# LAST: Bridging Vision-Language and Action Manifolds via Gromov-Wasserstein Alignment

arXiv: https://arxiv.org/abs/2606.11221 · HTML: https://mlsanity.com/paper/2606.11221/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Placed in 3D and spatial-aware VLA, one of 13 directions in Vision-language-action models.
- Builds on 13 of its 32 references that are in the map.
- First posted on arXiv 27 May 2026.

## Placed in

- Vision-language-action models / [3D and spatial-aware VLA](https://mlsanity.com/vla/3d-and-spatial-aware-vla/)

## Builds on (most cited first, 13 of 13)

- [Learning Transferable Visual Models From Natural Language Supervision](https://mlsanity.com/paper/2103.00020/) (2021)
- [Qwen2.5-VL Technical Report](https://mlsanity.com/paper/2502.13923/) (2025)
- [π0: A Vision-Language-Action Flow Model for General Robot Control](https://mlsanity.com/paper/2410.24164/) (2024)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Fine-Tuning Vision-Language-Action Models: Optimizing Speed and Success](https://mlsanity.com/paper/2502.19645/) (2025)
- [FAST: Efficient Action Tokenization for Vision-Language-Action Models](https://mlsanity.com/paper/2501.09747/) (2025)
- [SpatialVLA: Exploring Spatial Representations for Visual-Language-Action Model](https://mlsanity.com/paper/2501.15830/) (2025)
- [Evaluating Real-World Robot Manipulation Policies in Simulation](https://mlsanity.com/paper/2405.05941/) (2024)
- [Latent Space Oddity: on the Curvature of Deep Generative Models](https://mlsanity.com/paper/1710.11379/) (2017)
- [A micro Lie theory for state estimation in robotics](https://mlsanity.com/paper/1812.01537/) (2018)
- [What Matters in Building Vision-Language-Action Models for Generalist Robots](https://mlsanity.com/paper/2412.14058/) (2024)
- [Cross the Gap: Exposing the Intra-modal Misalignment in CLIP via Modality Inversion](https://mlsanity.com/paper/2502.04263/) (2025)
- [BEAST: Efficient Tokenization of B-Splines Encoded Action Sequences for Imitation Learning](https://mlsanity.com/paper/2506.06072/) (2025)

## Built on by (most cited first, 1 of 1)

- [Orca: The World is in Your Mind](https://mlsanity.com/paper/2606.30534/) (2026)

## Abstract (arXiv)

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.