Authors: Dimitrios Loukrezis, Zachary J. Grey, Dimitris G. Giovanis
Organizations: Scientific Computing Group Centrum Wiskunde & Informatica Amsterdam, The Netherlands · Applied & Computational Mathematics Division National Institute of Standards & Technology Boulder, CO, USA · Dpt. of Civil & Systems Engineering Dpt. of Applied Mathematics and Statistics Johns Hopkins University Baltimore, MD, USA
We introduce a conformal prediction framework for regression with Grassmann-valued responses. The proposed method constructs prediction regions in the form of metric balls directly on the Grassmann manifold, using basis-invariant geometric distances as non-conformity measures and scaling the corresponding non-conformity scores through a learned prediction difficulty estimator. This yields locally adaptive, input-dependent prediction regions that respect the underlying manifold geometry, while retaining the finite-sample marginal coverage guarantee of split conformal prediction under exchangeability. We evaluate several Grassmannian distance metrics and compare their prediction-region efficiency on synthetic benchmarks and an airfoil inverse shape-design problem. The results highlight the advantages of geometry-aware non-conformity scores, while local adaptation improves empirical coverage uniformity across prediction difficulty levels and, in strongly heteroscedastic settings, can substantially reduce prediction-region size while maintaining the target coverage.
Figures & tables
Figure 1: Data generation for the unit-sphere benchmark. Left: Sampling of the input domain. Right: Responses on the unit sphere.
Distance metric
Avg. Cov. ( ↑ )
Worst Cov. ( ↑ )
Bin Cov. Std. ( ↓ )
Equiv. Radius ( ↓ )
Vol. Frac. ( ↓ )
L∞ (standard)
0.902±0.022
0.7390
0.0867
N/A
0.07222
L∞ (adaptive)
0.901±0.022
0.8142
0.0527
N/A
0.06920
S2 geodesic (standard)
0.901±0.022
0.7251
0.0924
0.5257
0.06774
S2 geodesic (adaptive)
0.900±0.022
0.8096
0.0544
0.4997
0.06609
Gr. geodesic (standard)
0.901±0.022
0.7251
0.0924
0.5257
0.06774
Gr. geodesic (adaptive)
0.900±0.022
0.8096
0.0544
0.4997
0.06609
Table 1: Performance of standard and adaptive CP on the unit sphere, using different distance metrics (naive L∞ , S2 geodesic, Grassmannian). Results for α=0.1 over 500 seeds.
Figure 2: Empirical coverage provided by standard and adaptive CP over the six difficulty bins, for all considered distance metrics. Average results and percentiles over 500 seeds are shown.
Figure 3: Left: L∞ rectangle versus metric ball (here, spherical cap) based on the Grassmannian geodesic distance for a point near the equator. The rectangle covers a significantly larger area for the same empirical coverage. Right: Distortion of the L∞ rectangle near the poles.
Distance metric
Avg. Cov. ( ↑ )
Worst Cov. ( ↑ )
Bin Cov. Std. ( ↓ )
Equiv. Radius ( ↓ )
Vol. Frac. ( ↓ )
Gr. geodesic (standard)
0.905±0.021
0.5908
0.1538
1.0783
0.25998
Gr. geodesic (adaptive)
0.905±0.023
0.7958
0.0644
0.7436
0.14065
Projection (standard)
0.906±0.023
0.5980
0.1511
0.9888
0.19178
Projection (adaptive)
0.905±0.022
0.8200
0.0521
0.7025
0.11168
Procrustes (standard)
0.905±0.022
0.5932
0.1527
1.0601
0.24509
Procrustes (adaptive)
0.906±0.024
0.8028
0.0616
0.7351
0.13448
Table 2: Performance of standard and adaptive CP on the rotating plane, using different Grassmannian distance metrics. Results for α=0.1 over 100 seeds.
