Geometry-Aware Uncertainty Quantification via Conformal Prediction on Manifolds

2026-02-17

AI Summary

This work addresses the challenge of applying conformal prediction to response variables residing on Riemannian manifolds, where traditional methods fail due to their reliance on Euclidean assumptions. The authors propose an adaptive geodesic conformal prediction framework—the first to extend conformal prediction to manifold-structured data—by introducing geodesic nonconformity scores and a cross-validation-based difficulty estimator. This approach constructs geodesic spherical cap prediction regions that are invariant to area and location while adapting to local prediction difficulty. To enhance conditional coverage consistency, the method incorporates geodesic distance and an adaptive normalization mechanism. Evaluated on synthetic spherical data and the IGRF-14 geomagnetic field prediction task, the proposed framework substantially reduces conditional coverage variability and elevates worst-case coverage close to the nominal level, outperforming existing coordinate-based baseline methods.

Technology Category

• Machine Learning: Learning with Manifolds
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Consent frameworks and practices on the web
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• conformal prediction
• Riemannian manifolds
• uncertainty quantification
• geodesic residuals
• conditional coverage

Innovation

• conformal prediction
• Riemannian manifolds
• geodesic nonconformity
• heteroscedasticity
• uncertainty quantification