The Lie We Tell: Correcting the Euclidean Fallacy in Vision Language Action Policies via Score Matching on Tangent Space

Published 1 Jun 2026 in cs.RO and cs.LG | (2606.01847v1)

Abstract: Diffusion-based Vision-Language-Action policies achieve remarkable success in robotic manipulation, yet commit a fundamental geometric error we term the $\textbf{Euclidean Fallacy}$: representing SE(3) poses as flat $\mathbb{R}^{12}$ vectors. This approximation induces (1) manifold drift violating SO(3) constraints, (2) broken equivariance under coordinate transformations, and (3) non-geodesic trajectories with excessive kinematic cost. We introduce $\textbf{Lie Diffuser Actor (LDA)}$, a diffusion framework operating intrinsically on SE(3). Our method injects noise through left-invariant SDEs, predicts scores in the tangent space, and retracts samples via the exponential map. This formulation eliminates manifold drift by construction while guaranteeing coordinate-frame equivariance and geodesic optimality. On CALVIN ABC$\rightarrow$D, LDA improves average task length from $3.27$ to $3.51$ ($+7.3\%$). We further validate our method on real robot and the results show that our methodology outperforms the baseline on majority tasks.

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Summary

The paper introduces the Lie Diffuser Actor, which corrects the Euclidean fallacy by performing score matching on the tangent space of SE(3).
It employs noise injection in the se(3) tangent space via the exponential map, ensuring all generated poses remain valid and geometrically consistent.
Empirical evaluations on CALVIN and OpenVLA-OFT benchmarks demonstrate improved trajectory smoothness, reduced jitter, and robust, physically realizable manipulation.

Correcting the Euclidean Fallacy in Vision-Language-Action Policies: Lie Diffuser Actor on SE(3)

Problem Formulation and Geometric Limitations of Euclidean Parameterization

The work addresses a foundational flaw in contemporary diffusion-based Vision-Language-Action (VLA) policies used in robotic manipulation. Existing approaches parameterize SE(3) end-effector poses as $\mathbb{R}^{12}$ vectors—flattened representations of rotation and translation—enabling them to perform additive Gaussian noise injection. This practice, identified as the "Euclidean Fallacy," introduces three systematic violations: (1) manifold drift, as additive noise generically maps valid rotations out of $\mathrm{SO}(3)$ ; (2) broken equivariance, as noise distributions are not preserved under rigid workspace transformations; and (3) non-geodesic trajectories, yielding excessive kinematic cost and erratic motion profiles. As such, these methods force the denoising network to learn correction mappings to repair geometric invalidity rather than core manipulation semantics.

Prior efforts in the field—trajectory-level policies, autoregressive token-based policies, and hierarchical approaches—remain encumbered by this geometric misalignment due to their Euclidean pose models. While SVD-based or quaternion re-projection partially remedies validity at inference, this incurs training-inference mismatch, error amplification under near-degeneracy, and non-differentiable projections that disrupt generative consistency. Consequently, these architectures cannot guarantee frame invariance or kinematically natural motion—key requirements for physical robot deployment.

Lie Diffuser Actor: Intrinsic Manifold Diffusion on SE(3)

The Lie Diffuser Actor (LDA) framework is introduced to resolve these geometric inconsistencies by formulating the diffusion process directly on the SE(3) manifold. The core mechanism involves:

Noise Injection in the Lie Algebra: Forward noising is performed via left-invariant SDEs, injecting Gaussian velocity twists $\xi \in \mathfrak{se}(3)$ and re-mapping via the exponential map.
Tangent-Space Score Prediction: Instead of outputting ambient-space noise, the network directly predicts the noise twist per step, ensuring that all updates are valid rigid transformations by construction.
Exponential Retraction: Each denoising update is composed via group multiplication, ensuring output remains on SE(3) and satisfying $\det(R) = 1$ , $R^T R = I$ without projection.

Three major theoretical guarantees are rigorously established:

Elimination of Manifold Drift: All forward and reverse samples reside on SE(3) for all time steps (cf. Proposition 4.1).
Left-Invariant Equivariance: The learned score function satisfies $s(hg, t) = \operatorname{Ad}_h[s(g,t)]$ for all $h \in \mathrm{SE}(3)$ (cf. Theorem 4.2), ensuring invariance to coordinate-frame changes.
Geodesic Generation: Deterministic probability flow ODEs in the limit generate geodesics on SE(3)—kinematically optimal screw motions (cf. Proposition 4.3).

Architectural Framework and Training Objective

The model architecture builds on 3D Diffuser Actor with key geometric extensions:

Geometric Scene Encoding: Multimodal context fuses RGB-D observations (back-projected to point clouds processed by GAT-based transformers) and natural language instructions (CLIP-based transformer encoders). Cross-attention is employed for spatial-semantic grounding.
Denoising Backbone: A transformer refines SE(3) trajectory waypoints using time-embedded, SE(3)-structured pose tokens, maintaining explicit trajectory coupling and stepwise context grounding.
Tangent-Space Output Head: The action head predicts a twist $(\omega, v) \in \mathbb{R}^6$ in $\mathfrak{se}(3)$ per trajectory step, applying the exponential map for each update.
Score Matching in $\mathfrak{se}(3)$ : Training uses denoising score matching in the Lie algebra, with auxiliary translation and gripper state loss terms.

