Adaptive conformal prediction on the Grassmannian

Dimitrios Loukrezis; Zachary J. Grey; Dimitris G. Giovanis

stat.MLOct 16, 2026

Adaptive conformal prediction on the Grassmannian

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

Abstract

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. ( $\uparrow$ )	Worst Cov. ( $\uparrow$ )	Bin Cov. Std. ( $\downarrow$ )	Equiv. Radius ( $\downarrow$ )	Vol. Frac. ( $\downarrow$ )
$L_{\infty}$ (standard)	$0.902\pm 0.022$	$0.7390$	$0.0867$	N/A	$0.07222$
$L_{\infty}$ (adaptive)	$0.901\pm 0.022$	$0.8142$	$0.0527$	N/A	$0.06920$
$\mathbb{S}^{2}$ geodesic (standard)	$0.901\pm 0.022$	$0.7251$	$0.0924$	$0.5257$	$0.06774$
$\mathbb{S}^{2}$ geodesic (adaptive)	$0.900\pm 0.022$	$0.8096$	$0.0544$	$0.4997$	$0.06609$
Gr. geodesic (standard)	$0.901\pm 0.022$	$0.7251$	$0.0924$	$0.5257$	$0.06774$
Gr. geodesic (adaptive)	$0.900\pm 0.022$	$0.8096$	$0.0544$	$0.4997$	$0.06609$

Table 1: Performance of standard and adaptive CP on the unit sphere, using different distance metrics (naive

L_{\infty}

\mathbb{S}^{2}

geodesic, Grassmannian). Results for

\alpha=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_{\infty}

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_{\infty}

rectangle near the poles.

Distance metric	Avg. Cov. ( $\uparrow$ )	Worst Cov. ( $\uparrow$ )	Bin Cov. Std. ( $\downarrow$ )	Equiv. Radius ( $\downarrow$ )	Vol. Frac. ( $\downarrow$ )
Gr. geodesic (standard)	$0.905\pm 0.021$	$0.5908$	$0.1538$	$1.0783$	$0.25998$
Gr. geodesic (adaptive)	$0.905\pm 0.023$	$0.7958$	$0.0644$	$0.7436$	$0.14065$
Projection (standard)	$0.906\pm 0.023$	$0.5980$	$0.1511$	$0.9888$	$0.19178$
Projection (adaptive)	$0.905\pm 0.022$	$0.8200$	$0.0521$	$0.7025$	$0.11168$
Procrustes (standard)	$0.905\pm 0.022$	$0.5932$	$0.1527$	$1.0601$	$0.24509$
Procrustes (adaptive)	$0.906\pm 0.024$	$0.8028$	$0.0616$	$0.7351$	$0.13448$

Table 2: Performance of standard and adaptive CP on the rotating plane, using different Grassmannian distance metrics. Results for

\alpha=0.1

over 100 seeds.

Figure 4: Empirical coverage across the input domain (partitioned into

N_{\mathrm{bins}}=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. ( $\uparrow$ )	Worst Cov. ( $\uparrow$ )	Bin Cov. Std. ( $\downarrow$ )	Equiv. Radius ( $\downarrow$ )
Gr. geodesic (standard)	$0.901\pm 0.008$	$0.8414$	$0.0400$	$0.1678$
Gr. geodesic (adaptive)	$0.901\pm 0.009$	$0.8453$	$0.0267$	$0.1687$
Projection (standard)	$0.901\pm 0.008$	$0.8414$	$0.0400$	$0.1674$
Projection (adaptive)	$0.901\pm 0.008$	$0.8454$	$0.0269$	$0.1683$
Procrustes (standard)	$0.901\pm 0.008$	$0.8414$	$0.0400$	$0.1677$
Procrustes (adaptive)	$0.901\pm 0.009$	$0.8453$	$0.0268$	$0.1686$

Table 3: Performance of standard and adaptive CP on the airfoil inverse shape-design problem. Results for

\alpha=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

C_{L}

C_{D}

, and

C_{M}

and then averaged over the three coefficient orderings. Worst-bin coverage is the minimum coverage over all coefficient bins and orderings, averaged across splits.

Panel of Figure 6 — 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 $d_{\mathrm{Gr}}/R^{*}$ 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 $\mathbf{x}=(C_{L},C_{D},C_{M})$ .

Conformal Prediction

Conformal prediction wraps predictive models with calibrated sets or intervals that provide statistical coverage guarantees.

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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.

Abstract

Conformal Prediction

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