Figure 4: Empirical coverage across the input domain (partitioned into Nbins=12 difficulty bins) for the standard and adaptive CP and for all Grassmannian distance metrics. Average results and percentiles over 100 seeds are shown.
Figure 5: Equivalent geodesic radius over the input domain for the standard and adaptive CP and for all Grassmannian distance metrics. Average results and percentiles over the 100 seeds are shown. The true noise function over the input domain is shown as well.
Distance metric
Avg. Cov. ( ↑ )
Worst Cov. ( ↑ )
Bin Cov. Std. ( ↓ )
Equiv. Radius ( ↓ )
Gr. geodesic (standard)
0.901±0.008
0.8414
0.0400
0.1678
Gr. geodesic (adaptive)
0.901±0.009
0.8453
0.0267
0.1687
Projection (standard)
0.901±0.008
0.8414
0.0400
0.1674
Projection (adaptive)
0.901±0.008
0.8454
0.0269
0.1683
Procrustes (standard)
0.901±0.008
0.8414
0.0400
0.1677
Procrustes (adaptive)
0.901±0.009
0.8453
0.0268
0.1686
Table 3: Performance of standard and adaptive CP on the airfoil inverse shape-design problem. Results for α=0.1 and 100 random splits of the original dataset. For each split, the bin-coverage standard deviation is computed separately using equal-frequency partitions of CL , CD , and CM and then averaged over the three coefficient orderings. Worst-bin coverage is the minimum coverage over all coefficient bins and orderings, averaged across splits.
Figure 6: Validation predictions of the conformalized Grassmannian regression model mapped to physical airfoil coordinates. Each panel shows the predicted airfoil shape, the pointwise min/max envelopes obtained from airfoil shapes sampled from the Grassmannian CP region, and the true airfoil shape. The quantity dGr/R∗ is the ratio of the Grassmannian prediction error to the conformal radius, where values not exceeding one indicate coverage. Every prediction uses the complete aerodynamic covariate vector x=(CL,CD,CM) .
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.
Marzieh Amiri Shahbazi, Ali Baheri
Rochester Institute of Technology, Rochester, NY, USA.
We introduce the conditional expected local radius (CELR) and develop a new conformal prediction method for non-Euclidean responses in general metric spaces. CELR measures local conditional probability concentration, allowing prediction sets to accommodate heterogeneity and multimodality without requiring densities or a vector space structure. Its conditional distribution yields a nonconformity score that is comparable across covariate values and avoids costly transport comparisons between distance profiles. We further adapt a metric distributional random forest that handles multivariate covariates and nonlinear interactions, while a single forest fit supplies all score components, reducing computation by eliminating repeated local fitting. We establish uniform consistency of the estimators and asymptotic conditional validity under suitable regularity conditions, together with finite-sample marginal coverage under exchangeability. Numerical studies demonstrate effective uncertainty quantification across different metric spaces and substantial computational savings over the conditional profile score method.
Rui Qiu, Fang Yao, Zhou Yu
School of Statistics, KLATASDS-MOE, East China Normal University · School of Mathematical Sciences, Center for Statistical Science, Peking University
Full conformal prediction is a framework that implicitly formulates distribution-free confidence prediction regions for a wide range of estimators. However, a classical limitation of the full conformal framework is the computation of the confidence prediction regions, which is usually impossible since it requires training infinitely many estimators (for real-valued prediction for instance). The main purpose of the present work is to describe a generic strategy for designing a tight approximation to the full conformal prediction region that can be efficiently computed. Along with this approximate confidence region, a theoretical quantification of the tightness of this approximation is developed, depending on the smoothness assumptions on the loss and score functions. The new notion of thickness is introduced for quantifying the discrepancy between the approximate confidence region and the full conformal one.
Laboratoire SAMM Université Paris 1 Panthéon-Sorbonne 90, rue de Tolbiac 75634 PARIS CEDEX 13, FRANCE · Safran Aircraft Engines Rond-Point René Ravaud, Réau, 77550 Moissy-Cramayel CEDEX, France