Geometry-Aware Uncertainty Quantification via Conformal Prediction on Manifolds
arXiv:2602.16015v1 Announce Type: new 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-cas
arXiv:2602.16015v1 Announce Type: new 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.
Executive Summary
The article introduces 'adaptive geodesic conformal prediction,' a novel framework for uncertainty quantification in regression tasks involving Riemannian manifolds. By replacing Euclidean residuals with geodesic nonconformity scores and normalizing them with a cross-validated difficulty estimator, the method adapts prediction regions to local prediction difficulty, ensuring more uniform conditional coverage. The study demonstrates significant improvements in coverage variability and worst-case coverage in both synthetic and real-world datasets, addressing the limitations of existing methods that assume Euclidean output spaces.
Key Points
- ▸ Introduction of adaptive geodesic conformal prediction for Riemannian manifolds
- ▸ Use of geodesic nonconformity scores and cross-validated difficulty estimators
- ▸ Improved conditional coverage and reduced coverage variability
- ▸ Application to synthetic and real-world datasets with notable success
Merits
Innovative Approach
The framework addresses a critical gap in conformal prediction by extending it to non-Euclidean spaces, which is a significant advancement in the field of uncertainty quantification.
Empirical Validation
The study provides strong empirical evidence through both synthetic and real-world datasets, demonstrating the practical efficacy of the proposed method.
Adaptability
The method's ability to adapt prediction regions to local prediction difficulty ensures more uniform and accurate coverage, which is a substantial improvement over non-adaptive methods.
Demerits
Complexity
The computational complexity of the method, particularly the cross-validation process, may limit its applicability in real-time or resource-constrained environments.
Generalizability
While the method shows promise, its generalizability to other types of manifolds and datasets beyond those tested remains to be thoroughly explored.
Implementation Challenges
The practical implementation of geodesic nonconformity scores and difficulty estimators may require specialized knowledge and tools, potentially hindering widespread adoption.
Expert Commentary
The article presents a significant advancement in the field of uncertainty quantification by addressing the limitations of conformal prediction in non-Euclidean spaces. The proposed adaptive geodesic conformal prediction framework demonstrates a nuanced understanding of the challenges posed by Riemannian manifolds and offers a robust solution. The empirical validation through synthetic and real-world datasets lends credibility to the method's effectiveness. However, the computational complexity and potential implementation challenges should be carefully considered. Future research should explore the generalizability of this method to other types of manifolds and datasets, as well as its integration with other geometric deep learning techniques. Overall, this study sets a strong foundation for further advancements in uncertainty quantification in complex geometric spaces.
Recommendations
- ✓ Further research should focus on optimizing the computational efficiency of the method to make it more accessible for real-time applications.
- ✓ Exploring the application of this framework to other types of manifolds and datasets will enhance its generalizability and robustness.
