# Normalizing Flows on the Product Space of SO(3) Manifolds for Probabilistic Human Pose Modeling

arXiv: https://arxiv.org/abs/2404.05675 · HTML: https://mlsanity.com/paper/2404.05675/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 41 references that are in the map.
- First posted on arXiv 8 Apr 2024.

## Builds on (most cited first, 28 of 28)

- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Expressive Body Capture: 3D Hands, Face, and Body from a Single Image](https://mlsanity.com/paper/1904.05866/) (2019)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [AMASS: Archive of Motion Capture as Surface Shapes](https://mlsanity.com/paper/1904.03278/) (2019)
- [On the Continuity of Rotation Representations in Neural Networks](https://mlsanity.com/paper/1812.07035/) (2018)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [Keep it SMPL: Automatic Estimation of 3D Human Pose and Shape from a Single Image](https://mlsanity.com/paper/1607.08128/) (2016)
- [Masked Autoregressive Flow for Density Estimation](https://mlsanity.com/paper/1705.07057/) (2017)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [HuMoR: 3D Human Motion Model for Robust Pose Estimation](https://mlsanity.com/paper/2105.04668/) (2021)
- [MADE: Masked Autoencoder for Distribution Estimation](https://mlsanity.com/paper/1502.03509/) (2015)
- [Probabilistic Modeling for Human Mesh Recovery](https://mlsanity.com/paper/2108.11944/) (2021)
- [Weakly Supervised 3D Human Pose and Shape Reconstruction with Normalizing Flows](https://mlsanity.com/paper/2003.10350/) (2020)
- [Pose-NDF: Modeling Human Pose Manifolds with Neural Distance Fields](https://mlsanity.com/paper/2207.13807/) (2022)
- [Implicit-PDF: Non-Parametric Representation of Probability Distributions on the Rotation Manifold](https://mlsanity.com/paper/2106.05965/) (2021)
- [GFPose: Learning 3D Human Pose Prior with Gradient Fields](https://mlsanity.com/paper/2212.08641/) (2022)
- [normflows: A PyTorch Package for Normalizing Flows](https://mlsanity.com/paper/2302.12014/) (2023)
- [Hierarchical Kinematic Probability Distributions for 3D Human Shape and Pose Estimation from Images in the Wild](https://mlsanity.com/paper/2110.00990/) (2021)
- [Probabilistic 3D Human Shape and Pose Estimation from Multiple Unconstrained Images in the Wild](https://mlsanity.com/paper/2103.10978/) (2021)
- [Adversarial Parametric Pose Prior](https://mlsanity.com/paper/2112.04203/) (2021)
- [Motron: Multimodal Probabilistic Human Motion Forecasting](https://mlsanity.com/paper/2203.04132/) (2022)
- [HuManiFlow: Ancestor-Conditioned Normalising Flows on SO(3) Manifolds for Human Pose and Shape Distribution Estimation](https://mlsanity.com/paper/2305.06968/) (2023)
- [Risk-Sensitive Sequential Action Control with Multi-Modal Human Trajectory Forecasting for Safe Crowd-Robot Interaction](https://mlsanity.com/paper/2009.05702/) (2020)
- [ProtoRes: Proto-Residual Network for Pose Authoring via Learned Inverse Kinematics](https://mlsanity.com/paper/2106.01981/) (2021)
- [Delving into Discrete Normalizing Flows on SO(3) Manifold for Probabilistic Rotation Modeling](https://mlsanity.com/paper/2304.03937/) (2023)
- [SMPL-IK: Learned Morphology-Aware Inverse Kinematics for AI Driven Artistic Workflows](https://mlsanity.com/paper/2208.08274/) (2022)
- [Unifying supervised learning and VAEs -- coverage, systematics and goodness-of-fit in normalizing-flow based neural network models for astro-particle reconstructions](https://mlsanity.com/paper/2008.05825/) (2020)

## Built on by (most cited first, 7 of 7)

- [DPoser-X: Diffusion Model as Robust 3D Whole-body Human Pose Prior](https://mlsanity.com/paper/2508.00599/) (2025)
- [Utilizing Uncertainty in 2D Pose Detectors for Probabilistic 3D Human Mesh Recovery](https://mlsanity.com/paper/2411.16289/) (2024)
- [Neural Human Pose Prior](https://mlsanity.com/paper/2507.12138/) (2025)
- [LieHMR: Autoregressive Human Mesh Recovery with SO(3) Diffusion](https://mlsanity.com/paper/2509.25739/) (2025)
- [BabyFlow: 3D modeling of realistic and expressive infant faces](https://mlsanity.com/paper/2512.19560/) (2025)
- [Learning Manifold and Ito Dynamics with Branched Neural Rough Differential Equations](https://mlsanity.com/paper/2606.05272/) (2026)
- [Naturalness-guided Manifold Flow Matching for Sign Language Production](https://mlsanity.com/paper/2609.33339/) (2026)

## Abstract (arXiv)

Normalizing flows have proven their efficacy for density estimation in Euclidean space, but their application to rotational representations, crucial in various domains such as robotics or human pose modeling, remains underexplored. Probabilistic models of the human pose can benefit from approaches that rigorously consider the rotational nature of human joints. For this purpose, we introduce HuProSO3, a normalizing flow model that operates on a high-dimensional product space of SO(3) manifolds, modeling the joint distribution for human joints with three degrees of freedom. HuProSO3's advantage over state-of-the-art approaches is demonstrated through its superior modeling accuracy in three different applications and its capability to evaluate the exact likelihood. This work not only addresses the technical challenge of learning densities on SO(3) manifolds, but it also has broader implications for domains where the probabilistic regression of correlated 3D rotations is of importance.