This design yields strictly lower rotation constraint violations—including median, P90, P95, P99 orthogonality errors—compared to those of SVD-projected, Euclidean-diffusion baselines.

Empirical Validation: Simulation, Real-Robot, and Cross-Architecture Studies

Simulation Benchmarks

Evaluations on CALVIN (long-horizon, language-conditioned manipulation) confirm the efficacy of the LDA. In the canonical ABC→D protocol, the average chain length increased from 3.27 (Euclidean baseline) to 3.51 (+7.3%)—a direct reflection of enhanced trajectory consistency and reduced error accumulation. Similar, or larger, gains are observed in more diverse ABCD→D settings, where Euclidean models degrade due to compounded frame sensitivity.

Real-Robot Deployment

Comprehensive real robot experiments test four tasks spanning diverse pose constraints (platform transport, precision insertion, spatial sorting, vertical stacking). LDA delivers perfect success on tasks with high pose coupling, and substantial improvements on fine placement and sorting, confirming that geometric consistency yields practical reliability. Qualitative analyses show reduced trajectory jerk, more stable gripper alignment, and fewer recovery maneuvers.

Constraint and Stability Analysis

Direct comparison of the denoising trajectories in quaternion space reveals the Euclidean model's off-manifold excursions and significant orthogonality/determinant violations (up to $\mathrm{SO}(3)$ 0), while LDA remains on-manifold (deviations at $\mathrm{SO}(3)$ 1 or better). Denoising step stability is improved by orders of magnitude—a critical property for closed-loop or receding-horizon deployment.

Cross-Architecture Validation

Porting SE(3) score matching to OpenVLA-OFT, a 7B-parameter VLA transformer with no 3D encoding, yields consistent gains (success rate from 93.87 to 94.13 on LIBERO-Long), underscoring that improvements derive from intrinsic geometric formulation, not only architectural details.

Relation to Prior Work

LDA closes essential gaps left by prior research:

Trajectory-Level Diffusion Policies: All high-performing baselines (e.g., 3D Diffuser Actor, RDT-1B, DP3) use ambient-space parameterizations, inheriting manifold drift and equivariance loss.
Equivariant Architectures: Works such as EquiBot, Diffusion-EDFs, ET-SEED encode equivariance at the input or feature level, but generative processes remain Euclidean and thus asymmetric.
Riemannian Generative Modeling: The Riemannian diffusion/flow matching literature provides mathematical tools for manifold-valued generation, but LDA's left-invariant SDE on SE(3) confers stronger equivariance and is essential for the stochastic, multimodal action distributions required in manipulation policy learning. Recent works in Lie group diffusion [Bertolini et al., (Bertolini et al., 4 Feb 2025)], Riemannian flow matching [Chen & Lipman, 2024], and equivariant policies [Yang et al., 2024; Ryu et al., 2024] are closely related but do not integrate tangent-space score prediction and full action-level intrinsic diffusion.

Implications and Future Directions

This work demonstrates that respecting the intrinsic manifold structure of SE(3) in both noise injection and inference is essential for high-fidelity, robust robot manipulation policies. By ensuring geometric validity and coordinate-invariance, the method yields sample-efficient learning, task transferability, and stable real-world deployment. Critically, the approach generalizes across architectures—including non-spatial (MLP-head) VLA transformers—suggesting broad applicability.

Future directions include:

Integration with Stronger Foundation Models: Combining intrinsic SE(3) diffusion with vision-language architectures trained at web scale.
Extension to Other Manifolds: Adapting left-invariant SDE-based diffusion to Sim(3) for scale-aware manipulation, or to product manifolds for bimanual and articulated systems.
Closed-Loop Control and Online Adaptation: Leveraging geometric consistency for receding-horizon or feedback policies under dynamic environmental change.
Sample-Efficient Imitation and Generalization: Exploiting the theoretical equivariance for fine-tuning with few demonstrations and zero-shot adaptation in novel workspaces.

Conclusion

Lie Diffuser Actor formalizes and corrects the geometric mis-specification endemic to Euclidean-diffusion VLA policies by operating intrinsically on SE(3) via left-invariant SDEs and tangent-space score matching. The method eliminates manifold drift, achieves strict equivariance under coordinate transformation, and produces kinematically natural, geodesic trajectories. Empirical validation on diverse benchmarks and real-robot systems establishes its superiority in both numerical and behavioral metrics. This architecture provides a foundation for robust, physically consistent generative manipulation policies and motivates broader manifold-aware policy learning across robotic applications.

Citation:

"The Lie We Tell: Correcting the Euclidean Fallacy in Vision Language Action Policies via Score Matching on Tangent Space" (2606.01847)

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