Geometry-Aware Uncertainty Quantification via Conformal Prediction on Manifolds

TL;DR

This work addresses uncertainty quantification for manifold-valued responses, where conventional conformal prediction can be miscalibrated due to chart distortions and heteroscedasticity. It introduces adaptive geodesic conformal prediction (AGCP), which uses geodesic distances and a cross-validated difficulty estimator to produce geometry-respecting prediction regions in the form of geodesic caps with radii that adapt to local uncertainty. The approach offers a distribution-free coverage guarantee via split conformal prediction and a concrete mechanism to achieve near-uniform conditional coverage across inputs, validated on a synthetic sphere with strong heteroscedasticity and an IGRF-14 based geomagnetic forecasting task. Results show substantial improvements in conditional coverage uniformity and worst-case coverage compared to standard geodesic and coordinate-based baselines, with a practical diagnostic to decide when adaptivity is beneficial. The framework lays groundwork for extensions to streaming settings and anisotropic prediction regions that align with manifold geometry and local error structures.

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.

Figures (6)

• Figure 1: Standard geodesic CP uses fixed-radius caps for every test point (left), while adaptive geodesic CP varies cap size by local difficulty (right).
• Figure 2: 300-trial Monte Carlo distributions. (a) Valid marginal coverage for all methods. (b) Naive coordinate wastes 26% more area. (c) Adaptive method achieves lowest conditional coverage variance. (d) Adaptive worst-case coverage stays closest to the 0.90 target.
• Figure 3: Conditional coverage vs. estimated difficulty $\hat{\sigma}(x)$ (left) and true vMF concentration $\kappa(x)$ (right). The adaptive method stays flat near 0.90 across all difficulty levels; the standard and naive methods over-cover easy (high-$\kappa$) regions while under-covering hard (low-$\kappa$) regions.
• Figure 4: (a) Secular variation rate across the globe. Equatorial and South Atlantic regions exhibit the fastest field evolution, creating natural heteroscedasticity. (b) Distribution of secular variation rates, spanning a $274\times$ range.
• Figure 5: IGRF-14: 100-trial results. The adaptive method achieves $3.5\times$ lower conditional coverage standard deviation and raises worst-case coverage by 0.166.