# Sparse Compositional Flow Matching by geometric assembly from motion primitives

arXiv: https://arxiv.org/abs/2605.23341 · HTML: https://mlsanity.com/paper/2605.23341/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Diffusion and flow policies for robotics, one of 14 directions in Diffusion models.
- Builds on 42 of its 62 references that are in the map.
- First posted on arXiv 22 May 2026.

## Placed in

- Diffusion models / [Diffusion and flow policies for robotics](https://mlsanity.com/diffusion/diffusion-and-flow-policies-for-robotics/)

## Builds on (most cited first, 30 of 42)

- [Diffusion Policy: Visuomotor Policy Learning via Action Diffusion](https://mlsanity.com/paper/2303.04137/) (2023)
- [Compositional Visual Generation with Composable Diffusion Models](https://mlsanity.com/paper/2206.01714/) (2022)
- [MotionGPT: Human Motion as a Foreign Language](https://mlsanity.com/paper/2306.14795/) (2023)
- [Social-STGCNN: A Social Spatio-Temporal Graph Convolutional Neural Network for Human Trajectory Prediction](https://mlsanity.com/paper/2002.11927/) (2020)
- [It Is Not the Journey but the Destination: Endpoint Conditioned Trajectory Prediction](https://mlsanity.com/paper/2004.02025/) (2020)
- [Reduce, Reuse, Recycle: Compositional Generation with Energy-Based Diffusion Models and MCMC](https://mlsanity.com/paper/2302.11552/) (2023)
- [Leapfrog Diffusion Model for Stochastic Trajectory Prediction](https://mlsanity.com/paper/2303.10895/) (2023)
- [SR-LSTM: State Refinement for LSTM towards Pedestrian Trajectory Prediction](https://mlsanity.com/paper/1903.02793/) (2019)
- [Compositional Foundation Models for Hierarchical Planning](https://mlsanity.com/paper/2309.08587/) (2023)
- [PartCrafter: Structured 3D Mesh Generation via Compositional Latent Diffusion Transformers](https://mlsanity.com/paper/2506.05573/) (2025)
- [Optimal Flow Matching: Learning Straight Trajectories in Just One Step](https://mlsanity.com/paper/2403.13117/) (2024)
- [ActFormer: A GAN-based Transformer towards General Action-Conditioned 3D Human Motion Generation](https://mlsanity.com/paper/2203.07706/) (2022)
- [Trajectory Prediction with Latent Belief Energy-Based Model](https://mlsanity.com/paper/2104.03086/) (2021)
- [Neural Dynamic Policies for End-to-End Sensorimotor Learning](https://mlsanity.com/paper/2012.02788/) (2020)
- [Bridging Past and Future: End-to-End Autonomous Driving with Historical Prediction and Planning](https://mlsanity.com/paper/2503.14182/) (2025)
- [Multimodal Deep Generative Models for Trajectory Prediction: A Conditional Variational Autoencoder Approach](https://mlsanity.com/paper/2008.03880/) (2020)
- [Non-Probability Sampling Network for Stochastic Human Trajectory Prediction](https://mlsanity.com/paper/2203.13471/) (2022)
- [Human Trajectory Prediction via Counterfactual Analysis](https://mlsanity.com/paper/2107.14202/) (2021)
- [PoCo: Policy Composition from and for Heterogeneous Robot Learning](https://mlsanity.com/paper/2402.02511/) (2024)
- [Generative Trajectory Stitching through Diffusion Composition](https://mlsanity.com/paper/2503.05153/) (2025)
- [Social-Implicit: Rethinking Trajectory Prediction Evaluation and The Effectiveness of Implicit Maximum Likelihood Estimation](https://mlsanity.com/paper/2203.03057/) (2022)
- [MoFlow: One-Step Flow Matching for Human Trajectory Forecasting via Implicit Maximum Likelihood Estimation based Distillation](https://mlsanity.com/paper/2503.09950/) (2025)
- [EnergyMoGen: Compositional Human Motion Generation with Energy-Based Diffusion Model in Latent Space](https://mlsanity.com/paper/2412.14706/) (2024)
- [Fast Inference and Update of Probabilistic Density Estimation on Trajectory Prediction](https://mlsanity.com/paper/2308.08824/) (2023)
- [ProDMPs: A Unified Perspective on Dynamic and Probabilistic Movement Primitives](https://mlsanity.com/paper/2210.01531/) (2022)
- [How Compositional Generalization and Creativity Improve as Diffusion Models are Trained](https://mlsanity.com/paper/2502.12089/) (2025)
- [Trajectory Flow Matching with Applications to Clinical Time Series Modeling](https://mlsanity.com/paper/2410.21154/) (2024)
- [Causal Composition Diffusion Model for Closed-loop Traffic Generation](https://mlsanity.com/paper/2412.17920/) (2024)
- [Euro-PVI: Pedestrian Vehicle Interactions in Dense Urban Centers](https://mlsanity.com/paper/2106.12442/) (2021)
- [State-Covering Trajectory Stitching for Diffusion Planners](https://mlsanity.com/paper/2506.00895/) (2025)

## Abstract (arXiv)

Embodied trajectories, such as the executable motion sequences of robotic manipulators, underwater vehicles, and mobile robots, are a fundamental output of embodied AI. Modern generative models often treat them as a dense, monolithic signal generated point by point, fitting an intricate high-dimensional posterior while leaving the data's latent structure unmodeled, the same sample inefficiency long identified by the structured generative model literature. We argue that a compositional latent structure is a natural choice: many embodied tasks share recurring motion fragments that can be made explicit as a finite repertoire of reusable motion primitives, and compositional units naturally align with subtask boundaries to support task decomposition. Existing compositional generators, however, compose in a latent space and rely on post-hoc decoding to relate sampled units to actual trajectory segments. We instead compose directly in the physical trajectory space through a flow-matching framework with two coupled designs. Motion-Primitive Dictionary Learning equips each atom with a learnable length mask and binary starting indicators so the atom itself is the primitive, reused verbatim wherever it is placed. Structural Sparse Flow Matching with Geometric Constraints then generates a binary placement matrix using duration-aware tokenization and a differentiable geometric loss that enforces spatial continuity and temporal contiguity where adjacent primitives meet. On Open X-Embodiment and 3DMoTraj, the framework attains state-of-the-art accuracy and reduces the FDE/ADE ratio from 1.8 to 1.07, improving ADE by 19.2% and FDE by 21.0% over the strongest baseline.