stat.MLOct 22, 2026

Metric Conformal Prediction via the Conditional Expected Local Radius

Authors: Rui Qiu, Fang Yao, Zhou Yu

Organizations: School of Statistics, KLATASDS-MOE, East China Normal University · School of Mathematical Sciences, Center for Statistical Science, Peking University

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Abstract

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.

Figures & tables

Figure 1: Empirical surfaces of the ELR functional ω↦Rβ(ω)\omega\mapsto R_{\beta}(\omega) for a three-component Gaussian mixture with β=0.05,0.2\beta=0.05,0.2 , and 0.50.5 . Smaller values of β\beta emphasize fine local concentration, whereas larger values shift the functional toward global dispersion. Colors represent ELR values, white curves show level sets, and black points denote observations.
 C^1−α(X0)←{ω∈Ω:S^(R^β(ω∣X0)∣X0)≤Q^1−α}.\hat{C}_{1-\alpha}(X_{0})\leftarrow\Big\{\omega\in\Omega:\hat{S}\big(\hat{R}_{\beta}(\omega\mid X_{0})\mid X_{0}\big)\leq\hat{Q}_{1-\alpha}\Big\}. 
Algorithm 1 ELR-based split conformal prediction for object-valued responses
Figure 2: Average conditional coverage (top) and prediction-set length (bottom) under Settings 1–2 with sample size n=2000n=2000 . The left and right columns correspond to Settings 1 and 2, respectively. Red and black curves denote ELR and CPS, respectively. The dashed horizontal line marks the nominal coverage level 0.90.9 .
Figure 3: Average conditional coverage for ELR (top) and CPS (bottom) under the single-index model in Setting 4 with n=2000n=2000 . The left and right columns correspond to p=5p=5 and p=10p=10 , respectively. Each panel shows coverage at 100100 test points. The nominal coverage level is 0.90.9 .
Figure 4: Average conditional coverage (left) and prediction-set size (right) for the sphere-valued model in Setting 6, with p=5,10,20p=5,10,20 and n=500,1000n=500,1000 . Each boxplot summarizes the results at 243243 test points. The dashed red line marks the nominal coverage level 0.90.9 .
Figure 5: Average conditional coverage for the distribution-valued model in Setting 7, with p=5,10,20p=5,10,20 and n=500,1000n=500,1000 . Each boxplot summarizes the results at 243243 test points. The dashed red line marks the nominal coverage level 0.90.9 .
SettingResponse spacennppELR (s)CPS (s)Speedup
1R\mathbb{R}2000162.2187.73.023.02
3S2\mathbb{S}^{2}4001276.82,515.59.099.09
4R\mathbb{R}20001073.322,208.7303.14303.14
Table 1: Mean elapsed times (seconds) for ELR and CPS over five Monte Carlo replications. Speedup is the ratio of mean times, T‾CPS/T‾ELR\overline{T}_{\mathrm{CPS}}/\overline{T}_{\mathrm{ELR}} .
Figure 6: Local empirical coverage as a function of time of day in the New York City taxi application. Left: weekday versus weekend/holiday bins. Right: rainy versus non-rainy conditions. The solid red horizontal line indicates the nominal coverage level 0.950.95 .
Figure 7: Representative local network patterns in the New York City taxi application. For each traffic regime, we display the local Fréchet mean and the observations with the smallest and largest CRS within the corresponding local training sample. Heatmaps display the matrix logarithms of the corresponding SPD matrices.
Figure 8: Arizona versus Alaska in the U.S. energy data. Blue regions denote ELR-based conformal prediction sets with nominal coverage 0.90.9 across selected years; red crosses mark the observed energy compositions.
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