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Metric Conformal Prediction via the Conditional Expected Local Radius

Rui Qiu, Fang Yao, Zhou Yu

arXiv:2610.31336v1stat.MLcs.LG

TL;DR

For non-Euclidean responses, informative prediction sets must represent conditional spread and shape without relying on densities or vector-space structure. The paper uses CELR-based conformal scores and a metric distributional random forest; it establishes finite-sample marginal coverage under exchangeability and asymptotic conditional validity under regularity conditions. The method’s kernel choice depends on the response space, and sample splitting leaves only part of the data for conditional-law estimation.

  • Problem

    Informative prediction sets for metric-space responses must describe conditional spread and shape despite the possible absence of densities and vector operations.

    Source: §1 · pp. 2–3

  • Method

    The method transforms conditional expected local radii by their conditional distribution and estimates score components using weights from a metric distributional random forest.

    Source: §1 · p. 3, §7 · p. 25

  • Results

    Split conformal calibration gives finite-sample marginal coverage under exchangeability, and the method establishes asymptotic conditional validity under stated regularity conditions.

    Source: §7 · p. 25

  • Takeaways & Limitations

    The CELR score ranks candidates by local concentration and can reflect heteroscedasticity and multimodality without requiring densities or vector-space structure.

    Source: §7 · p. 25

    Sample splitting leaves only part of the data for estimating the conditional law; out-of-bag scoring is proposed as a possible fuller-sample alternative requiring a coverage argument.

    Source: §7 · p. 25

Abstract

from arXiv · show

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.

1 Introduction

Conformal prediction for non-Euclidean responses must describe conditional spread and shape without relying on densities or vector-space operations. The paper introduces CELR-based scores and a metric distributional random forest, with consistency and validity guarantees.

1 Introduction

  • Conditional response laws may be multimodal and vary with predictors, while conventional residual- or density-based constructions need structures that general metric spaces may lack.A ball around a single regression estimate may poorly describe likely responses.Source: §1 · pp. 2–3
  • CELR averages conditional distance quantiles around a candidate, and transforming it by its conditional distribution yields scores comparable across covariate values.The resulting prediction sets can accommodate heterogeneity and multimodality without densities or vector-space structure.Source: §1 · p. 3
  • The CELR score avoids transport comparisons between conditional distance profiles required by Zhou and Müller (2025).It uses a candidate’s conditional distance profile.Source: §1 · p. 3
  • One forest fit supplies weights for distance profiles, CELRs, and their conditional distributions, reducing computation relative to separate local linear fits.The local linear alternative also relies on a single-index assumption for multivariate covariates.Source: §1 · pp. 3–4
  • A metric distributional random forest uses response-adaptive splits and partitions multivariate covariates directly, allowing nonlinear interactions and local variation.Its response-informed weights adapt to variation in the conditional law.Source: §1 · p. 4
  • The paper establishes uniform consistency for estimated distance profiles, CELRs, and score ranks, and derives asymptotic conditional validity.The proof treats the forest as a growing-order U-statistic and uses a maximal inequality with leaf-shrinkage and stability bounds.Source: §1 · p. 4

2 Expected local radius

The expected local radius (ELR) summarizes how quickly probability mass accumulates in metric balls around a candidate. Its locality parameter controls the scale of concentration being measured.

2 Expected local radius

  • ELR averages distance quantiles over probability levels up to β, measuring concentration around a candidate in a metric space.Smaller values indicate greater concentration at that scale.Source: §2 · pp. 4–5
  • Using ELR in a conformal score avoids the transport comparisons between conditional distance profiles required by Zhou and Müller (2025).The L1 DTM formulation studied by Brécheteau (2019) agrees with ELR under the paper’s convention.Source: §2 · p. 5
  • ELR is nondecreasing in β, is bounded by distance quantiles, and changes by at most the distance moved when its center changes.The stated properties assume finite R1 for the monotonicity and quantile results, and β ∈ (0, 1] for the Lipschitz bound.Source: §2 · p. 5
  • As β decreases, ELR emphasizes local concentration; on a Riemannian manifold with continuous positive density near the candidate, rescaled ELR is proportional to f(ω)^−1/k.Higher local density therefore corresponds to smaller rescaled ELR values.Source: §2 · p. 6
  • In a general metric space, ELR remains defined through metric-ball probabilities even when no natural density is available.The small-β expansion uses local small-ball probabilities and their local dimension and concentration coefficient.Source: §2 · p. 6
  • Very small β can make empirical ELR estimates unstable because only a few observations contribute; the numerical studies consider β in [0.1, 0.25].Figure 1 encodes ELR values with colors, level sets with white curves, observations with black points, and contrasts β = 0.05, 0.2, and 0.5.Source: §2 · p. 6, Figure 1 · p. 7

3 ELR-based conformal prediction in metric spaces

The paper turns conditional local concentration into a conformal score for metric-space responses. It combines split conformal calibration with a metric distributional random forest whose weights support the score’s component estimates.

