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

Authors
Bing-Cheng Chuang, I-Hsuan Chu, Bor-Jiun Lin, YuanFu Yang, Min Sun, Chun-Yi Lee
Source
arXiv
Published
Collection
Embodied AI
Vision Language ActionDiffusion PolicyLie GroupsSe3 RoboticsRobot Manipulation
Open arXivMarkdown source

Summary

Lie Diffuser Actor keeps diffusion-based robot pose generation on SE(3) instead of flattening poses into Euclidean vectors. The paper claims this reduces invalid rotations, preserves coordinate-frame equivariance, and improves manipulation benchmarks over 3D Diffuser Actor and OpenVLA-OFT variants.

Problem

  • Diffusion VLA policies often encode a robot gripper pose as a 12D Euclidean vector, using a flattened rotation matrix plus translation.
  • Adding Gaussian noise to rotation matrices can produce non-orthogonal rotations, frame-dependent scores, and inefficient pose paths.
  • This matters for manipulation because invalid or unstable pose trajectories can raise control cost and hurt transfer across camera or workspace frames.

Approach

  • The method, Lie Diffuser Actor, adds noise in the tangent space se(3) as a 6D twist with angular and linear velocity components.
  • It maps each noisy twist back to SE(3) with the exponential map, so every denoising step stays a valid rigid transform.
  • The reverse diffusion model predicts score vectors in se(3), then updates poses through group multiplication rather than vector addition.
  • The architecture extends 3D Diffuser Actor with RGB-D point-cloud encoding, CLIP text features, a Transformer denoiser, and a tangent-space prediction head.
  • The paper gives theoretical claims for manifold closure, left-invariant equivariance, and geodesic-like screw-motion trajectories.

Results

  • On CALVIN ABC→D, Lie Diffuser Actor reaches an average task length of 3.512 versus 3D Diffuser Actor at 3.27, a reported +7.3% gain.
  • On CALVIN ABC→D, step success rates are SR1 93.7, SR2 83.4, SR3 70.3, SR4 57.6, and SR5 46.2 for Lie Diffuser Actor.
  • On CALVIN ABCD→D, Lie Diffuser Actor reaches average task length 3.584 versus 3D Diffuser Actor at 3.288; SR5 improves from 41.6 to 53.7.
  • In OpenVLA-OFT cross-architecture validation, SE(3) score matching improves LIBERO Long success rate from 92.20% to 94.13%.
  • In rotation constraint analysis, the method reports lower orthogonality violation than 3D Diffuser Actor: 5.7% lower at median, 11.8% at P90, 5.4% at P95, and 2.6% at P99.
  • Real-robot experiments are said to outperform the baseline on most tasks, but the excerpt does not provide task-level counts or success rates.

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