Geometry-Aware Uncertainty Quantification via Conformal Prediction on Manifolds
This work addresses uncertainty quantification for regression on manifolds, which is incremental as it adapts existing conformal prediction to handle non-Euclidean geometries.
The paper tackled the problem of uncertainty quantification for regression on Riemannian manifolds, where existing conformal prediction methods assume Euclidean spaces and produce poorly calibrated regions. The proposed adaptive geodesic conformal prediction method reduced conditional coverage variability and raised worst-case coverage closer to the nominal level in synthetic and real-world experiments.
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.