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

Source: [The Lie We Tell: Correcting the Euclidean Fallacy in Vision Language Action Policies via Score Matching on Tangent Space](<https://www.sota2.com/research/paper/6a1e6438dbd7150506f85881>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2026-06-02T05:03:19.763Z

Authors: Bing-Cheng Chuang, I-Hsuan Chu, Bor-Jiun Lin, YuanFu Yang, Min Sun, Chun-Yi Lee

Published: 2026-06-01T07:59:29.000Z

## 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.

## Recorded benchmark results

A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage.

- [Move Doll Platform on Real Robot Move Doll Platform](<https://www.sota2.com/research/sota/move-doll-platform-on-real-robot-move-doll-platform.md>): Move Doll Platform; Real Robot Move Doll Platform; Success Rate: 100.

- [Sort Blocks on Real Robot Sort Blocks](<https://www.sota2.com/research/sota/sort-blocks-on-real-robot-sort-blocks.md>): Sort Blocks; Real Robot Sort Blocks; Success Rate: 75.

- [Stack Cups on Real Robot Stack Cups](<https://www.sota2.com/research/sota/stack-cups-on-real-robot-stack-cups.md>): stack cups; Real Robot Stack Cups; Success Rate: 60.

- [Put Block in Box on Real Robot Put Block in Box](<https://www.sota2.com/research/sota/put-block-in-box-on-real-robot-put-block-in-box.md>): Put Block in Box; Real Robot Put Block in Box; Success Rate: 0.75.

- [Long-horizon language-conditioned manipulation on Calvin ABC→D](<https://www.sota2.com/research/sota/long-horizon-language-conditioned-manipulation-on-calvin-abc-d.md>): Long-horizon language-conditioned manipulation; Calvin ABC-\>D; Average Sequence Length: 3.584.

## Original sources and implementation

[Original paper](<https://arxiv.org/abs/2606.01847>)

[PDF](<https://arxiv.org/pdf/2606.01847.pdf>)