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

Fangran Miao; Jian Huang; Ting Li

cs.CVMar 16, 2026

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

Authors: Fangran Miao, Jian Huang, Ting Li

Organizations: Department of Data Science and Artificial Intelligence, The Hong Kong Polytechnic University · Department of Applied Mathematics, The Hong Kong Polytechnic University · Department of Data Science and Statistics, Southern University of Science and Technology

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.

Figures & tables

Panel of Figure 1 — Figure 1: Text-to-motion samples under our Riemannian Motion Generation framework.

Format	${T}$	${R}_{\mathrm{Orientation}}$	${R}_{\mathrm{Joint}}$	${P}$	$\mathrm{d}{T}$	$\mathrm{d}{R}_{\mathrm{Orientation}}$	$\mathrm{d}{R}_{\mathrm{Joint}}$	$\mathrm{d}{P}$	Foot	Ambient
									Contact	Dimension
HumanML3D	✗	✗	✓	✓	✓	✓	✗	✓	✓	$12J-1$
MotionStreamer	✗	✗	✓	✓	✓	✓	✗	✓	✗	$12J+8$
DART	✓	✓	✓	✓	✓	✓	✗	✓	✗	$12J+12$
HY-Motion	✓	✓	✓	✓	✗	✗	✗	✗	✗	$9J+3$
RMG (Ours)	✓	✓	✓	✗	✗	✗	✗	✗	✗	$4J+3$

Table 1: Comparison of motion representations.

{T}

: global translation;

{R}_{\mathrm{Orientation}}

: global orientation;

{R}_{\mathrm{Joint}}

: per-joint rotation;

{P}

: joint position;

\mathrm{d}\cdot

: temporal difference of the variable. Compared to existing formats, our method adopts the most concise format, with each factor represented on its natural manifold.

Figure 2: ( Top ) Illustration of the unified Riemannian representation for articulated motion. Each motion frame can be factorized into global translation $(\mathcal{M}_{{T}})$ , global orientation and per-joint rotations $(\mathcal{M}_{{R}})$ , and local pose $(\mathcal{M}_{{P}})$ along with the temporal differences $(T\mathcal{M}_{{F}}\ \text{for}\ {F}\in\{{T},{R},{P}\}$ ) . 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 Equation 3 . The red line is the geodesic between $\bm{x}_{0}$ and $\bm{x}_{1}$ while the yellow line with arrow is the velocity at $\bm{x}_{t}$ .

Method	HumanML3D Format				MotionStreamer Format
	FID $\downarrow$	R@1 $\uparrow$	Div $\rightarrow$	MModality $\uparrow$	FID $\downarrow$	R@1 $\uparrow$	Div $\rightarrow$	MModality $\uparrow$
GT	0.002	0.511	9.503	2.799	0.002	0.702	27.492	–
MLD ( 2023 )	0.473	0.481	9.724	2.413	18.236	0.546	26.352	–
T2M-GPT ( 2023 )	0.116	0.492	9.761	1.856	12.475	0.606	27.275	–
MotionGPT ( 2023 )	0.232	0.492	9.528	2.008	14.375	0.456	27.114	–
MoMask ( 2024 )	0.045	0.521	–	1.241	12.232	0.621	27.127	–

Table 2: Evaluation of text-based motion generation on HumanML3D ( Guo et al., 2022 ) dataset. Reported metrics are FID, R@1, Diversity, and MModality. The models in bold are the optimal models, and the models in underline are the sub-optimal models. The guidance scale is set to

6.5

Method	G.S.	FID $\downarrow$	R@1 $\uparrow$
ScaMo ( 2025 )	–	89.0	0.67
MotionMillion-3B	–	10.8	0.79
MotionMillion-7B	–	10.3	0.79
Ours (0.5B+1.7B)	2.0	5.6	0.81
Ours (0.5B+1.7B)	3.0	7.8	0.86

Table 3: Evaluation of text-based motion generation on MotionMillion ( Fan et al., 2025 ) dataset. Reported metrics are FID and R@1. In our method, 0.5B and 1.7B refer to the parameter sizes of our Riemannian flow matching model and Qwen3-1.7B, respectively. G.S. refers to the guidance scale.

