Coordinate Singularities Break Conformal Coverage for Gaze and Head Pose

2026-06-29

AI Summary

This work addresses the severe degradation of conditional coverage in conformal prediction for spherical or rotation manifold outputs—such as gaze direction and head pose—when using coordinate charts like Euler angles, where geometric distortions induce singular regions (e.g., pitch angles exceeding 70°) with coverage dropping below 40%. The study theoretically establishes, for the first time, how coordinate singularities structurally undermine conformal coverage guarantees. To resolve this, the authors propose a computationally efficient, training-free, coordinate-invariant geodesic scoring method that replaces flat-coordinate-based conformal scores with geodesic distances on the manifold. Extensive evaluation across four benchmark datasets demonstrates that the proposed approach restores conditional coverage in singular regions to the nominal 90% level while preserving stable marginal coverage.

Abstract

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.

Problem

• conformal prediction
• coordinate singularities
• gaze estimation
• head pose estimation
• coverage failure

Innovation

• conformal prediction
• coordinate singularities
• geodesic scoring
• gaze estimation
• head pose estimation