Geometry-Aware Uncertainty Quantification via Conformal Prediction on Manifolds

TL;DR

This paper introduces an adaptive conformal prediction method for data on Riemannian manifolds, improving the calibration and uniformity of prediction regions in non-Euclidean spaces.

Contribution

It develops geodesic conformal prediction with heteroscedastic noise handling, providing more uniform and accurate coverage on manifolds compared to existing Euclidean-based methods.

Findings

Significantly reduces coverage variability on the sphere

Improves worst-case coverage near the nominal level

Outperforms coordinate-based baselines in experiments

Abstract

Conformal prediction provides distribution-free coverage guaranties for regression; yet existing methods assume Euclidean output spaces and produce prediction regions that are poorly calibrated when responses lie on Riemannian manifolds. We propose \emph{adaptive geodesic conformal prediction}, a framework that replaces Euclidean residuals with geodesic nonconformity scores and normalizes them by a cross-validated difficulty estimator to handle heteroscedastic noise. The resulting prediction regions, geodesic caps on the sphere, have position-independent area and adapt their size to local prediction difficulty, yielding substantially more uniform conditional coverage than non-adaptive alternatives. In a synthetic sphere experiment with strong heteroscedasticity and a real-world geomagnetic field forecasting task derived from IGRF-14 satellite data, the adaptive method markedly reduces conditional coverage variability and raises worst-case coverage much closer to the nominal level, while coordinate-based baselines waste a large fraction of coverage area due to chart distortion.