hPGA-DP: Hybrid Geometric Diffusion Policy

• hPGA-DP is a robot manipulation learning approach that embeds explicit spatial inductive bias by integrating Projective Geometric Algebra into diffusion policy frameworks.
• It uses a dedicated Projective Geometric Algebra Transformer for state encoding and action decoding, separating geometric reasoning from stochastic denoising.
• Empirical results demonstrate higher task success, faster training convergence, and improved robustness in both simulated and real-world robotic environments.

hPGA-DP (hybrid Projective Geometric Algebra Diffusion Policy) is a robot manipulation learning approach that integrates Projective Geometric Algebra (PGA) into diffusion policy architectures, embedding explicit geometric inductive biases into neural representations. By leveraging the Projective Geometric Algebra Transformer (P-GATr) for state encoding and action decoding—while employing conventional denoising networks such as U-Net or Transformer backbones—hPGA-DP achieves efficient training and improved task success in both simulated and real-world robotics environments (Sun et al., 8 Jul 2025).

1. Conceptual Foundations and Motivation

Diffusion policies are a class of methods in robot learning that generate actions by progressively denoising trajectories perturbed by stochastic noise, conditioned on observations. However, standard diffusion architectures require networks to redundantly re-learn core spatial operations—such as translation and rotation—in every new task and environment. hPGA-DP addresses this inefficiency by directly embedding the algebraic structure of 3D Euclidean space into the model architecture using PGA. This approach removes the need to re-learn basic geometric properties and strengthens the model's spatial inductive bias, ultimately leading to faster convergence and more reliable policy learning. PGA provides a unified algebraic framework for encoding points, directions, translations, rotations, and rigid motions, represented as multivectors in $\mathbb{G}_{3,0,1}$.

2. Integration of Projective Geometric Algebra via P-GATr

Central to hPGA-DP is the use of the Projective Geometric Algebra Transformer (P-GATr), an architecture operating directly on multivector representations to maintain $E(3)$-equivariance across state encodings and decoded actions. In the system:

• State Encoding: The robot’s proprioceptive data and 6D object poses are mapped into PGA multivectors. For a point at position $(x_1, x_2, x_3)$, the dual PGA form is

$$\mathbf{x} = \mathbf{e}_{123} - x_1 \mathbf{e}_{023} + x_2 \mathbf{e}_{013} - x_3 \mathbf{e}_{012}$$

These multivectors are tensorized and processed by the P-GATr encoder, ensuring that equivariant geometric structure is preserved and exploiting the symmetries of Euclidean space.

• Action Decoding: After the denoising pass (see Section 4), the processed latent action representations are decoded by a P-GATr module, reconstructing positions and orientations as PGA multivectors. These are then mapped to control signals (e.g., Cartesian positions, unit quaternions) for robot actuators.

3. Hybrid Diffusion Policy Architecture

hPGA-DP employs a hybrid modular structure that separates geometric encoding/decoding from the stochastic denoising process:

• Denoising Backbone: The core denoising is performed using either U-Net or Transformer networks, which have demonstrated high efficiency for reversing the Gaussian forward process. The transition from noisy to clean latent actions follows

$$\mathbf{z}_{\mathbf{a}, k} = \sqrt{\bar{\alpha}_k}\, \mathbf{z}_{\mathbf{a}, 0} + \sqrt{1-\bar{\alpha}_k}\, \boldsymbol{\varepsilon}, \quad \boldsymbol{\varepsilon} \sim \mathcal{N}(0, \mathbf{I})$$

where $\mathbf{z}_{\mathbf{a},0}$ is the latent action, $\bar{\alpha}_k$ is the cumulative data-retention factor at step $k$, and $\boldsymbol{\varepsilon}$ is injected noise.

• Geometric Module Specialization: P-GATr is reserved exclusively for input encoding and output decoding. This architectural separation avoids the slow convergence observed when P-GATr is used for iterative denoising, as its geometric regularization is not optimized for the noise-reversal task.
• Staged Decoder Supervision: To preserve geometric structure, supervision on the decoder is applied only when denoising has progressed sufficiently (i.e., for denoising steps where $k \geq K_{\text{thresh}}$, with

$$K_{\text{thresh}} = K_{\max} - \lfloor \eta \cdot K_{\max} \rfloor$$

where $\eta$ is a hyperparameter). This restricts geometric supervision to late-stage latents that are more coherent, enforcing correct spatial interpretation without impeding denoising efficacy early in the process.

