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

Vision-Language-Action (VLA) models have emerged as a powerful paradigm for generalist robot manipulation, yet they lack geometric inductive biases: policies trained at specific orientations require substantially more data to generalize across rotational configurations. We present \textsc{EquiVLA}, the first general framework for end-to-end $\mathrm{SO}(2)$-equivariant VLA models, applicable to any architecture coupling a frozen vision-language backbone with a flow-matching Diffusion Transformer action head. \textsc{EquiVLA} introduces \textsc{EquiPerceptor}, which produces approximately $\mathrm{SO}(2)$-equivariant visual representations from frozen ViT features; and \textsc{EquiActor}, an exactly $\mathrm{SO}(2)$-equivariant flow-matching Diffusion Transformer action head. Together, they establish an approximate $\mathrm{SO}(2)$ equivariance chain from camera observations to predicted action sequences. Instantiated on GR00T~N1.5 and evaluated across four LIBERO suites, CALVIN ABCD$\to$D, and five real-robot tasks on Mobile ALOHA, \textsc{EquiVLA} achieves $92.6\%$ average success on LIBERO (vs. $78.1\%$ baseline), an average sequence length of $4.03$ on CALVIN (vs. $3.45$), and improves real-robot success from $54\%$ to $72\%$.

This paper tackles spatial perception and manipulation challenges in Vision-Language-Action (VLA) models. To address depth ambiguity from monocular input, we leverage a pre-trained multi-view diffusion model to synthesize latent novel views and propose a Geometry-Guided Gated Transformer (G3T) that aligns multi-view features under 3D geometric guidance while adaptively filtering occlusion noise. To improve action learning efficiency, we introduce Action Manifold Learning (AML), which directly predicts actions on the valid action manifold, bypassing inefficient regression of unstructured targets like noise or velocity. Experiments on LIBERO, RoboTwin 2.0, and real-robot tasks show our method achieves superior success rate and robustness over SOTA baselines. Project page: https://junjxiao.github.io/Multi-view-VLA.github.io/.

We take a Gromov-Wasserstein perspective on Vision-Language-Action (VLA) learning, where the goal is to make the relational geometry of action representations compatible with the semantic geometry of VL embeddings. However, this alignment is non-trivial due to the mathematical heterogeneity between the domains: the semantic space of vision-language is topologically linear and isotropic, whereas the physical manifold of robotic action is non-Euclidean and anisotropic. Their disjoint metric structures render direct regression ill-posed. To resolve this incompatibility, we introduce LAST (Lie-algebraic Action Space Tokenizer), which reconstructs the action space to establish local metric compatibility with the VL modality via a two-stage transformation: (1) Global Topological Linearization: linearizing the action manifold via Lie-algebraic mapping, converting trajectories into a fixed-length, physically additive representation. (2) Local Metric Discretization: hierarchically discretizing the representation into schemas and whitened residuals, yielding approximately isotropic local charts that are statistically aligned with the semantic metric. By resolving the structural mismatch at both global and local levels, LAST enables VLA models with superior convergence and generalizability.