Conformal prediction gives finite-sample coverage guarantees for regression, but most standard constructions are designed for Euclidean output spaces. When the response lies on a Riemannian manifold, Euclidean residuals and coordinate-based regions can ignore the geometry that defines meaningful error. We propose adaptive geodesic conformal prediction, a simple framework that builds nonconformity scores from geodesic distances and normalizes them with a cross-validated estimate of local prediction difficulty. On the sphere, this produces geodesic caps whose area is independent of position, while their radii still adapt to heteroscedastic noise. In both a synthetic sphere experiment and an IGRF-14 geomagnetic field forecasting task, the adaptive method preserves valid marginal coverage, reduces variation in conditional coverage, and improves worst-case coverage relative to non-adaptive and coordinate-based baselines.
Figures & tables
Figure 1: Visual comparison of the two geodesic conformal methods on the synthetic S2 benchmark. (a) The standard geodesic method assigns a fixed-radius cap to every test point, so easy and hard cases receive identical-size regions. (b) The proposed adaptive method varies cap size by local difficulty σ^(x) , producing tighter regions where prediction is easier and wider regions where prediction is harder.
Figure 2: 300-trial Monte Carlo distributions on the synthetic S2 benchmark. (a) All methods achieve valid marginal coverage near 0.90 . (b) The naive coordinate method requires ≈26% more area. (c) The adaptive method attains the lowest conditional coverage standard deviation. (d) Adaptive worst-case coverage stays closest to the 0.90 target.
Figure 3: Conditional coverage on the synthetic benchmark. (a) Coverage vs. estimated difficulty σ^(x) : standard and naive methods over-cover easy bins ( >0.97 ) and drop below 0.70 on hard bins; the adaptive method stays close to 0.90 across the range. (b) Coverage vs. true vMF concentration κ(x) : at low κ (hard, dispersed targets), both non-adaptive methods fall below 0.90 , while the adaptive method remains comparatively stable.
Figure 4: IGRF-14 data overview. (a) Secular variation rate ( ∘ /yr) at the 352 surface sites. The South Atlantic Anomaly (SAA, dashed ellipse) and magnetic equator exhibit the fastest field evolution, creating intrinsic spatial heteroscedasticity. (b) Distribution of the secular variation rate, spanning a 274× range from 0.0017∘ /yr to 0.467∘ /yr. No synthetic noise is added.
Figure 5: IGRF-14 100-trial Monte Carlo distributions. (a) All methods achieve valid marginal coverage near 0.90 . (b) The naive coordinate method uses substantially more area. (c) The adaptive method reduces conditional coverage standard deviation by ∼3.5× relative to standard geodesic. (d) Worst-case bin coverage improves by 0.166 relative to standard geodesic.
Figure 6: IGRF-14 conditional coverage and diagnostic. (a) Coverage vs. estimated difficulty σ^(x) : the standard geodesic method drops below 0.70 in the hardest bin, while the adaptive method stays near 0.90 . (b) Coverage vs. geographic latitude: the standard method under-covers equatorial latitudes where secular variation is fastest. (c) Difficulty estimator quality: σ^(x) vs. actual prediction error ( r=0.516 , p≈5.6×10−123 ).