Panel of Figure 3 — 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.

Appendix figures & tables7 assets

Supplementary material from the paper’s appendix.

Appendix

Method	HumanML3D Format
	FID $\downarrow$	R@1 $\uparrow$	R@2 $\uparrow$	R@3 $\uparrow$	MM-Dist $\downarrow$	Div $\rightarrow$	MM $\uparrow$
GT	0.002	0.511	0.703	0.797	2.974	9.503	2.799
MLD	0.473	0.481	0.673	0.772	3.196	9.724	2.413
T2M-GPT	0.116	0.492	0.679	0.775	3.121	9.761	1.856
MotionGPT	0.232	0.492	0.681	0.733	3.096	9.528	2.008
MoMask	0.045	0.521	0.713	0.807	2.958	–	1.241

Appendix

Table 4: Full results on the HumanML3D dataset in H3D format.

Method	MotionStreamer Format
	FID $\downarrow$	R@1 $\uparrow$	R@2 $\uparrow$	R@3 $\uparrow$	MM-Dist $\downarrow$	Div $\rightarrow$	MM $\uparrow$
GT	0.002	0.702	0.864	0.914	15.151	27.492	–
MLD	18.236	0.546	0.730	0.792	16.638	26.352	–
T2M-GPT	12.475	0.606	0.774	0.838	16.812	27.275	–
MotionGPT	14.375	0.456	0.598	0.628	17.892	27.114	–
MoMask	12.232	0.621	0.784	0.846	16.138	27.127	–

Appendix

Table 5: Full results on the HumanML3D dataset in MotionStreamer format.

Method	Guidance Scale	FID $\downarrow$	R@1 $\uparrow$	R@2 $\uparrow$	R@3 $\uparrow$
ScaMo	–	89.0	0.67	0.81	0.87
MotionMillion-3B	–	10.8	0.79	0.91	0.94
MotionMillion-7B	–	10.3	0.79	0.90	0.94
Ours (0.5B+1.7B)	2.0	5.6	0.81	0.92	0.95
Ours (0.5B+1.7B)	3.0	7.8	0.86	0.95	0.97

Appendix

Table 6: Full results of MotionMillion dataset.

Figure 4: Training loss curves for MotionMillion.

Figure 5: Gradient norm curves for MotionMillion.

Hyperparameter	RMG-base	RMG
Input Dim.	91	91
Hidden Dim.	384	1024
No. of Layers	6	24
No. of Heads	8	8
FFN Multiplier	8	4
Max Learning Rate	1e-4	1e-4

Appendix

Table 7: Training hyperparameters for different model scales.

Figure 6: The conversion functions between different motion representations.

Human Motion Generation

Human motion generation creates articulated movement from language, audio, video, spatial constraints, or interaction goals.

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Open paper AskPDF arXiv

Recent advances in diffusion and flow models have substantially improved text-driven human motion generation. Yet most methods generate in low-dimensional, temporally downsampled latent spaces learned primarily for reconstruction, a bottleneck that can limit generation quality and preclude direct manipulation of individual frames and joints. We introduce MotionSpaceFlow (MSFlow), a representation-aware flow-matching framework that predicts clean motion directly in continuous motion space without a learned encoder or decoder. To account for the anisotropic structure of direct motion representations, we propose representation-aware noise scaling and show how the initial Gaussian source scale governs the covariance of intermediate probability-path marginals. We further introduce a Representation-Aware Multimodal Diffusion Transformer (RA-MMDiT), which jointly updates token-level language and full-resolution motion features through joint attention while adapting temporal information flow to the motion representation: causal attention for incremental features defined by frame-to-frame changes, and bidirectional attention for global features such as absolute joint coordinates. Across different datasets and motion representations, MSFlow achieves state-of-the-art text-to-motion performance. Its global representation variant additionally enables zero-shot, inference-time control over any joint or frame through projection sampling without control-conditioned training, delivering leading motion quality with exact constraint satisfaction.

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