3 ELR-based conformal prediction in metric spaces

  • CELR measures conditional probability concentration around a candidate, providing the basis for prediction sets in metric spaces.At a fixed locality level, smaller CELR indicates greater concentration, while larger values indicate weaker concentration.Source: §3 · p. 7, §3.1 · p. 7

3.1 Split conformal prediction

  • Transforming CELR through its conditional distribution gives the conditional radius score, which is uniformly distributed and independent of X under continuity of the conditional CELR distribution.Split conformal calibration uses exchangeable calibration and test scores to guarantee marginal coverage.Source: §3.1 · p. 8

3.2 Metric distributional random forests

  • A metric distributional random forest adapts distributional random forests to metric-space responses by choosing a kernel on the response space.A bounded, continuous, characteristic kernel gives a mean embedding that uniquely determines the conditional law.Source: §3.2 · p. 8, §3.2.1 · pp. 8–9

3.2.1 Conditional mean embeddings for metric-space responses

  • The kernel must suit the response space because the Gaussian distance kernel need not be positive definite for a general metric.The affine-invariant Riemannian distance between positive-definite matrices is one cited example.Source: §3.2.1 · p. 9

3.2.2 Estimation via metric distributional random forests

  • MMD selects forest splits to capture changes in the full response distribution, including spread or shape, rather than primarily targeting conditional centers.After sorting and feature construction, the appendix approximations allow each coordinate to be scanned in linear time.Source: §3.2.2 · p. 9
  • The same forest weights estimate conditional distance profiles, CELRs, and their conditional distribution functions, so no additional forest fits are required.The weighted empirical measure supplies conditional expectations, including metric-ball probabilities used for distance profiles.Source: §3.2.2 · p. 10

4 Theoretical guarantees

The theoretical analysis establishes uniform consistency for the metric distributional random forest and its conditional distance-profile and CELR estimators, then derives asymptotic conditional validity under additional regularity conditions. The results rely on forest, covariate, response-space, and score-continuity assumptions.

4.1 Uniform consistency of the forest estimator

  • Theorem 5 establishes uniform convergence of the MDRF estimator in RKHS norm under forest regularity, bounded covariate density, and a Lipschitz target map.The rate reflects subsampling variability and cell shrinkage; the √log n factor accounts for uniformity over covariates.Source: §4.1 · p. 12
  • Uniform consistency of the forest estimator in RKHS norm supports reuse of its weights, but does not directly establish consistency of conditional distance profiles defined by nonsmooth ball indicators.The analysis therefore separately establishes profile convergence under regularity controlling covariate variation and metric-ball complexity.Source: §4.1 · pp. 11–12, §4.2 · p. 13

4.2 Asymptotic conditional validity for conformal prediction

  • Theorem 6 extends the conditional distance-profile convergence rate uniformly over covariates, candidate centers, and radii, with a √log n factor for uniformity.The proof uses covariate-space discretization and a maximal inequality for growing-order U-processes.Source: §4.2 · p. 13
  • Forest predictions are convex combinations of ball indicators with weights independent of center and radius, yielding a uniform stochastic bound of order Op(sn log n/n).Response complexity affects the bound's constant, while subsampling and shrinking-leaf bias determine its convergence exponent; these bounds do not establish a general forest rate advantage.Source: §4.2 · p. 14
  • Theorem 7 establishes uniform CELR consistency over covariates and candidate centers, adding a (log n)^(1/ν) factor when transferring profile control to integrated quantiles.Lighter conditional tails reduce this logarithmic penalty; uniform CELR consistency helps prevent asymptotic distortion of candidate rankings.Source: §4.2 · pp. 14–15
  • Theorem 8 gives coverage approaching 1 − α conditional on X0 under regularity conditions, complementing finite-sample marginal validity from exchangeability.Uniform estimation of score components supports the asymptotic conditional guarantee.Source: §4.2 · p. 15

5 Simulation studies

The simulations assess conditional coverage, prediction-set size, applicability across response spaces, and computation against CPS. ELR performs competitively across settings and is faster in all three timing configurations.

