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Riemannian Motion Generation: A Unified Framework for Human Motion Representation and Generation via Riemannian Flow Matching

Fangran Miao, Jian Huang, Ting Li

Abstract

Human motion generation is often learned in Euclidean spaces, although valid motions follow structured non-Euclidean geometry. We present Riemannian Motion Generation (RMG), a unified framework that represents motion on a product manifold and learns dynamics via Riemannian flow matching. RMG factorizes motion into several manifold factors, yielding a scale-free representation with intrinsic normalization, and uses geodesic interpolation, tangent-space supervision, and manifold-preserving ODE integration for training and sampling. On HumanML3D, RMG achieves state-of-the-art FID in the HumanML3D format (0.043) and ranks first on all reported metrics under the MotionStreamer format. On MotionMillion, it also surpasses strong baselines (FID 5.6, R@1 0.86). Ablations show that the compact $\mathscr{T}+\mathscr{R}$ (translation + rotations) representation is the most stable and effective, highlighting geometry-aware modeling as a practical and scalable route to high-fidelity motion generation.

Riemannian Motion Generation: A Unified Framework for Human Motion Representation and Generation via Riemannian Flow Matching

Abstract

Human motion generation is often learned in Euclidean spaces, although valid motions follow structured non-Euclidean geometry. We present Riemannian Motion Generation (RMG), a unified framework that represents motion on a product manifold and learns dynamics via Riemannian flow matching. RMG factorizes motion into several manifold factors, yielding a scale-free representation with intrinsic normalization, and uses geodesic interpolation, tangent-space supervision, and manifold-preserving ODE integration for training and sampling. On HumanML3D, RMG achieves state-of-the-art FID in the HumanML3D format (0.043) and ranks first on all reported metrics under the MotionStreamer format. On MotionMillion, it also surpasses strong baselines (FID 5.6, R@1 0.86). Ablations show that the compact (translation + rotations) representation is the most stable and effective, highlighting geometry-aware modeling as a practical and scalable route to high-fidelity motion generation.
Paper Structure (44 sections, 2 theorems, 30 equations, 6 figures, 7 tables)

This paper contains 44 sections, 2 theorems, 30 equations, 6 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Related Works
  3. Human Motion Generation
  4. Riemannian Manifold
  5. General Flow Matching
  6. Methodology
  7. Notation.
  8. Motion Representation and Manifold
  9. Global Translation ($\mathscr{T}$).
  10. Global Orientation and Per-Joint Rotations ($\mathscr{R}$).
  11. Local Pose ($\mathscr{P}$).
  12. Temporal Differentiation ($\mathrm{d} \cdot$).
  13. Unified View.
  14. Prior Distribution
  15. Riemannian Gaussian Distribution.
  16. ...and 29 more sections

Key Result

Proposition 1

Let $\mathcal{M}$ be a smooth $m$-dimensional manifold and let $\pi: U \subset \mathbb{R}^d \to \mathcal{M}$ be a smooth submersion with $d > m$. For any loss $L(z) = \ell(\pi(z))$, the gradient $\nabla L(z)$ is orthogonal to $\ker(D\pi(z))$, and the Hessian $\nabla^2 L(z)$ has at least $d - m$ zero

Figures (6)

  • Figure 1: Text-to-motion samples under our Riemannian Motion Generation framework.
  • Figure 2: (Top) Illustration of the unified Riemannian representation for articulated motion. Each motion frame can be factorized into global translation$(\mathcal{M}_{\mathscr{T}})$, global orientation and per-joint rotations$(\mathcal{M}_{\mathscr{R}})$, and local pose$(\mathcal{M}_{\mathscr{P}})$ along with the temporal differences$(T\mathcal{M}_{\mathscr{F}}\ \text{for}\ \mathscr{F}\in\{\mathscr{T},\mathscr{R},\mathscr{P}\}$). Each factor is represented on its natural Riemannian manifold, yielding a scale-free representation with intrinsic normalization. (Bottom) Illustration of the Riemannian flow matching process in the RMG manifold. $\mathcal{M}$ is defined by our proposed manifold \ref{['eq:rmg_manifold']}. The red line is the geodesic between $\bm x_0$ and $\bm x_1$ while the yellow line with arrow is the velocity at $\bm x_t$.
  • Figure 3: Ablation study on different factors of our framework. All the models are trained with the same setting (including the parameter size, random seed) and evaluated on the HumanML3D benchmark.
  • Figure 4: Training loss curves for MotionMillion.
  • Figure 5: Gradient norm curves for MotionMillion.
  • ...and 1 more figures

Theorems & Definitions (4)

  • Proposition 1: Redundancy induces flat directions
  • proof
  • Proposition 2: Informal statistical advantage of Riemannian Flow Matching
  • proof : Proof sketch