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

• URL: http://arxiv.org/abs/2606.01847v1
• Date: Mon, 01 Jun 2026 07:59:29 GMT
• Title: The Lie We Tell: Correcting the Euclidean Fallacy in Vision Language Action Policies via Score Matching on Tangent Space
• Abstract summary: We introduce a diffusion framework operating intrinsically on SE(3).<n>Our method injects noise through left-invariant SDEs, predicts scores in the space tangent, and retracts samples via the exponential map.<n>On CALVIN ABC$rightarrow$D, LDA improves average task length from $3.27$ to $3.51$ ($+7.3%$)
• Score: 19.950575615573978
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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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