5.1 Univariate covariates

  • Both methods track 0.9 coverage across much of Settings 1–2; ELR sharpens the length drop at x=0 and reduces nearby deviations, though both under-cover just right of zero in Setting 2.CPS has the more pronounced undercoverage at the right endpoint in Setting 2.Source: §5.1 · pp. 16–17, Figure 2 · p. 16
  • Across the additional sphere-valued Setting 3, both methods remain close to nominal coverage, and ELR remains competitive across Euclidean and non-Euclidean univariate responses.Setting 3 is reported in the supplement.Source: §5.1 · p. 17

5.2 Multivariate covariates

  • In the single-index Setting 4, CPS has smaller absolute conditional-coverage errors, while ELR avoids the single-index structural assumption and shows similar patterns for p=5 and p=10.Both methods are close to nominal coverage overall, with local deviations near the noise-level change.Source: §5.2 · p. 17, Figure 3 · p. 18
  • In vector-valued mixture Setting 5, empirical coverage remains close to target across two-component, three-component, and covariate-dependent mixtures; CPS is not implemented because its multivariate extension assumes a single index.Source: §5.2 · pp. 17–18
  • For sphere-valued Setting 6, coverage stays near nominal as p increases from 5 to 20; prediction-set size decreases with larger samples and rises only slightly with added covariates.The results summarize 243 test points for n=500 and n=1000.Source: §5.2 · pp. 18–19, Figure 4 · p. 19
  • For distribution-valued Setting 7, empirical coverage remains stable across the tested covariate dimensions and sample sizes.Together with Setting 6, this supports calibration for non-Euclidean responses.Source: §5.2 · p. 19, Figure 5 · p. 20

5.3 Computation time comparison

  • ELR is faster than CPS in all three timing configurations.The comparison covers heteroscedastic scalar, sphere-valued, and multivariate single-index settings.Source: §5.3 · p. 20, Table 1 · p. 20

6 Real data applications

The real-data applications use the method for SPD-valued taxi-flow networks and compositional energy responses, assessing calibration and how prediction sets adapt across conditions. The examples show stable taxi coverage and energy sets that vary over time and across states.

6.1 New York City taxi data

  • The 10 × 10 SPD taxi-flow responses use ten joint predictors, extending the earlier time-of-day-only taxi analysis.Responses summarize trips among Manhattan subregions in consecutive 1,000-trip bins and use the log-Cholesky metric.Source: §6.1 · p. 21
  • 0.9502 overall empirical coverage at a 0.95 target, with daytime coverage centered near nominal across weekday/weekend and rainy/non-rainy conditions.Larger deviations occur mainly in sparse or more variable regimes.Source: §6.1 · p. 21, Figure 6 · p. 22
  • Weekday morning networks are more concentrated and recurrent than the more diffuse, less regular weekend/holiday nightlife networks.Within each regime, the local Fréchet mean summarizes responses, while minimum- and maximum-CRS matrices illustrate relatively typical and atypical observations.Source: §6.1 · p. 22, Figure 7 · p. 23

6.2 U.S. energy data

  • The energy responses are three-part compositions mapped to the positive orthant of the unit sphere, with prediction sets constructed using geodesic distance.The predictors include year, energy-use and price measures, climate variables, and a coastal indicator, enabling comparisons among states within a year.Source: §6.2 · pp. 23–24
  • Arizona’s prediction sets move toward the non-fossil corner, while Alaska’s remain near traditional energy as natural-gas shares gradually increase.The sets adapt in location, size, and shape to covariates rather than imposing a fixed-width band.Source: §6.2 · p. 24, Figure 8 · p. 25
  • Energy prediction sets generally contain or lie near observed compositions, shift toward larger non-fossil shares over time, and vary across geographic and structural contrasts.Aggregated comparisons also show a common temporal transition alongside state-level differences.Source: §6.2 · p. 24

7 Discussion

The discussion presents CELR’s conditional-distribution transform as a comparable local-concentration score for metric-space prediction sets. It also notes the data-use cost of sample splitting and identifies possible applications beyond prediction.

7 Discussion

  • CELR’s conditional-distribution transform yields CRS, comparable across covariate values and ranking candidates by local concentration rather than distance from a point predictor.This supports sets that reflect heteroscedasticity and multimodality without requiring a density or vector-space structure.Source: §7 · p. 25
  • A single metric distributional random forest supplies all score components, while split conformal calibration gives finite-sample marginal coverage under exchangeability.Asymptotic conditional validity holds under the stated regularity conditions.Source: §7 · p. 25
  • Sample splitting leaves only part of the data for estimating the conditional law; out-of-bag reuse would require accounting for score dependence.The discussion also identifies anomaly detection, change-point testing, and decision problems under distributional ambiguity as possible uses for conditional local radii.Source: §7 · p. 25

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