# Learning 4D Geometric Priors for Inference-Efficient World Action Models

arXiv: https://arxiv.org/abs/2607.05468 · HTML: https://mlsanity.com/paper/2607.05468/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- Placed in Embodied and robot world models, one of 11 directions in World models.
- Builds on 25 of its 42 references that are in the map.
- First posted on arXiv 6 Jul 2026.

## Placed in

- World models / [Embodied and robot world models](https://mlsanity.com/wm/embodied-and-robot-world-models/)

## Builds on (most cited first, 25 of 25)

- [Wan: Open and Advanced Large-Scale Video Generative Models](https://mlsanity.com/paper/2503.20314/) (2025)
- [π0: A Vision-Language-Action Flow Model for General Robot Control](https://mlsanity.com/paper/2410.24164/) (2024)
- [π0.5: a Vision-Language-Action Model with Open-World Generalization](https://mlsanity.com/paper/2504.16054/) (2025)
- [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)
- [World Action Models are Zero-shot Policies](https://mlsanity.com/paper/2602.15922/) (2026)
- [Fast-WAM: Do World Action Models Need Test-time Future Imagination?](https://mlsanity.com/paper/2603.16666/) (2026)
- [Causal World Modeling for Robot Control](https://mlsanity.com/paper/2601.21998/) (2026)
- [X-VLA: Soft-Prompted Transformer as Scalable Cross-Embodiment Vision-Language-Action Model](https://mlsanity.com/paper/2510.10274/) (2025)
- [Cosmos Policy: Fine-Tuning Video Models for Visuomotor Control and Planning](https://mlsanity.com/paper/2601.16163/) (2026)
- [mimic-video: Video-Action Models for Generalizable Robot Control Beyond VLAs](https://mlsanity.com/paper/2512.15692/) (2025)
- [GigaWorld-Policy: An Efficient Action-Centered World--Action Model](https://mlsanity.com/paper/2603.17240/) (2026)
- [Unified Vision-Language-Action Model](https://mlsanity.com/paper/2506.19850/) (2025)
- [Discrete Diffusion VLA: Bringing Discrete Diffusion to Action Decoding in Vision-Language-Action Policies](https://mlsanity.com/paper/2508.20072/) (2025)
- [DiT4DiT: Jointly Modeling Video Dynamics and Actions for Generalizable Robot Control](https://mlsanity.com/paper/2603.10448/) (2026)
- [Unified 4D World Action Modeling from Video Priors with Asynchronous Denoising](https://mlsanity.com/paper/2604.26694/) (2026)
- [GeoVLA: Empowering 3D Representations in Vision-Language-Action Models](https://mlsanity.com/paper/2508.09071/) (2025)
- [Motubrain: An Advanced World Action Model for Robot Control](https://mlsanity.com/paper/2604.27792/) (2026)
- [World Action Models: The Next Frontier in Embodied AI](https://mlsanity.com/paper/2605.12090/) (2026)
- [Next Forcing: Causal World Modeling with Multi-Chunk Prediction](https://mlsanity.com/paper/2606.11187/) (2026)
- [WAM4D: Fast 4D World Action Model via Spatial Register Tokens](https://mlsanity.com/paper/2606.14048/) (2026)
- [Any3D-VLA: Enhancing VLA Robustness via Diverse Point Clouds](https://mlsanity.com/paper/2602.00807/) (2026)
- [DeMaVLA: A Vision-Language-Action Foundation Model for Generalizable Deformable Manipulation](https://mlsanity.com/paper/2605.31286/) (2026)
- [Towards High-Consistency Embodied World Model with Multi-View Trajectory Videos](https://mlsanity.com/paper/2511.12882/) (2025)

## Built on by (most cited first, 15 of 15)

- [4D-WAM: Infusing Spatiotemporal Awareness into World Action Models through Trajectory Fields](https://mlsanity.com/paper/2608.08023/) (2026)
- [ST-WAM: Semantic-Temporal World Action Model for Robust Manipulation under Visual Distribution Shifts](https://mlsanity.com/paper/2607.28993/) (2026)
- [StageWAM: Joint-Embedding Stage Prediction for World-Action Models in Robot Manipulation](https://mlsanity.com/paper/2608.10780/) (2026)
- [DC-WAM: Dynamic-Centric Visual Supervision and Reasoning for World-Action Models](https://mlsanity.com/paper/2607.25918/) (2026)
- [WNM-3D: A World Navigation Model with 3D Scene Conditioning for Closed-Loop VLN](https://mlsanity.com/paper/2608.07267/) (2026)
- [DIDO: Distilling Interaction-Centric Dynamics into One-Step Denoising for World Action Models](https://mlsanity.com/paper/2609.15570/) (2026)
- [Acting in Meters: Learning Metric Interactions for Precise Robotic Manipulation](https://mlsanity.com/paper/2609.18243/) (2026)
- [CSWAM: Better Causal Semantic Representations for Out-of-Distribution Generalization in World Action Models](https://mlsanity.com/paper/2609.18462/) (2026)
- [MaskHarness-WAM: Instance-Grounded Harnessing for Long-Horizon Robot Manipulation](https://mlsanity.com/paper/2609.19974/) (2026)
- [MT-WAM: Reorienting the One-Pass Predictive Representation Toward Action Generation](https://mlsanity.com/paper/2609.21474/) (2026)
- [An Action Is Worth One Patch: Unified World-Action Modeling with PatchWAM](https://mlsanity.com/paper/2609.25961/) (2026)
- [Copper-Policy: Focus on the Representation for Robust Robot Manipulation](https://mlsanity.com/paper/2609.32779/) (2026)
- [Spatial Grafting: Grounding 3D Features for Flow-Matching Robot Policies](https://mlsanity.com/paper/2609.35249/) (2026)
- [MVG-WAM: Multiple View Geometry-Aware World-Action Modeling for Robotic Manipulation](https://mlsanity.com/paper/2609.37793/) (2026)
- [What Visual Representations Do World Action Models Need?](https://mlsanity.com/paper/2610.13024/) (2026)

## Abstract (arXiv)

World Action Models (WAMs) have shown strong potential for robotic manipulation by jointly modeling visual future dynamics and executable action sequences. However, existing video-action co-training methods primarily optimize appearance-oriented video latents, which may insufficiently capture the temporally evolving geometry required for precise manipulation. We propose MECo-WAM, a Multi-Expert Co-Training World Action Model that injects action-relevant 4D geometric priors into video-action representations while preserving the original lightweight inference graph. During training, MECo-WAM combines video and action experts with a lightweight 4D expert supervised by relational targets from a frozen VGGT encoder. Asymmetric expert visibility prevents non-causal shortcuts from auxiliary geometry to action generation. To transfer geometric knowledge into the deployed video-action pathway, we introduce decayed 4D read-mask attention, which provides restricted current-frame geometric guidance early in training and progressively removes this dependency. We further propose action-aware temporal geometric distillation, which aligns within-frame geometric relations and their temporal evolution while emphasizing visual regions most relevant to robot actions. At deployment, all auxiliary 4D components are removed. Experiments on LIBERO (98.2%), RoboTwin 2.0 (92.6%), and challenging real-world manipulation tasks show that MECo-WAM improves manipulation performance without increasing inference cost.