What question did this study set out to answer?

The aim is to improve action decoding in vision-language-action (VLA) systems using a geometric approach.

April 25, 2026Open Access

WAM-Flow: Wasserstein Action Manifolds for Vision-Language-Action Policy Alignment and Optimal-Transport Flow-Matching Decoding

Key Points

The aim is to improve action decoding in vision-language-action (VLA) systems using a geometric approach.
Developed a Wasserstein Action Manifold for action decoding using Riemannian geometry.
Evaluated through theory-linked simulations that link action distributions to task conditions.
Derived a Wasserstein-regularized flow-matching objective to enhance decoding accuracy.
Demonstrated quantization distortion is minimized by using Wasserstein distance metrics.
Provided Lipschitz bounds showing learned flow sensitivity under perturbations.
Established a deployment-shift bound relating decoder error to training and deployment distribution discrepancies.

Abstract

WAM-Flow technical report / preprint. Vision-language-action (VLA) policies commonly decode actions either through discrete tokenization or through continuous Euclidean regression. Both choices can be mismatched to robot control semantics: tokenization introduces irreversible quantization, while Euclidean losses ignore task-dependent temporal, cross-joint, and embodiment structure Brohan et al. , 2023; Kalashnikov et al. , 2023; Team et al. , 2024; Black et al. , 2024. This paper develops a geometric formulation of action decoding and evaluates it through theory-linked simulations. We define a Wasserstein Action Manifold for action chunks as a task-conditioned Riemannian geometry on trajectory space. In the default construction, the manifold is \ (RT dₐ\) equipped with a context-dependent symmetric positive definite metric \ (G_ \), where \ (= (x, , s) \) denotes visual, language, and optional proprioceptive context. This metric induces a task-dependent quadratic transport cost and corresponding Wasserstein discrepancy between action distributions Villani, 2009; Peyre and Cuturi, 2019. Building on conditional flow matching and optimal transport, we derive a Wasserstein-regularized flow-matching objective for VLA action decoding Lipman et al. , 2023; Tong et al. , 2024. Under explicit assumptions, we state: (i) a coupling-based upper bound showing that discrete decoders inherit quantization distortion controlled by the Wasserstein distance between the true continuous action law and its quantized reconstruction; (ii) a sufficient optimization statement showing that Wasserstein regularization directly penalizes this distortion surrogate; (iii) Lipschitz rollout-sensitivity bounds for the learned flow under observation perturbations; and (iv) a deployment-shift bound in which decoder error depends on Wasserstein distance between training and deployment distributions. Existing OSF archival DOI: 10. 17605/OSF. IO/2J765; Existing OSF archival page: https: //osf. io/2j765/. Files include the technical report PDF and the LaTeX source tarball when available.

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