Generative Modeling with Orbit-Space Particle Flow Matching

Flow matching has recently emerged as a flexible and efficient framework for generative modelling by learning deterministic transport dynamics between probability measures. In this work, we extend flow matching to the space of probability measures over probability measures, introducing a Wasserstein-on-Wasserstein (WoW) formulation. Leveraging the nested Wasserstein geometry, we show that measures over transport plans naturally induce velocity fields that realize metameasure flows. This yields a principled generalization of Wasserstein flow matching via coupled outer and inner transport plans. To address the substantial computational cost of WoW transport, we propose scalable approximations based on sliced and linear Wasserstein distances, enabling efficient training while promoting numerically stable, near-straight trajectories. Our framework unifies and extends existing approaches to point cloud and set generation, providing a practical and theoretically grounded method for generative modelling in WoW spaces.

Recovering complete 3D representations of objects from few casual image captures remains a significant challenge. Recent 3D generative models, particularly those based on Flow-Matching (FM), can synthesize high-quality textured assets; however, they often suffer from ''synthetic bias'' where learned priors override observational evidence, alongside a lack of alignment with the observed instance. Conversely, optimization-based methods like 3D Gaussian Splatting (3DGS) provide high fidelity on visible surfaces but fail to reason about unobserved geometry. In this paper, we present FlowObject, a framework that reformulates sparse-view 3D reconstruction as a training-free, guided inverse problem. Our approach applies a dual-space guidance strategy to steer the Ordinary Differential Equation (ODE) trajectory of a flow-matching model, enabling the completion of unseen regions through learned generative priors while enforcing strict consistency with real-world observations. By integrating a 3DGS refinement stage, FlowObject further bridges the gap between ''synthetic-looking'' generative outputs and photorealistic reconstructions. Comprehensive benchmarks on synthetic and real-world datasets demonstrate that current state-of-the-art methods often struggle to achieve geometric completeness and observational consistency simultaneously, especially under severe occlusions. In contrast, our method significantly outperforms state-of-the-art generative models and optimization-based frameworks in both geometric completeness and view-dependent appearance fidelity.

Generating high-quality 3D point clouds requires capturing both global shape topology and local geometric details. Existing flow-based methods rely on continuous normalizing flows (CNFs) that demand expensive ODE solving and trace estimation during training, while diffusion models require hundreds of iterative denoising steps. Moreover, most approaches adopt single-level generation directly in point space, disregarding the hierarchical structure natural to 3D shapes. We propose Hierarchical Flow Matching (HFM) that extends flow matching to bilevel structure for unconditional 3D point cloud generation. HFM decomposes the task into two levels via optimal-transport flow matching: a \textit{Latent Flow Matching} models the global shape manifold in a compact latent space, and a \textit{Conditional Point Flow Matching} reconstructs detailed point clouds conditioned on the latent code. Both flows are trained with simple MSE regression losses. The resulting straight OT paths enable efficient sampling with as few as 15 Euler steps per flow, while the structured latent space supports downstream tasks including classification. Extensive experiments on ShapeNet and ModelNet benchmarks demonstrate that HFM achieves competitive or even best performance compared with prior state-of-the-art methods.