerful demonstration of the scene-level understanding enabled by the hierarchical EDF architecture, something impossible for object-centric models.

• Bottles-on-a-shelf: The robot successfully handled a task with a multimodal distribution of solutions (multiple identical bottles to choose from), showcasing the strength of using a generative diffusion model. It also generalized to different numbers of bottles.

  Overall, the experiments provide compelling evidence that Diffusion-EDFs are not just a theoretical construct but a practical, robust, and highly efficient solution for real-world robotic manipulation.

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7. Conclusion & Reflections

7.1. Conclusion Summary

The paper presents Diffusion-EDFs, a novel bi-equivariant diffusion model for learning 6-DoF visual robotic manipulation tasks. By combining the generative capabilities of diffusion models with the geometric principles of $SE(3)$ equivariance, the method achieves a trifecta of desirable properties:

1. High Data Efficiency: It learns complex tasks from only 5-10 human demonstrations.

2. Fast Training: It trains in under an hour, a ~15x improvement over its energy-based predecessor, EDFs.

3. Superior Generalization: It significantly outperforms state-of-the-art methods in generalizing to unseen objects, poses, and cluttered environments, critically without requiring object segmentation.

   The work's core contributions are the development of a novel bi-equivariant diffusion process on $SE(3)$, a corresponding bi-equivariant score network architecture, and a hierarchical model structure that enables scene-level understanding. These contributions are rigorously validated through both simulation and challenging real-world hardware experiments.

7.2. Limitations & Future Work

The authors candidly acknowledge several limitations and propose future research directions:

• Lack of Trajectory/Control-Level Inference: Diffusion-EDFs only infer the final goal pose. It does not generate the intermediate motion trajectory or handle low-level control. The authors suggest integrating it with motion planners or geometric control frameworks to create end-to-end motion generation.
• Dependency on Separate Grasp Observation: The current pipeline requires a separate step where a wrist-mounted camera observes the grasped object. This prevents fully closed-loop control where the robot might adjust its plan mid-action. Future work could involve incorporating segmentation techniques to distinguish the held object from the scene in a single observation, enabling a more fluid workflow.

7.3. Personal Insights & Critique

• Strengths and Inspirations:

  • Elegant Synthesis: The paper is an excellent example of successfully synthesizing two powerful but distinct paradigms—geometric deep learning and generative diffusion models. The theoretical work to make the diffusion process itself bi-equivariant is particularly elegant and principled.
  • Pragmatism: The focus on data efficiency and training speed addresses major pain points in practical robotics research. Learning from 5 demonstrations in under an hour is a significant step towards making robot learning accessible and scalable.
  • Focus on Raw Scenes: The ability to work on unsegmented, cluttered point clouds is a massive practical advantage. Many academic methods assume perfect perception, which is an unrealistic bottleneck in the real world. This work's robustness makes it far more applicable.

• Potential Issues and Areas for Improvement:

  • Heuristic for Diffusion Origin: The contact-based heuristic for selecting the diffusion origin is clever and works well for the tested tasks. However, it is still a heuristic. For tasks where non-contact geometry is critical for alignment (e.g., aligning two parts based on non-contact features), this heuristic might be suboptimal. A learnable attention mechanism to determine the most "important" region for diffusion could be a more general solution.
  • Inference Speed: While training is fast, inference relies on an iterative denoising process (annealed Langevin dynamics). The paper reports inference times of 5-17 seconds. While acceptable for many tasks, this could be too slow for dynamic environments or tasks requiring rapid responses. Exploring faster sampling techniques from the diffusion model literature (e.g., DDIM, consistency models) could be a valuable extension.
  • Task Scope: The experiments are focused on pick-and-place tasks. While these are foundational, the method's applicability to other classes of manipulation, such as continuous control tasks (sliding, pushing, stirring) or deformable object manipulation, remains an open question. Adapting the $SE(3)$ framework to these more complex dynamics would be a challenging but impactful direction for future research.