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

arXiv:2606.01847v1 Announce Type: cross Abstract: Diffusion-based Vision-Language-Action policies achieve remarkable success in robotic manipulation, yet commit a fundamental geometric error we term the extbf{Euclidean Fallacy}: representing SE(3) poses as flat 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 extbf{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 ABCrightarrowD, 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.

Source: arXiv cs.LG | 2026-06-02