Exact anisotropic volumes for Grassmannian prediction sets

Extend the equivalent-radius and volume measures used to compare Grassmannian metrics to exact anisotropic prediction-set volume calculations in higher-dimensional Grassmannians, thereby enabling more precise efficiency comparisons.

Background

The framework compares prediction-region efficiency across Grassmannian metrics using equivalent-radius and volume measures. For higher-dimensional Grassmannians, these measures are based on practical normalization procedures rather than exact volumes of generally anisotropic metric balls. The authors identify exact anisotropic prediction-set volumes as an unresolved direction for improving such comparisons.

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.

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

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.

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