4. Training Procedure and Loss Functions

Training follows the standard paradigm for diffusion models but with modifications to accommodate geometric decoding:

• Noise Prediction Loss: The denoising network is supervised to predict the injected noise at each diffusion step using a mean squared error loss:

$$\mathcal{L}_{\text{Denoise}} = \left\| \epsilon_\theta(\mathbf{z}_{\mathbf{a},k}, \mathbf{z}_\mathbf{o}) - \boldsymbol{\varepsilon} \right\|^2$$

where $\epsilon_\theta$ is the learned denoiser, $\mathbf{z}_{\mathbf{a},k}$ is the current noisy action latent, and $\mathbf{z}_\mathbf{o}$ is the observation embedding.

• Latent Reconstruction: The estimated clean latent is recovered as

$$\hat{\mathbf{z}}_{\mathbf{a},0} = \frac{1}{\sqrt{\bar{\alpha}_k}} \left( \mathbf{z}_{\mathbf{a},k} - \sqrt{1 - \bar{\alpha}_k} \, \hat{\boldsymbol{\varepsilon}} \right)$$

Supervision on geometric decoding is restricted to $\hat{\mathbf{z}}_{\mathbf{a},0}$ in the final $\eta$-fraction of the diffusion trajectory.

• Action Decoding Loss: The P-GATr decoder’s outputs are compared (typically using an MSE or task-specific spatial loss) to ground-truth action multivectors, enforcing geometric fidelity.

5. Benefits of Geometric Inductive Bias and Empirical Findings

The integration of PGA and P-GATr as geometric inductive biases yields pronounced benefits:

• Task Performance: hPGA-DP achieves higher success rates on complex manipulation problems (Lift, Can, Stack, Square, and Mug) relative to policies trained with non-geometric (U-Net or Transformer) or misapplied geometric (P-GATr-only denoising) architectures.
• Training Efficiency: Empirical results report faster convergence, reaching high success within approximately 30 epochs—a roughly threefold reduction in epochs required compared to non-hybrid baselines. While each epoch may be marginally longer, total wall-clock training time is reduced due to greater learning efficiency.
• Robustness: Performance is robust to moderate changes in the decoder loss masking hyperparameter $\eta$. Ablation studies confirm that the improvements are due to the explicit geometric integration and not merely architectural rearrangement.
• Transfer to Physical Systems: Real-world experiments with dual-arm robots performing tasks such as block stacking and drawer manipulation reflect increased success and reduced total training time.

Architecture	Geometric Bias	Denoising Module	Success Rate	Convergence Epochs
U-Net/Transformer	None	U-Net/Transformer	Moderate	High
P-GATr (all modules)	Strong	P-GATr	Low/Slow	Very High
hPGA-DP (hybrid)	Targeted	U-Net/Transformer	High	Low

6. Architectural Design Choices and Practical Implications

• Specialization of Modules: Assigning geometric reasoning to P-GATr and stochastic denoising to standard architectures exploits the strengths of both approaches while avoiding degeneration in learning speed or generalization.
• Supervision Scheduling: Supervising the decoder only on the final denoising steps ensures that action representations are geometrically interpretable without creating competing optimization pressures in early, heavily-noised stages.
• Representation: Using PGA multivectors to encode both observations and actions allows the policy to manipulate positions, rotations, and other geometric constructs in a theoretically grounded and practically effective manner.
• Scalability: Faster convergence and robust real-world performance suggest that hPGA-DP is applicable to a range of robot learning tasks, especially those requiring high-fidelity spatial reasoning and efficient adaptation to new environments or tasks.

7. Summary and Outlook

hPGA-DP introduces a hybrid architecture for robotic manipulation learning that brings the mathematical rigor of Projective Geometric Algebra to deep generative policy models. By dedicating P-GATr modules to encoding and decoding the latent spaces and leveraging proven diffusion-denoising architectures for stochastic processing, it achieves improved sample efficiency, higher task success, and faster training. The empirical evidence supports that explicit geometric inductive biases, when appropriately modularized, are highly beneficial for domains where spatial reasoning is paramount. This method demonstrates an effective pathway for integrating advanced algebraic frameworks with modern generative policy learning for robotics (Sun et al., 8 Jul 2025).