# A Smooth Representation of Belief over SO(3) for Deep Rotation Learning with Uncertainty

arXiv: https://arxiv.org/abs/2006.01031 · HTML: https://mlsanity.com/paper/2006.01031/

## Key facts

- Cited by 30 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Jun 2020.

## Built on by (most cited first, 30 of 30)

- [A Survey of Uncertainty in Deep Neural Networks](https://mlsanity.com/paper/2107.03342/) (2021)
- [Deep Regression on Manifolds: A 3D Rotation Case Study](https://mlsanity.com/paper/2103.16317/) (2021)
- [Theseus: A Library for Differentiable Nonlinear Optimization](https://mlsanity.com/paper/2207.09442/) (2022)
- [LoopSplat: Loop Closure by Registering 3D Gaussian Splats](https://mlsanity.com/paper/2408.10154/) (2024)
- [Motron: Multimodal Probabilistic Human Motion Forecasting](https://mlsanity.com/paper/2203.04132/) (2022)
- [Intraoperative 2D/3D Image Registration via Differentiable X-ray Rendering](https://mlsanity.com/paper/2312.06358/) (2023)
- [Orienting Novel 3D Objects Using Self-Supervised Learning of Rotation Transforms](https://mlsanity.com/paper/2105.14246/) (2021)
- [Active Learning of Discrete-Time Dynamics for Uncertainty-Aware Model Predictive Control](https://mlsanity.com/paper/2210.12583/) (2022)
- [Bingham Policy Parameterization for 3D Rotations in Reinforcement Learning](https://mlsanity.com/paper/2202.03957/) (2022)
- [Learning to Predict 3D Rotational Dynamics from Images of a Rigid Body with Unknown Mass Distribution](https://mlsanity.com/paper/2308.14666/) (2023)
- [Learning Generalizable Tool-use Skills through Trajectory Generation](https://mlsanity.com/paper/2310.00156/) (2023)
- [Unified framework for diffusion generative models in SO(3): applications in computer vision and astrophysics](https://mlsanity.com/paper/2312.11707/) (2023)
- [Pose Representations for Deep Skeletal Animation](https://mlsanity.com/paper/2111.13907/) (2021)
- [Learning Interpretable Dynamics from Images of a Freely Rotating 3D Rigid Body](https://mlsanity.com/paper/2209.11355/) (2022)
- [Learning to Localize in Unseen Scenes with Relative Pose Regressors](https://mlsanity.com/paper/2303.02717/) (2023)
- [Forecasting Continuous Non-Conservative Dynamical Systems in SO(3)](https://mlsanity.com/paper/2508.07775/) (2025)
- [Stable Object Reorientation using Contact Plane Registration](https://mlsanity.com/paper/2208.08962/) (2022)
- [Continuous-Time SO(3) Forecasting with Savitzky--Golay Neural Controlled Differential Equations](https://mlsanity.com/paper/2506.06780/) (2025)
- [Geometry-Aware Uncertainty Quantification via Conformal Prediction on Manifolds](https://mlsanity.com/paper/2602.16015/) (2026)
- [Exploring the Adjugate Matrix Approach to Quaternion Pose Extraction](https://mlsanity.com/paper/2205.09116/) (2022)
- [Deep Projective Rotation Estimation through Relative Supervision](https://mlsanity.com/paper/2211.11182/) (2022)
- [Self-supervised learning of object pose estimation using keypoint prediction](https://mlsanity.com/paper/2302.07360/) (2023)
- [ARC-Flow : Articulated, Resolution-Agnostic, Correspondence-Free Matching and Interpolation of 3D Shapes Under Flow Fields](https://mlsanity.com/paper/2503.02606/) (2025)
- [Object Pose Distribution Estimation for Determining Revolution and Reflection Uncertainty in Point Clouds](https://mlsanity.com/paper/2512.07211/) (2025)
- [Cryo-forum: A framework for orientation recovery with uncertainty measure with the application in cryo-EM image analysis](https://mlsanity.com/paper/2307.09847/) (2023)
- [On the power of data augmentation for head pose estimation](https://mlsanity.com/paper/2407.05357/) (2024)
- [Training Without Orthogonalization, Inference With SVD: A Gradient Analysis of Rotation Representations](https://mlsanity.com/paper/2604.05414/) (2026)
- [ELMP: Efficient Learning for Motion Planning via Analytical Policy Gradients](https://mlsanity.com/paper/2607.00215/) (2026)
- [Revisiting Euler-Angle Regression with Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2607.09650/) (2026)
- [Towards a Modular Bin-picking Framework for Handling Object Pose Uncertainties](https://mlsanity.com/paper/2607.13698/) (2026)

## Abstract (arXiv)

Accurate rotation estimation is at the heart of robot perception tasks such as visual odometry and object pose estimation. Deep neural networks have provided a new way to perform these tasks, and the choice of rotation representation is an important part of network design. In this work, we present a novel symmetric matrix representation of the 3D rotation group, SO(3), with two important properties that make it particularly suitable for learned models: (1) it satisfies a smoothness property that improves convergence and generalization when regressing large rotation targets, and (2) it encodes a symmetric Bingham belief over the space of unit quaternions, permitting the training of uncertainty-aware models. We empirically validate the benefits of our formulation by training deep neural rotation regressors on two data modalities. First, we use synthetic point-cloud data to show that our representation leads to superior predictive accuracy over existing representations for arbitrary rotation targets. Second, we use image data collected onboard ground and aerial vehicles to demonstrate that our representation is amenable to an effective out-of-distribution (OOD) rejection technique that significantly improves the robustness of rotation estimates to unseen environmental effects and corrupted input images, without requiring the use of an explicit likelihood loss, stochastic sampling, or an auxiliary classifier. This capability is key for safety-critical applications where detecting novel inputs can prevent catastrophic failure of learned models.