# Coordinate Singularities Break Conformal Coverage for Gaze and Head Pose

arXiv: https://arxiv.org/abs/2607.02565 · HTML: https://mlsanity.com/paper/2607.02565/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 28 of its 47 references that are in the map.
- First posted on arXiv 29 Jun 2026.

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

- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [What Uncertainties Do We Need in Bayesian Deep Learning for Computer Vision?](https://mlsanity.com/paper/1703.04977/) (2017)
- [On the Continuity of Rotation Representations in Neural Networks](https://mlsanity.com/paper/1812.07035/) (2018)
- [A Gentle Introduction to Conformal Prediction and Distribution-Free Uncertainty Quantification](https://mlsanity.com/paper/2107.07511/) (2021)
- [Distribution-Free Predictive Inference For Regression](https://mlsanity.com/paper/1604.04173/) (2016)
- [Conformalized Quantile Regression](https://mlsanity.com/paper/1905.03222/) (2019)
- [Conformal Prediction Under Covariate Shift](https://mlsanity.com/paper/1904.06019/) (2019)
- [Adaptive Conformal Inference Under Distribution Shift](https://mlsanity.com/paper/2106.00170/) (2021)
- [Gaze360: Physically Unconstrained Gaze Estimation in the Wild](https://mlsanity.com/paper/1910.10088/) (2019)
- [Fine-Grained Head Pose Estimation Without Keypoints](https://mlsanity.com/paper/1710.00925/) (2017)
- [ETH-XGaze: A Large Scale Dataset for Gaze Estimation under Extreme Head Pose and Gaze Variation](https://mlsanity.com/paper/2007.15837/) (2020)
- [Conformal Prediction: a Unified Review of Theory and New Challenges](https://mlsanity.com/paper/2005.07972/) (2020)
- [Gaze Estimation using Transformer](https://mlsanity.com/paper/2105.14424/) (2021)
- [L2CS-Net: Fine-Grained Gaze Estimation in Unconstrained Environments](https://mlsanity.com/paper/2203.03339/) (2022)
- [6D Rotation Representation For Unconstrained Head Pose Estimation](https://mlsanity.com/paper/2202.12555/) (2022)
- [Conformal Uncertainty Sets for Robust Optimization](https://mlsanity.com/paper/2105.14957/) (2021)
- [Deep Directional Statistics: Pose Estimation with Uncertainty Quantification](https://mlsanity.com/paper/1805.03430/) (2018)
- [Probabilistic orientation estimation with matrix Fisher distributions](https://mlsanity.com/paper/2006.09740/) (2020)
- [DAD-3DHeads: A Large-scale Dense, Accurate and Diverse Dataset for 3D Head Alignment from a Single Image](https://mlsanity.com/paper/2204.03688/) (2022)
- [Conformal Prediction: A Data Perspective](https://mlsanity.com/paper/2410.06494/) (2024)
- [An End-to-End Review of Gaze Estimation and its Interactive Applications on Handheld Mobile Devices](https://mlsanity.com/paper/2307.00122/) (2023)
- [Owl and Lizard: Patterns of Head Pose and Eye Pose in Driver Gaze Classification](https://mlsanity.com/paper/1508.04028/) (2015)
- [HHP-Net: A light Heteroscedastic neural network for Head Pose estimation with uncertainty](https://mlsanity.com/paper/2111.01440/) (2021)
- [Quantifying the Impact of Motion on 2D Gaze Estimation in Real-World Mobile Interactions](https://mlsanity.com/paper/2502.10570/) (2025)
- [MAC-Gaze: Motion-Aware Continual Calibration for Mobile Gaze Tracking](https://mlsanity.com/paper/2505.22769/) (2025)
- [Confidence-aware 3D Gaze Estimation and Evaluation Metric](https://mlsanity.com/paper/2303.10062/) (2023)
- [Conformal inference for regression on Riemannian Manifolds](https://mlsanity.com/paper/2310.08209/) (2023)

## Abstract (arXiv)

Conformal prediction provides distribution-free reliability guarantees for vision systems, but these guarantees depend on how prediction errors are measured in the output space. Many vision tasks produce outputs on curved spaces (e.g. gaze directions on the sphere or 3D head rotations), yet intermediate prediction heads, residuals, uncertainty estimates, or conformal scores are often defined in flat coordinate charts such as yaw-pitch or Euler angles. We show that this scoring choice introduces systematic geometric distortion near coordinate singularities (large pitch angles on the sphere and poses approaching gimbal lock in 3D rotations). Across four datasets (ETH-XGaze, Gaze360, BIWI, AFLW2000-3D), slice-conditional coverage at a nominal 90% target drops by 30-50 percentage points in these regions, falling to 38.9% on ETH-XGaze and 42.0% on Gaze360 at gaze pitch above 70 degrees, and to 57.5% on BIWI and 55.2% on AFLW2000-3D at head pose pitch above 60 degrees near gimbal lock, despite marginal coverage remaining near 90%. We prove that this is structural. Scalar thresholding changes the size of chart-coordinate prediction sets but leaves their distorted axis ratios unchanged. To diagnose this hidden failure mode, we show that a simple geometric quantity, the Riemannian volume density, strongly correlates with where coverage collapse occurs. Finally, we show that coordinate-free geodesic scoring removes this distortion. It requires no retraining and adds negligible computational cost.