Extension beyond Grassmann manifolds

Extend the locally adaptive conformal prediction framework beyond Grassmann manifolds to broader classes of Riemannian and quotient manifolds, thereby establishing a general foundation for distribution-free uncertainty quantification of structured geometric outputs.

Background

The proposed framework is specifically constructed for subspace-valued responses on Grassmann manifolds, using basis-invariant distances and metric balls in that geometry. A broader theory covering other Riemannian and quotient manifolds would make the approach applicable to a wider range of structured geometric outputs. The authors explicitly identify this generalization as an open direction.

References

Several directions remain open. The equivalent-radius and volume measures were used in this work to provide a practical normalization for comparing different Grassmannian metrics, but their extension to exact anisotropic prediction-set volumes in higher-dimensional Grassmannians would enable more precise efficiency comparisons. Locally anisotropic conformal regions whose geometry adapts not only in radius but also along different tangent directions, may further improve efficiency when uncertainty is strongly directional. Other extensions include alternative estimators of local geometric difficulty, data-driven selection or combination of Grassmannian metrics, and conformal methods designed for weaker forms of exchangeability or distribution shift. Finally, extending the framework beyond Grassmann manifolds to broader classes of Riemannian and quotient manifolds would provide a general foundation for distribution-free uncertainty quantification of structured geometric outputs.

— Adaptive conformal prediction on the Grassmannian  (2610.23508 - Loukrezis et al., 16 Oct 2026) in Section 7, Conclusions